Four manifolds with tree like ends

T.Kato

Introduction

In this paper we study anti self dual connections over the Casson handles by equipping complete Riemannian metrics on them. Casson handles are effectively used by M.Freedman, to develop four manifold theory. For example they can be used to construct topological connected sums for closed four manifolds. This leads to the fact that simply connected oriented topological 4 manifolds are determined by their intersection forms and the Kirby Siebenmann classes.

Let Q_{1} ⊕ Q_{2} be a representation of the intersection form of a simply connected smooth closed 4 manifold M. Then as above M decomposes as W_{1}#W_{2} topologically and compatibly with respect to the decomposition of intersection forms. By the classification of the intersection forms plus Donaldson’s theorem for definite forms on M ([Do1]), at least one of Q_{1} or Q_{2}, say Q_{1}, may be representable by the intersection form of a smooth closed 4 manifold (e.g.

Q_{1} = lH = l (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) or k(1) ⊕ l(−1)). We denote the manifold by |Q_{1}|. H

is called the hyperbolic matrix. Then one finds that on a complement of a topological disk D ⊂ W_{1}, W_{1}\D may be regarded as a smooth submanifold in |Q_{1}|, and the end of it admits the handle body with infinitely many 1 handles in W_{1}#W_{2} = M. They are called Casson handles.

In this paper we study open 4 manifold which corresponds to S ≡ W_{1}\D part in the above. Recall that any Casson handle is expressed by an infinite tree (embedded in R^{2}) with one end point, and with signs ± on every edge of it. We say that such a tree is a signed tree. In this paper we will define a subclass of infinite trees which is called homogeneous trees of finite type. These are constructed by iterating to attatch infinitely many another half periodic trees. A connected infinite subtree of a homogeneous tree of finite type is called a tree of finite type. Any homogeneous tree can be approximated by a family of homogeneous trees of finite type.

We say that an open four manifold S has tree-like end if there are finite number of signed trees T_{1}, …, T_{l}, such that S is diffeomorphic to D^{4}∪_{j} CH(T_{j}), where CH(T_{j}) is the Casson handle corresponding T_{j}, and every CH(T_{j}) is attatched to the 4 ball D^{4} along the attatching S^{1} of the first kinky handle (which corresponds to the end point in T_{j}). Notice that there is natural smooth structure on it induced from the handle body.

Let M be an oriented closed smooth four manifold with π_{1}(M) = 1. Let us take a marking of the standard form (M, φ):

φ : (H_{2}(M; Z), <, >) ≅ (⊕Z, k(−E_{8}) ⊕ lH, or k(−1) ⊕ l(1)).

Then by Freedman, one finds an open four manifold S with tree-like end, and finds a smooth embedding S ↪ M which induces an embedding of the form:

(H^{2}(S : Z), < , >) ≅ (⊕Z, lH or l(1)) ↪ (H^{2}(M : Z), < , >).

In order to construct the moduli space of SO(3) ASD connections over M, one takes α̂ ∈ H^{2}(M; Z_{2}) which is mod 2 reduction of α ∈ H^{2}(M; Z). Let us take α̂ ∈ H^{2}(M; Z_{2}) and a positive integer k. Suppose the intersection form of M is isomorphic to L = (−E_{8})⊕(k−1)(−E_{8})⊕lH or (−1)⊕(k−1)(−1)⊕l(1) for k ≥ 2 and l ≥ 1. Then one has the corresponding decomposition α̂ = α̂_{1} ⊕ α̂_{2} ⊕ α̂_{3}. Let Σ_{1}, …, Σ_{d} ⊂ S be embedded surfaces which represents homology classes [Σ_{i}] ∈ H_{2}(S; Z) ↪ H_{2}(M; Z).

Theorem 0.1 Suppose α̂_{1}, α̂_{2}, α̂_{3} ≠ 0, and S ⊂ M consists of signed homogeneous trees. Then if the SO(3) Donaldson’s invariant Q_{k,α̂}(M; [Σ_{1}], …, [Σ_{d}]) does not vanish, then (1) at least one of the trees is of infinite type, (2) or L^{2} ASD moduli space over S is generically w-separated.

Here w is a weight function over S. Our main construction, when S is of finite type, is to equip a complete Riemannian metric on S and construct Atiyah, Hitchin and Singer complex as a Fredholm operator between w weighted Sobolev spaces. Then one can use any L^{2} ASD connection over S as a base point, and can construct L_{w}^{2} ASD moduli space. w separated property is some discontinuity of L_{w}^{2} ASD moduli space in the larger space, L^{2} ASD moduli space over S, under generic deformation of metrics (see 1.E). If one gets an L^{2} ASD connection as a limit of elements in 𝔐(P, (M, h_{i}); [Σ_{1}], …, [Σ_{m}]), then it cannot be a base of any generic L_{w}^{2} ASD moduli space over S. Here h_{i} are Riemannian metrics on M which approach to a specified complete Riemannian metric on S on each compact subset.

Vanishing of Q is well known when M decomposes as a connected sum M = W_{1}♯W_{2} diffeomorphically with respect to the decomposition of the intersection form L = L_{1}⊕L_{2} ([Do2]). S is homeomorphic to the interior of l(S^{2}×S^{2})\D^{4}, or l(CP^{2})\D^{4}. In particular one has a topological connected sum M ≅ (M\S)♯S. Our theorem states that the vanishing result holds if the decomposition is not so complicated smoothly around the end.

Let us outline the proof of it as follows. Suppose the invariant is non zero. Let L_{2} = lH which is the intersection form of l(S^{2} × S^{2}). Let us remove a small disk D^{4} from this, and denote S′ = l(S^{2} × S^{2})\D^{4}. Then one has a smooth immersion S′ → M which is compatible with the intersection form. By Casson and Freedman’s tower construction, one finds an open subset S′′ ⊂ S′ and a smooth embedding S′′ ↪ M which realizes the form L_{2}. S′′ has tree-like end. Notice that one may assume that all Σ_{i} are embedded in S′′. Our main construction is to admit a complete Riemannian metric g on S′′ so that one can do gauge theory over (S′′, g), when S′′ is of finite type. Let us choose a family of Riemannian metrics g_{i} on M so that g_{i}|S′′ converges to g on every compact subsets of S′′. Let M_{k,α̂}((M, g_{i}); [Σ_{1}], …, [Σ_{d}]) be the oriented 0 dimensional moduli space of ASD connections. Its number is invariant for any i, since it is invariant under generic deformation of Riemannian metrics. In particular for all i, the above spaces are all non empty. Let us take a sequence [A_{i}] ∈ M_{k,α̂}((M, g_{i}); [Σ_{1}], …, [Σ_{d}]). Then one finds a subsequence such that its restriction on S′′ converges on every compact subset, to [A_{∞}] ∈ M_{k′,α̂′}((S′′, g); [Σ_{1}], …, [Σ_{d′}]) which should be a smooth manifold. Since α̂_{i} ≠ 0, this space has a negative formal dimension. This is a contradiction.

One has an application of the main theorem (of the proof). Let T_{1}, …, T_{k} be signed homogeneous trees defining open four manifolds S with tree-like end, not necassarily of finite type. In the general case, one doesnot get moduli spaces as smooth manifolds over S. However in the situation of the main theorem, one has some estimate on the decay of the ASD connections {A_{i}} over S. This can be obtained by approximating S by a family of another open manifolds such that ASD moduli spaces are constructed over the approximations.

Let S be an open four manifold with tree-like end, smoothly embedded into a compact manifold M. Suppose the conditions in the above theorem are satisfied except on trees. Notice that any tree with one end point admits a natural distance from the fixed point. Let us use e to denote the edges in T_{j}.

Let F ({T_{j}}) : N → N be a function:

F ({T_{j}})(n) = Sup_{v,j}#{e| one end of e is v, and v ∈ T_{j}, |v| = n}.

For any tree T, let us put T (n) = {x ∈ T : |x| ≤ n}. For any infinite homogeneous tree, one gets a family of another homogeneous trees of finite type {T^{n}}_{n} such that T (n) and T^{n}(n) are isomorphic. Let us put:

Y_{n} = D^{4} ∪_{j=1}^{k} CH(T_{j}^{n}), W_{n} = D^{4} ∪_{j=1}^{k} CH(T_{j}(n)).

Each Y_{n} admits a compatible Riemannian metric g_{n}, g_{n+1}|W_{n} = g_{n}, and a weight function w_{n} such that one can do gauge theory over (Y_{n}, g_{n}, w_{n}).

Let us choose a sequence A_{i} ∈ 𝔐((M, g_{i}), [Σ_{1}], …, [Σ_{m}]) ≠ ϕ. Then a subsequence {A_{i}} converges to a L^{2} ASD connection A_{∞} over S on each compact subset. Let us define G({A_{i}}) : N × N ↦ (0, C] and λ_{i} by:

G(\{A_{i}\})(n, n′) = |F_{A_{n}}|L_{w_{n}}^{2}(W_{n}\W_{n−n′}, g; Λ_{+}^{2}), λ_{n} = inf\frac{|(d_{A_{n}}^{+})_{w_{n}}^{∗}(u)|L_{w_{n}}^{2}(W_{n})}{|u|L_{w_{n}}^{2}(W_{n})}.

Let F = {F : N ↦ N}, G = {h : N × N ↦ (0, C(h)]} and H = {h : N ↦ (0, C(h)]} be the sets of functions.

Suppose the restriction of the Donaldson’s invariant Q|H_{2}(S : Z) is non zero for a principal bundle P with α̂_{j} ≠ 0 as before. Then one can choose a sequence of ASD connections A_{i} ∈ 𝔐((M, h_{i}), P) which converges to A_{∞} as before. The following estimate is independent of any data on M\S:

Theorem 0.2 There exists a universally defined map 𝔉 : F × H ↦ G and a constant C such that one has the pointwise estimate at (i, j) with j ≥ Cλ_{i}^{−1}:

𝔉(F ({T_{j}}), {λ_{i}}) ≤ G({A_{i}}).

Notice that all Y_{n}, Riemannian metrics and weight functions are determined by the asymptotics of {T_{j}}_{j}. Thus the above statement saids that all the constants to choose 𝔉 above is estimated by the growth function F ({T_{j}}_{j}) and {λ_{i}}.

The estimate is independent of M\S and embedded surfaces. One will use the fact that the invariant is polynomial. In order to verify this, one can use the above argument. If G({A_{i}})(n, n′) is sufficiently small, then one gets A′_{n} on Y_{n} by cutting off A_{n} around W_{n}\W_{n−n′}. Since A′_{n} has a sufficiently small L_{w_{n}}^{2} norm on its curvature, one can find an ASD connection A′′_{n} on Y_{n} around A′_{n}. Since Y_{n} has tree-like end of finite type, one can construct ASD moduli space 𝔐(Y_{n}, [Σ_{1}], …, [Σ_{m′}]). Its formal dimension is negative. This is a contradiction.

One has an application of the vanishing results above from some relationship between Kahler forms and Picard lattice over algebraic surfaces. Let M be a simply connected complex algebraic surface with p_{g} > 0. Let us take a marking:

ψ : H^{2}(M; Z) ≅ ⊕Z ⊕ (⊕Z) ⊕ (⊕Z)

with respect to the decomposition of the form as (−E_{8}) ⊕_{k−1} (−E_{8}) ⊕^{l} H or (−1) ⊕ (k − 1)(−1) ⊕ l(1), k ≥ 2. Then one has an open manifold with tree-like end S ⊂ M, which corresponds to lH or l(1). Let h be a hyperplane class of M, and decompose h = h_{1} ⊕ h_{2} ⊕ h_{3} with respect to above. Suppose there is an integral (1, 1) form α = α_{1} ⊕ α_{2} ⊕ α_{3} ∈ H^{1,1}(M; R) ∩ H^{2}(M; Z), such that the mod 2 reduction α̂_{1}, α̂_{2}, α̂_{3} ≠ 0 ∈ H^{2}(M; Z_{2}). Corollary 0.1 Let M be as above, and α satisfy the above condition. Then if there is a hyperplane class h ∈ H^{2}(M; Z) with h_{1} = h_{2} = 0, then at least one of T_{j} for S have infinite type, or L^{2} ASD moduli space over S is generically w-separated. M can be holomorphically embeddable into a projective space CP^{N}. Let us take such embedding i : M ↪ CP^{N}. Then one has a natural homology class h ∈ H_{2}(M; Z), the hyperplane class. This is defined as the intersection of M with the hyperplane CP^{N−1} ⊂ CP^{N}. Let ω be the Fubini Study form over CP^{N}. Then the Poincaré dual to h is the restriction of ω = ω|M. This form gives a Kahler (Hodge) structure on M. There are some restriction of ω on the position in H^{2}(M; R). For example, let y be a closed analytic subvariety in M. Then one has a positivity < ω, y >> 0. The above theorem is a generalization of this fact. It follows from the first theorem before, and the non vanishing result: Theorem 0.3 (Do) Let M be as before and take any hyperplane class h. Then there is k_{0} = k(M, h) such that for any k ≥ k_{0}, and any α̂ which is mod 2 reduction of an integral (1, 1) class, the Donaldson’s invariant Q_{α̂,k}(h) is positive. The non realizable result above, gives some restrictions of the growth of trees, from relation between the Kahler cone and the Picard lattice.

Recall that the Picard number is the rank of the image of the Picard group Pic(M) in H^{2}(M : Z). Let M be a complex algebraic surface with even type of the form, k(−E_{8}) ⊕ lH. Let us say that the Picard group of M is large if the Picard number is grater than max{8k + 1, 2l + 1}. Thus the Picard numbers of all projective Kummer surfaces are large (more than 17).

Let M be as above, and suppose the Picard group of M is large.

Corollary 0.2 Suppose a complex algebraic surface M of even type of the form has large Picard group. Let T_{1}, …, T_{2l} be signed homogeneous trees which represents S ⊂ M. Then at least one of them must be of infinite type, or L^{2} ASD moduli space over S is generically w-separated.

In order to see this, one uses the following fact. Let M be an oriented smooth closed 4 manifold with π_{1}(M) = 1. Then H^{2}(M; Z) is a free abelian group. Let us take a marking φ : (H^{2}(M; Z, < , >) ↦ (⊕Z, L) where L is a unimodular form on a free Z module, and < , > is the intersection form. In general there are many choices of markings. For example suppose L is of even type which contains at least two hyperbolic matrices. Then for any primitive sublattices l, l′ ⊂ L of rank 1, there is an automorphism on L which sends l to l′. In particular there exists a marking ϕ as above, such that ϕ(h) ∈ ⊕_{2l}Z, and for some α = α_{1} + α_{2} + α_{3} ∈ im Pic(M), one has [α_{1}]_{2}, [α_{2}]_{2}, [α_{3}]_{2} ≠ 0 ∈ ⊕Z_{2}.

The main construction in this paper is to obtain ASD moduli spaces over Casson handles of finite type. In order to compute the indices of the Atiyah, Hitchin and Singer (AHS) complex between weighted Sobolev spaces, one will use the analytic tool, developped by [T1]. In the paper, it is shown that one can make gauge theory over periodic open 4 manifolds, using Fourier Laplace transform between spaces. Here in order to get Fredholmness for AHS complex over Casson handles, one uses Fourier Laplace transform iteratedly. As a generalization of the result by [T1], one gets the following. Notice that it is known that any open 4 manifold admits some smooth structure ([Q]).

Proposition 0.1 Any smoothing on |kE_{8}|\ pt cannot admit tree-like end of finite type.

The author would like to express his gratitude to the Max Planck Institut for their hospitality, during his visit.

I Yang Mills moduli space

1 Moduli space

1.A Admissible four manifolds: In this section, one will construct Yang Mills moduli spaces over non compact four manifolds. We treat open four manifolds which can admit Riemannian metrics and weight functions such that they are able to construct Fredholm complex.

Let Y be a non compact smooth four manifold. Throughout this section, one always assumes that Y is simply connected, and simply connected at infinity.

Let g be a complete Riemannian manifold of bounded geometry and w : Y ↦ [0, ∞) be a smooth function. Recall that a complete Riemannian metric is of bounded geometry, if (1) the injectivity radious is more than a positive constant ϵ > 0 at any point, and (2) for any l ≥ 0, sup_{x∈Y} |∇^{l}R| < ∞, where ∇ and R are with respect to g. Throughout this paper, we always assume that g is of bounded geometry, and |∇^{l}(w)|C^{0}(Y) < ∞ for all l ≥ 1.

Let Λ^{i}(Y) and Λ_{+}^{2}(Y) be the exterior differentials on i forms and self dual 2 forms with respect to g, i = 0, 1. Then we have the Atiyah Hitchin Singer complex (without coefficient) as:

0 ⟶ C_{c}^{∞}(Y; Λ^{0}) ⟶^{d} C_{c}^{∞}(Y; Λ^{1}) ⟶^{d^{+}} C_{c}^{∞}(Y; Λ_{+}^{2}) ⟶0

where d^{+} = (1 + ∗) ∘ d. Using L^{2} adjoint operator (here we do not use any weight functions), we get the next elliptic operator:

P = d^{∗} ⊕ d^{+} : C_{c}^{∞}(Y; Λ^{1}) ↦ C_{c}^{∞}(Y; Λ^{0} ⊕ Λ_{+}^{2}).

Let us introduce weighted Sobolev k norms on Y by:

|u|W_{w}^{k} = (Σ_{l≤k} ∫ exp(w)|∇^{l}u|^{2}vol)^{\frac{1}{2}}.

We denote these Sobolev weighted spaces by W_{w}^{k} or L_{w}^{2} (for k = 0). Then one has a complex of bounded maps:

0 ⟶ W_{w}^{k+2}(Y; Λ^{0}) ⟶^{d} W_{w}^{k+1}(Y; Λ^{1}) ⟶^{d_{+}} W_{w}^{k}(Y; Λ_{+}^{2}) ⟶ 0

Now let us call that the pair (g, w) is admissible, if the following conditions are satisfied; (1) the above consists of a Fredholm complex, namely each differential has closed range, and its cohomology group is of finite dimension as a vector space over R. We denote their cohomology groups by H^{∗}(AHS), ∗ = 0, 1, 2. Notice that by the condition, one has L_{w}^{2} adjoint operators d_{w}^{∗} and (d_{+})_{w}^{∗}.

(2) Y admits a partition Y = K ∪ Y_{1} ∪ … Y_{l} such that (a) K is a compact subset and each Y_{j} is an open subset of Y, (b) let us put D = d_{w}^{∗} ⊕ d_{+} or = d ⊕ (d_{+})_{w}^{∗}. Then there is a positive constant C_{k} such that for any u ∈ C_{c}^{∞}(Y_{j}; Λ^{∗}), ∗ = 1, 2, one has the bound |u|W_{w}^{k+1} ≤ C|D(u)|W_{w}^{k}.

(3) For any f ∈ C^{1}(Y) with |df|L_{w}^{2}(Y) < ∞, there is f̄ ∈ R with f − f̄ ∈ L_{w}^{2}(Y).

(4) There is a compact subset K ⊂ Y and a homeomorphism Y \K ≅ S^{3} × [0, ∞).

In part II, we will construct admissible pairs for all Casson handles constructed from homogeneous signed trees of finite type.

Proposition 1.1 Let (g, w) be an admissible pair on Y. Then P_{w} = d_{w}^{∗} ⊕ d_{+} : W_{w}^{1}(Y; Λ^{1}) ↦ L_{w}^{2}(Y; Λ^{0}⊕Λ_{+}^{2}) has closed range with H^{0}(AHS) = H^{1}(AHS) = 0 and H^{2}(AHS) ≥ k = H_{+}^{2}(Y : R).

P roof : Clearly H^{0}(AHS) = H^{1}(AHS) = 0. We show dimH^{2}(AHS) ≥ k. For this, one takes two steps.

Let H_{cp}^{2}(Y : R)^{+} be a subspace of H_{cp}^{2}(Y : R) consisting of vectors with ∫_{Y} u ∧ u > 0. Then H_{cp}^{2}(Y : R)^{+} is a linear subspace of H^{2}(Y : R) of dimension k, since the natural map I : H_{cp}^{2}(Y : R) ↦ H^{2}(Y : R) gives an isomorphism.

Now by the above, one has dimH_{cp}^{2}(Y : R)^{+} = k. For any element u ∈ C^{∞}(Y; Λ^{2}), let us denote by u^{+} the projection to Λ_{+}^{2} part. Then one defines:

P r : H_{cp}^{2}(Y : R)^{+} ↦ H^{2}(AHS)

by assigning [u] → [u^{+}]. This map is well defined. We show that Pr is an injection.

Suppose [u^{+}] = 0 ∈ H^{2}(AHS). Thus one has α ∈ W_{w}^{1}(Y; Λ^{1}) such that u^{+} = d^{+}(α), or (u − d(α))^{+} = 0. Let us put v = u − d(α), and take the cup product < v, v >= ∫_{Y} v ∧ v. Notice that by the Stokes theorem, ∫_{Y} u ∧ d(α) = 0, since u is a closed form. Thus one has the equality ∫_{Y} v ∧ v = ∫_{Y} u ∧ u ≥ 0. On the other hand since v does not have self dual part, one has ∫_{Y} v ∧ v ≤ 0. This shows u = d(α). This implies [u] = 0 ∈ H_{cp}^{2}(Y : R)^{+} since I gives an isomorphism. Thus one concludes dimH^{2} ≥ k.

Example 1.1 In general, dimH_{cp}^{2}(Y; R)^{+} and dimH^{2}(AHS) = Ker(d_{+})_{w}^{∗} do not coincide. One takes Y = R^{2} × R^{2}, where one equips the standard metric on R^{2}, and the product one on Y. Let us choose a weight function w on Y. We show dimH^{2}(AHS) is non zero. Let u_{1} and u_{2} be 2 forms R^{2} respectively, such that |u_{i}| ≡ 1, i = 1, 2, pointwisely. Then clearly du_{i} = 0, i = 1, 2, and u_{1} + u_{2} gives a self dual 2 form with bounded pointwise norm. Then one puts v = exp(−w)(u_{1}+u_{2}) ∈ L_{w}^{2}(Y; Λ_{+}^{2}). Then v satisfies the equation d(exp(w)v) = 0. Thus v is a non trivial element in H^{2}(AHS).

1.B ASD moduli space: Let E → Y be a G (= SO(3) or U(1)) vector bundle such that except some compact subset K ⊂ Y, E|Y \K is trivial. One denotes the corresponding principal G bundle by P. For SO(3) bundles, E is determined by w_{2}(E) ∈ H^{2}(Y : Z_{2}). Let A be a smooth connection over E such that except some compact subset on Y, it satisfies the ASD equation:

F_{A} + ∗F_{A} = 0

where F_{A} is the curvature form of A. Let us denote by R̃(Y) the set of smooth connections as above satisfying |F_{A}|L^{2}(Y) < ∞. In particular, if A is the trivial connection except some compact subset, it is an element in this.

Lemma 1.1 For A ∈ R̃(Y), p_{1}(A) = −\frac{1}{4π^{2}} ∫_{Y} tr(F_{A} ∧ F_{A}) is an integer.

In order to show this, one uses the following:

Sublemma 1.1 (U) Let D ⊂ Y be δ ball with any point as the center, where δ > 0 is suficiently small. Then there exists another small ϵ > 0 with the following property; suppose |F_{A}|L^{2}(D) < ϵ. Then there exists a gauge transformation g over D, such that:

Sup_{x∈D}|∇^{l}(g^{∗}(A) − d)|(x) ≤ C_{l}|F_{A}|L^{2}(D)

where C_{l} are constants, independent of A.

Proof of lemma: Let us take any small ϵ > 0. Then there exists a compact subset K ⊂ Y, and a smooth gauge transformation g ∈ C^{∞}(Y; Aut(E)) such that |g^{∗}A − d|W^{1}(Y \K) < ϵ, where d is the trivial connection (Y is simply connected at infinity). One may assume that K is a compact sumanifold with boundary. Let D(K) be a double of K. Then using a cut off function φ around ∂K, one may extend A′ ≡ φ(g^{∗}A) over D(K), by putting d over the other side of K ⊂ D(K). By the Chern Weil theory, p_{1}(A′) is an integer. On the other hand, since p_{1}(A) − p_{1}(A′) may be arbitrarily small as above, one concludes that p_{1}(A) is an integer. This completes the proof.

Let us put Ad(P) = P ×_{G}𝔊, where 𝔊 is the Lie algebra of G. (P ×_{G}R^{3} = E). We also denote by 𝔊 as the trivial LieG bundle. Then we have the Atiyah Hitchin Singer complex (AHS complex) as:

0 ⟶ C_{c}^{∞}(Y; Ad(P)) ⟶^{d_{A}} C_{c}^{∞}(Y; Ad(P) ⊗ Λ^{1}) ⟶^{d_{A}^{+}} C_{c}^{∞}(Y; Ad(P) ⊗ Λ_{+}^{2}) ⟶ 0

where d_{A}^{+} = (1 + ∗) ∘ d_{A}. Using L_{w}^{2} adjoint operator, we get the next elliptic operator:

P (A)_{w} = (d_{A})_{w}^{∗} ⊕ d_{A}^{+} : W_{w}^{k+1}(Y; Ad(P) ⊗ Λ^{1}) ↦ W_{w}^{k}(Y; Ad(P) ⊗ (Λ^{0} ⊕ Λ_{+}^{2})).

Let us fix A_{0} ∈ R̃(Y). One defines an affine Hilbert space as:

𝔄_{k}(P)_{0} = {A_{0} + a|a ∈ W_{w}^{k}(Y; AdP ⊗ T^{∗}Y)}, k ≥ 3.

Let us take g ∈ C_{loc}^{1}(Y; AutE). By embedding as g ∈ C_{loc}^{1}(Y; Hom(E, E)), one may consider ∇_{A_{0}}g ∈ C^{0}(Y; Hom(E, E) ⊗ T^{∗}Y). Notice that if g is locally W^{4}, then it is of C^{1} class. Now one defines the weighted Sobolev gauge group:

\begin{matrix}𝔊_{l}(P) = \{h ∈ W_{loc}^{l}(Y; Aut(E))|∇_{A_{0}}h ∈ W_{w}^{l−1}(Y; Hom(E, E) ⊗ T^{∗}Y)\}, l ≥ 4. \\ 𝔊_{l}(P)_{0} = \{h ∈ W_{loc}^{l}(Y; Aut(E))|h − id ∈ W_{w}^{l−1}(Y; Hom(E, E) ⊗ T^{∗}Y)\}. \end{matrix}

First of all one has the following property of 𝔊_{l}(P). Suppose one of the following two conditions; (1) G = U(1), or (2) G = SO(3) and A|Y \K ∈ W_{w}^{k}(Y \K; Ad(P) ⊗ Λ^{1}) where E|Y \K admits a trivialization. Then:

Lemma 1.2 For each h ∈ 𝔊_{k+1}(P), there exists h̄ ∈ G such that h − h̄ ∈ W_{w}^{k}(Y \K; Aut(E)).

P roof : This follows from the Kato (toshio)’s inequality, |∇_{A_{0}}h| ≥ d|h| a.e., and admissibility condition (3).

The Lie algebras of 𝔊_{l}(P) and 𝔊_{l}(P)_{0} are correspondingly as follows:

\begin{matrix}𝔤_{l}(P) = \{h ∈ W_{loc}^{l}(Y; Ad(P))|∇_{A_{0}}h ∈ W_{w}^{l−1}(Y; Ad(P) ⊗ T^{∗}Y)\}, \\ 𝔤_{l}(P)_{0} = W_{w}^{l}(Y; Ad(P)). \end{matrix}

Lemma 1.3 (1) 𝔊_{k+1}(P)_{0} acts on 𝔄_{k}(P)_{0} by g^{∗}(A_{0} + a) = g^{−1}∇_{A_{0}}g + g^{−1}ag.

(2) Suppose P is a U(1) bundle with G = U(1). Then 𝔊_{k+1}(P) acts on 𝔄_{k}(P).

The proof is standard, and we omit it.

Notice that 𝔊_{k+1}(P) may not act on 𝔄_{k}(P)_{0}, since A_{0} may live only in L^{2}(Y). Let us define:

𝔐̂_{k}(P) = {A ∈ 𝔄_{k}(P)_{0} : F_{A} + ∗F_{A} = 0}.

Then 𝔊_{k+1}(P)_{0} acts on 𝔐̂_{k}(P). Thus one gets the quotient space:

𝔐_{k}(P) = 𝔐̂_{k}(P)/𝔊_{k+1}(P)_{0}.

Let us find a linear space which corresponds to the tangent space of 𝔐(P)_{k}.

Lemma 1.4 There exist natural continuous maps:

exp : 𝔤_{k+1}(P), (W_{w}^{k+1}(Y; Ad(P))) ↦ 𝔊_{k+1}(P) (𝔊_{k+1}(P)_{0}).

Moreover let us put 𝔊′_{k+1}(P)(ϵ) = {g ∈ 𝔊_{k+1}(P)||g − 1|W_{w}^{k+1}(Y) ≤ ϵ}. Then for sufficiently small ϵ > 0, there exists Log : 𝔊_{k+1}(P)_{0}(ϵ) ↦ W_{w}^{k+1}(Y; AdP) which inverts exp. One has a similar statement for exp|W_{w}^{k+1}(Y; Ad(P)). We also omit the proof.

Let us take any A ∈ 𝔄_{k}(P)_{0}, and consider the following continuous map:

d_{A} : 𝔤_{k+1}(P) ↦ W_{w}^{k}(Y; AdP ⊗ T^{∗}Y)

by u → \frac{d}{dt}(exp(tu)A)|_{t=0}. Locally this map is expressed as u → du + [u, a]. Lemma 1.5 Suppose an admissible pair (g, w) is given. Then for the AHS complex, d_{A} and d_{A}^{+} are closed operators. P roof : It is enough to show that d_{A}|W_{w}^{k+1}(Y; Ad(P)) has closed range.

Let A′_{0} ∈ 𝔄_{k}(P) be a smooth connection which coincides with A on some compact subset K, and is trivial except another compact subset. Then one has A = A′_{0} + a, a ∈ W^{k}(Y; AdP ⊗ T^{∗}Y). Let φ be a cut off function such that for some compact subsets K ⊂ K′ ⊂ K′′ ⊂ Y, one has φ|K′ ≡ 1 and φ|(K′′)^{c} ≡ 0. By the assumption, d : W_{w}^{k+1}(Y; Λ^{0}) ↦ W_{w}^{k}(Y; Λ^{1}) is invertible (satisfies |d(u)|W_{w}^{k} ≥ C|u|W_{w}^{k+1}). Then one sees d_{(1−φ)a} is invertible by choosing sufficiently large K′ ⊂ K′′, using the Sobolev embedding, W_{loc}^{k} ↪ C^{0}. Then one has the following estimate:

|u|W_{w}^{k+1} ≤ C(|d_{A}(u)|W_{w}^{k} + |u|W_{w}^{k})
≤ C(|d_{A}(u)|W_{w}^{k} + |φu|W_{w}^{k} + |(1 − φ)u|W_{w}^{k})
≤ C(|d_{A}(u)|W_{w}^{k} + |φu|W_{w}^{k} + |d_{(1−φ)a}((1 − φ)u)|W_{w}^{k})
≤ C(|d_{A}(u)|W_{w}^{k} + |φu|W_{w}^{k} + |(1 − φ)d_{(1−φ)a}(u)|W_{w}^{k} +
|[d_{(1−φ)a}, (1 − φ)]u|W_{w}^{k})
≤ C(|d_{A}(u)|W_{w}^{k} + |φu|W_{w}^{k} + |(1 − φ)d_{A}(u)|W_{w}^{k} +
|[d_{(1−φ)a}, (1 − φ)]u|W_{w}^{k} + |(1 − φ)[φa + A′_{0}, u]|W_{w}^{k})
≤ C(|d_{A}(u)|W_{w}^{k} + |ψu|W_{w}^{k})

where ψ is another compactly supported cut off function with ψ| Suppφ ≡ 1. The last inequality shows that d_{A} is a closed operator (see lemma 6.1).

Next let us take u ∈ W_{w}^{k+1}(Y; Ad(P) ⊗ Λ^{1}) with (d_{A})_{w}^{∗}(u) = 0. Then by admissibility condition, one has a similar inequality:

|u|W_{w}^{k+1} ≤ C(|d_{A}^{+}(u)|W_{w}^{k} + |ψu|W_{w}^{k})

where a cut off function ψ has compact support. This completes the proof.

Corollary 1.1 Let A ∈ 𝔄_{k}(P)_{0}. Then the AHS complex:

\begin{aligned}0 ⟶ W_{w}^{k+1}(Y; Ad(P)) & ⟶^{d_{A}} W_{w}^{k}(Y; Ad(P) ⊗ Λ^{1}) \\ & ⟶^{d_{A}^{+}} W_{w}^{k−1}(Y; Ad(P) ⊗ Λ_{+}^{2}) ⟶ 0 \end{aligned}

is a Fredholm complex of index −2p_{1}(P) + 3(dimH^{1}(AHS) − dimH^{2}(AHS)).

The index computation uses the excision principle, or relative index theorem ([GL]). One can also use [APS] and the method in section 8 here. One denotes their cohomology groups by H_{A}^{∗}(AHS), ∗ = 0, 1, 2.

Let us consider the restriction:

d_{A} : W_{w}^{k+1}(Y; AdP) ↦ W_{w}^{k}(Y; AdP ⊗ T^{∗}Y).

By the above corollary, one may consider the adjoint operator:

(d_{A})_{w}^{∗} : W_{w}^{k}(Y; AdP ⊗ T^{∗}Y) ↦ W_{w}^{k−1}(Y; AdP).

One understands (d_{A})_{w}^{∗} in the geometric way as follows. Let us take A ∈ 𝔄_{k}(P)_{0}. Then one has the ‘tangent space’ of 𝔄_{k}(P)_{0}/𝔊_{k+1}(P)_{0} at [A]:

\frac{W_{w}^{k}(Y; AdP ⊗ T^{∗}Y)}{d_{A}(W_{w}^{k+1}(Y; AdP))} ≅ Ker(d_{A})_{w}^{∗} : W_{w}^{k}(Y; AdP ⊗ T^{∗}Y) ↦ W_{w}^{k−1}(Y; AdP).

Let us take a smooth family of connections A(t) ∈ 𝔄_{k}(P)_{0}, t ∈ [0, 1]. Then one has:

\frac{∂F_{A(t)}^{+}}{∂t}|_{t=0} = (d_{A(0)} + ∗d_{A(0)})B ≡ d_{A(0)}^{+}B

where B = \frac{dA(t)}{dt}|_{t=0} ∈ W_{w}^{k}(Y; AdP ⊗ T^{∗}Y). Notice that near infinity, F_{A(t)} can be written as dA(t) + A(t) ∧ A(t). Each A(t) can be written as A(t) = A_{0} + a_{t}, where a_{t} ∈ W_{w}^{k}(Y; AdP ⊗ Λ^{1}). Since F_{A_{0}}^{+} = 0, one sees that F_{A_{t}}^{+} is a smooth family in W_{w}^{k−1}(Y; AdP ⊗ Λ_{+}^{2}). Now one gets the ‘tangent space’ of 𝔐_{k}(P) at [A] as:

Ker (d_{A})_{w}^{∗} ⊕ d_{A}^{+} : W_{w}^{k}(Y; AdP ⊗ T^{∗}Y) ↦ W_{w}^{k−1}(Y; AdP ⊗ (Λ^{0} ⊕ Λ_{+}^{2})).

In particular this space is isomorphic to H_{A}^{1}(AHS) which is of finite dimension.

Lemma 1.6 Ker d_{A} : W_{w}^{k+1}(Y; Ad(P)) ↦ W_{w}^{k}(Y; Ad(P) ⊗ Λ^{1}) is zero.

P roof : This follows from the Kato’s inequality, |∇_{A}u| ≥ |d|u|| a.e. Notice that there are also no L^{2} functions u with d_{A}(u) = 0. This is also seen as follows.

Suppose u ∈ L^{2}(Y; Ad(P)), |u|L^{2} = 1, satisfies d_{A}(u) = 0. Then for g = exp(u) ∈ 𝔊, one has g^{∗}(A) = A. Then it is known that for any path l between p and q in Y, one has P_{l}∘g(p) = g(q)∘P_{l}, where P_{l} is the parallel translation along l. Let q → ∞. Then g(q) approaches to the identity. On the other hand since P_{l} holds the inner product, one has < P_{l}(g(p)(u)), g(q)(P_{l}(u)) >_{q}∼< g(p)(u), u >_{p}. This shows that g(p) is near the identity. This impossible. This completes the proof.

Let us consider 𝔐_{k}(P) = 𝔐̂_{k}(P)_{0}/𝔊_{k+1}(P)_{0}, and show that under some assumptions, this space is a finite dimensional smooth manifold. For A ∈ 𝔄_{k}(P)_{0}, let us put:

CG_{A}^{k} = {A′ ∈ 𝔄_{k}(P)_{0}|(d_{A})_{w}^{∗}(A′ − A) = 0}.
I : CG_{A}^{k} × 𝔊_{k+1}(P)_{0} ↦ 𝔄_{k}(P)_{0} : (A, g) → g^{∗}A.

Let us calculate dI at (A, id). For u ∈ W_{w}^{k+1}(Y; Ad(P)) and v ∈ W_{w}^{k}(Y; Ad(P)⊗ Λ^{1}), one has:

\frac{d}{dt}(exp tu)^{∗}(A + tv)|_{t=0} = d_{A}u + v.

Thus dI is given by:

\begin{matrix}dI = 1 ⊕ d_{A} : Ker(d_{A})_{w}^{∗}∩ & W_{w}^{k}(Y; AdP ⊗ T^{∗}Y) × W_{w}^{k+1}(Y; AdP) \\ ↦ W_{w}^{k}(Y; AdP ⊗ T^{∗}Y). \end{matrix}

Since d_{A} has closed range, it is clear that the above map is an isomorphism at every (A, id).

Corollary 1.2 There are neighbourhoods U ⊂ 𝔄_{k}(P)_{0} of A and V ⊂ 𝔊_{k+1}(P)_{0} of id such that:

I : U ∩ CG_{A}^{k} × V ↦ 𝔄_{k}(P)_{0}

is a homeomorphism into its image. By restriction,

I : U ∩ CG_{A}^{k}(P) ∩ 𝔐̂_{k}(P)_{0} × V ↦ 𝔐̂_{k}(P)_{0}

is also a homeomorphism into 𝔐̂_{k}(P)_{0}.

Now one has the following:

Proposition 1.2 Suppose d_{A}^{+} : W_{w}^{k}(Y; AdP ⊗ T^{∗}Y) ↦ W_{w}^{k−1}(Y; AdP ⊗ Λ_{+}^{2}) is surjection for A ∈ 𝔐̂_{k}(P)_{0}. Then 𝔐_{k}(P) = 𝔐̂_{k}(P)_{0}/𝔊_{k+1}(P)_{0} is a finite dimensional smooth manifold at [A]. Its tangent space is naturally isomorphic to Kerd_{A}^{+}/ imd_{A}.

P roof : This follows from the general facts on group action ([FU] p48). From the above corollary, it is enough to verify that the above I gives a slice for the projection:

𝔐̂_{k}(P)_{0} ↦ 𝔐_{k}(P) = 𝔐̂_{k}(P)_{0}/𝔊_{k+1}(P)_{0}.

These can be verified by boostrapping as [FU]. We omit to write it.

1.C Perturbation of Riemannian metrics: In order to construct smooth moduli spaces, one uses K.Uhlenbeck’s generic metric theorem. Let (g, w) be an admissible pair for Y. Let us choose a smooth map h : Y ↦ [0, ∞) with h(x) ≥ \frac{w}{2}(x). Then one introduces the following Banach manifold:

ℭ = {ϕ ∈ C^{l}(Gl(T M)) : limsup_{K}(Σ_{j=0}^{l}e^{h}|∇^{j}(ϕ^{∗}g − g)|K = 0, K : compact}.

Let us take ϕ ∈ ℭ, and put g′ = ϕ^{∗}g. Then one has the AHS complex with respect to g′:

0 ⟶ W_{w}^{k+1}((Y, g′); Λ^{0}) ⟶^{d} W_{w}^{k}((Y, g′); Λ^{1}) ⟶^{d^{+}} W_{w}^{k−1}((Y, g′); Λ_{+}^{2}) ⟶ 0.

One sees that this is a Fredholm complex with the same index as the unpurturbed one. Our aim in this section, is to show the following:

Proposition 1.3 Suppose Y is indefinite. Then by a small perturbation of the Riemannian metric, one has no orbit of reducible connections in 𝔐_{k}(P). In particular 𝔐_{k}(P) is a smooth manifold.

In order to verify this, we follows [FU]. Let us take any L^{2} ASD connection A_{0} over Y, and consider 𝔄_{k}(P)_{0} = {A_{0} + a|a ∈ W_{w}^{k}(Y; Ad(P) ⊗ Λ^{1})}. Then one introduces a 𝔊_{k+1}(P)_{0} equivariant map:

P_{+} : 𝔄_{k}(P)_{0} × ℭ ↦ W_{w}^{k−1}((Y, g); Λ_{+}^{2})

by P_{+}(A, ϕ) = P_{+}(g)(ϕ^{−1}(F_{A})), where P_{+}(g) is the projection to the self dual part with respect to g. Notice that this map is well defined since A_{0} is L^{2} ASD with respect to g. Let us put 𝔐̄_{k}(P) = P_{+}^{−1}(0).

Proposition 1.4 𝔐̄_{k}(P) ∩ 𝔄_{k}^{∗}(P) × ℭ/𝔊_{k+1}(P)_{0} is a smooth Banach manifold.

We sketch its proof. First we see that dP_{+} is surjective at any (A, φ) with P_{+}(A, φ) = 0. Then it follows that 𝔐̄_{k}(P) is a Banach manifold on which 𝔊_{k+1}(P)_{0} acts. Then as before by making slice for the action, one gets the result.

Let 𝔠 be the Lie algebra of ℭ. Then dP_{+} splits as:

dP_{+} = d_{1}P_{+} ⊕ d_{2}P_{+} : W_{w}^{k}(Y; Ad(P) ⊗ Λ^{1}) ⊕ 𝔠 ↦ W_{w}^{k−1}(Y; Ad(P) ⊗ Λ_{+}^{2}),

where d_{1}P_{+}(α) = P_{+}(g)(φ^{−1}(d_{A}(α)))|(u, φ), and d_{2}P_{+}(r) = P_{+}((φ^{−1})^{∗}(r^{∗}F)). We show that the differential of P_{+} is surjective (notice Kerd_{A}∩W_{w}^{k+1}(Y; Ad(P)) = 0).

Let us consider the AHS complex:

0 ⟶ W_{w}^{k+1}(Ad(P)) ⟶^{d_{A}} W_{w}^{k}(Ad(P) ⊗ Λ^{1}) ⟶^{d_{A}^{+}} W_{w}^{k−1}(Ad(P) ⊗ Λ_{+}^{2}) ⟶ 0.

Since this is Fredholm, one sees dP_{+} has finite codimension. Let us take a representative u ∈ CokerdP_{+} with (d_{A}^{+})_{w}^{∗}(u) = 0. Then one has (d_{A}^{+})^{∗}(e^{w}u) = 0. Then one has the equations, d_{A}(F_{A}) = d_{A}^{∗}(F_{A}) = d_{A}(v) = d_{A}^{∗}(v) = 0, where v = φ^{∗}(e^{w}u). Then the same argument as [FU] p 56, shows that on open dense subset of Y, F_{A} can be expressed as α⊗a ∈ Λ_{+}^{2}⊗Ad(P), with |a| = 1 (pointwise norm) and d_{A}(a) = 0. By the irreducibility, it follows a = 0, which contradicts to non triviality of p_{1}(P). This completes the proof.

Proposition 1.5 Let us fix A_{0}, L^{2} ASD connection with respect to g. Then for a Baire set of ϕ ∈ ℭ, there are no reducible connection in 𝔄_{k}(P)_{0} with respect to ϕ^{∗}(g).

P roof : Suppose A_{0} is reducible, and denote the corresponding U(1) connection by the same A_{0}. Let P be a G = U(1) bundle. Then similarly as before one puts:

\begin{matrix}𝔄_{k}(P)_{0} = \{A_{0} + a|a ∈ W_{w}^{k}(Y; Λ^{1})\}, \\ 𝔊_{k+1}(P)_{0} = \{h ∈ W_{loc}^{k+1}(Y; Aut(E))|h − id ∈ W_{w}^{k+1}(Y; T^{∗}Y ⊗ C)\}. \end{matrix}

Then 𝔊_{k+1}(P)_{0} acts on 𝔄_{k}(P)_{0}. Now as above, one considers P_{+} : 𝔄_{k}(P)_{0}×ℭ ↦ W_{w}^{k−1}((Y, g′); Λ_{+}^{2}). As [FU], if dP_{+} is not surjective, then it follows F_{A_{0}} = 0. This shows that dP_{+} is surjective.

Now one has a smooth Banach manifold 𝔐̄_{k}(P) = P_{+}^{−1}(0)/𝔊_{k+1}(P)_{0}. Let us consider a Fredholm map between Banach manifolds π̄ : 𝔐̄_{k}(P) ↦ ℭ. Its Fredholm index is the one of the following:

0 ⟶ W_{w}^{k+1}(Y; Λ^{0}) ⟶^{d} W_{w}^{k}(Y; Λ^{1}) ⟶^{d^{+}} W_{w}^{k−1}(Y; Λ_{+}^{2}) ⟶ 0.

One has H^{0}(AHS) = H^{1}(AHS) = 0, and H^{2}(AHS) ≥ b_{+}^{2}. This is also the case when one purturbes the Riemannian metrics slightly. In particular, if b_{+}^{2} > 0, then one has dimH^{1}− dimH^{2} < 0. Then the Sard Smale theorem shows that for a Baire set of ℭ, ther are no L^{2} ASD connections over non trivial line bundles. This shows that there are no orbits of reducible ASD connections in 𝔐_{k}(P), for a Baire set of ℭ. This completes the proof.

Let us put the projection as π : 𝔐̄_{k}(P) ↦ ℭ. Then the direct application of [FU] to this case, shows: Corollary 1.3 Suppose b_{+}^{2} > 0. Then a Baire set of ϕ ∈ ℭ exists such that 𝔐̄_{k}(ϕ) ≡ π^{−1}(ϕ) are smooth finite dimensional manifolds.

1.D Large deformation of metrics: Let M be an oriented smooth closed four manifold, and Y ⊂ M be its connected open subset. Suppose Y can admit (g, w), an admissible pair. Here one considers to obtain ASD connections over Y, from sequences of another ASD connections over M.

Let us take a G bundle P ↦ M. Let us take an exaustion K_{0} ⊂ K_{1} ⊂ … Y by compact subsets. Let us take a family of Riemannian metrics {h_{i}}_{i} on M such that |h_{i}|K_{i} − g|K_{i}|C^{l}(K_{i}) → 0 as i → ∞. Let us take a sequence of ASD connections A_{i} over P with respect to (M, h_{i}). Proposition 1.6 (U1,2) After gauge transformation g_{i}^{∗}(A_{i}) and taking a subsequnce (which we also denote A_{i}), {A_{i}} converges to an ASD connection A_{∞} with respect to (Y, g), on every compact subsets of Y except finite set of points.

In order to construct an ASD moduli space 𝔐_{k}(Y, g), one wants to choose A_{∞} as the base connection. Notice that one knows only that F_{A_{∞}} lies in L^{2}((Y, g); Λ_{+}^{2}). In order to remove the case when A_{∞} is a flat connection, one will require some conditions on w_{2}(P) ∈ H^{2}(M : Z). This will be considered in the next section.

1.E Discontinuity of L^{2} ASD connection under a small deformation of metrics: In order to get a moduli space as a smooth manifold, one needs to make a small deformation of Riemannian metrics. Suppose for any generic metric h on M, ASD moduli space 𝔐_{k}(M, h) is non empty. Let us take a family of irreducible ASD connections {A_{i} ∈ 𝔐_{k}(M, h_{i})} such that h_{i}|K_{i} converges to g, and A_{i} converges to A_{∞} as above. Let us choose a generic small perturbation g′ = φ^{∗}(g) of g on Y, where φ ∈ ℭ. Then one addresses the following continuity problem for the limit ASD connection:

For any sequence {A_{i}} as above, there exist generic small perturbations h′_{i} of h_{i} with h′_{i} → φ^{∗}(g) on compact subsets, and another family {A′_{i} ∈ 𝔐_{k}(M, h′_{i})} such that A′_{i} converges to A′_{∞} ∈ 𝔐_{k}(Y, g′) on compact subset \ pts, where:

A′_{∞} ∈ 𝔄(P)_{0} = A_{∞} + W_{w}^{k}(Y; AdP ⊗ Λ^{1}). (∗∗)

Let us formulate w-separation property which states some discontinuity, contrary to (∗∗). Let (Y, g, w) be an admissible triple, and take an SO(3) bundle P → Y. Then one defines the L^{2} ASD moduli space 𝔐(g) as the set of smooth ASD connections A on P with ∫_{Y} |F_{A}|^{2} < ∞. By taking a trivialization near infinity, one assumes A ∈ W^{k}(Y; Λ^{1}) (after gauge transforming).

In 2.C,one will introduce the Donaldson’s invariant Q(M, P; [Σ_{1}], …, [Σ_{m}]) ∈ Z. For this, one will choose generic sections of line bundles Λ_{1}, …, Λ^{m} over spaces. Notice that these spaces are constructed only from embedded surfaces Σ_{j} ⊂ S.

Now let us take any A ∈ 𝔐(g). We say that A is generically w-separated under small perturbation, if there exists a Baire set 𝔅 ⊂ ℭ such that for any φ ∈ 𝔅 and any generic sections s_{1}, …, s_{m}, one has the following property; let us choose another generic sections s_{1}^{i}, …, s_{m}^{i} and h_{i} as 1.D so that (1) each s_{j}^{i} → s_{j} as i → ∞, and (2) 𝔐(P, (M, h_{i}); s_{1}^{i}, …, s_{m}^{i}) is generically defined, and (3) h_{i} → φ^{∗}(g).

Then for any convergent subsequence of A_{i} ∈ 𝔐(P, (M, h_{i}); s_{1}^{i}, …, s_{m}^{i}) as 1.D with the limit A_{∞}, A and A_{∞} has w-separated property:

A − A_{∞} ≠ L_{w}^{2}(Y; Λ^{1}).

If each A ∈ 𝔐(g) is generically w-separated, then the L^{2} moduli space 𝔐(g) is called generically w-separated (with respect to Q(M, P; [Σ_{1}], …, [Σ_{m}])).

1.F Asymptotics of ASD connections: Here one formulates asymptotics of L^{2} norms of F_{A} for ASD connections over a complete Riemannian manifold (Y, g). In the next section, one will get some lower bound of the sequence, if A is obtained as a limit under deformation of Riemannian metrics on closed manifold M. The estimate is independent of any data on M\Y.

Let Y ⊂ M be an open connected subset. Let us equip a complete Riemannanian metrtic g of bounded geometry. An admissible approximation of (M, Y, g) is a sequence of admissible triples {(Y_{i}, g_{i}, w_{i})} with the following property; there exists an exaustion of Y by compact subsets L_{0} ⊂ L_{1} ⊂ … Y and diffeomorphisms φ_{i}|L_{i} from L_{i} into Y_{i} such that (1) g|L_{i} = g_{i}|L_{i},

(2) d(∂L_{i+1}, L_{i}) ≥ 1, and

(3) π_{1}(L_{l+1}\L_{l}) ↦ π_{1}(L_{l+2}\L_{l}) has zero image.

We always assume that the addmissible approximation consists of uniformly bounded geometry.

Let us take a generic family of Riemannian metrics on M, {h_{i}} such that |h_{i} − g|C^{l}(L_{i}) < ϵ_{i} → 0. Then one chooses a sequence {A_{i} ∈ 𝔐_{k}(M, h_{i})}_{i}. By taking a subsequence, one may assume that {A_{i}} converges on each compact subsets, to A_{∞}, an ASD connection on (Y, g). One defines an asymptotics of convergent sequence {A_{i}} and a partial lowest spectrum:

\{a_{i,j} = |F_{A_{i}}|L_{w_{i}}^{2}(L_{i}\L_{i−j}; Λ_{+}^{2})\}_{i,j}, λ_{i} = λ(L_{i}, A_{i}) = inf\frac{|(d_{A_{i}}^{+})_{w_{i}}^{∗}(u)|L_{w_{i}}^{2}(L_{i})}{|u|L_{w_{i}}^{2}(L_{i})}.

Let us take an admissible approximation {(Y_{i}, g_{i}, w_{i})} of (M, Y, g) as above. Let {A_{i}}, {λ_{i}} and {a_{i,j}} be as above. Let us take cut off functions φ_{i,j} for each i, j on M with φ_{i,j}|M\L_{i} ≡ 0 and φ_{l}|L_{i−j} ≡ 1. Proposition 1.7 There is a function G : N × N × (0, 1] ↦ (0, 1] which only depends on the addmissible approximation with the following property; if one has bound G(i, j, λ_{i}) > a_{i,j} for (i, j) with j ≥ Cλ_{i}^{−1}, then one gets an ASD connection A′_{i} over Y_{i} with resepct to g_{i} such that (1) |A′_{i}|W_{w_{i}}^{k}(Y_{i}, g_{i}) < ∞, and

(2) its norm satisfies |A′_{i} − φ_{i,j}A_{i}|W_{w_{i}}^{k} < C(Y_{i})a_{i,j}. The proof follows by applying the continuous method over each (Y_{i}, g_{i}). Notice the inequality:

\begin{matrix}|u|L_{w_{i}}^{2}(Y_{i}) ≤ & |(1 − φ_{i,j})u|L_{w_{i}}^{2} + |φ_{i,j}u|L_{w_{i}}^{2} \\ ≤ & λ_{i}^{−1}(|d(1 − φ_{i,j})u|L_{w_{i}}^{2} + |d_{A_{i}}φ_{i,j}u|L_{w_{i}}^{2}) \\ ≤ & λ_{i}^{−1}(|(1 − φ_{i,j})d_{A′_{i}}u|L_{w_{i}}^{2} + |φ_{i,j}d_{A′_{i}}u|L_{w_{i}}^{2}) + \frac{C}{2} j^{−1}λ_{i}^{−1}|u|L_{w_{i}}^{2}(L_{i}\L_{i−j}). \end{matrix}

2 Casson handles and Donaldson’s invariant

2.A Classification: In this section, we recall several known facts and constructions on simply connected four manifolds. The realization problem of unimodular forms has been obtained completely by Freedman ([F]). Let λ be a non degenerate quadratic form over a free Z module Λ. We denote by |λ|, a topological closed oriented four manifold with the intersection form λ. More precisely this means that there exists an isomorphism:

(H^{2}(|λ|; Z), ( , )) ≅ (Λ, λ).

When Λ is a direct sum of Z, Λ = ⊕Z, then one saids that this isomorphism is a marking of |λ|.

Theorem 2.1 (F) Let λ be as above. If λ is of even type, we assume k ≡ \frac{σ(λ)}{8} mod 2. Then there is a closed oriented simply connected topological four manifold |λ| with the form λ, and its Kirby Siebennman invariant is equal to ks(|λ|) = k. Moreover |λ| satisfying above is unique up to homeomorphisms.

On the other hand, one of the striking result for smooth four manifolds is as follows:

Theorem 2.2 (Do) Let M be an oriented simply connected smooth closed four manifold with definite form λ. Then λ = ±kI.

For example |(−E_{8}) ⊕ (−E_{8})| cannot admit any smooth structure. Notice that K3 surface (Kummer surface) M is homeomorphic to |2(−E_{8})|♯3(S^{2} × S^{2}).

Finally one has the following classification of indefinite unimodular forms:

Theorem 2.3 Let λ be of indefinite odd type. Then λ is isomorphic to k(−1)⊕ l(1).

If λ is of indefinite even type, then it is isomorphic to kE_{8} ⊕ l (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) ≡ H.

2.B Decomposition of M: By the Donaldson’s theorem, one knows all types of intersection forms of simply connected smooth closed oriented four manifolds. Here we review the way to construct a homeomorphism M ≅ |k(−E_{8})| ⊕ l(S^{2} × S^{2}) for an even type M. For simplicity of the notation, one restricts to the K3 surface M. However the process will be applied to the general case. Let us fix a marking:

(H^{2}(M; Z), ( , )) ↦ (⊕_{22}Z, 2(−E_{8}) ⊕ 3H).

Let D be a closed topological disk in 3(S^{2} × S^{2}). Then from the above homeomorphism, one finds that there exists a smooth embedding S = 3(S^{2}×S^{2})\D ↪ M which is compatible with the forms respectively. In particular S admits a smooth structure on 3(S^{2} × S^{2})\D, induced from M. This smoothing on the end is far from the standard product S^{3} × [0, ∞).

In the Appendix, we recall the basic construction to find 3(S^{2} × S^{2})\D, due to Casson and Freedman’s tower construction. Here we have an outline of it. In general it is known that if one removes a point from a simply connected closed four manifold M, then it is homotopy equivalent to the wedges of S^{2} ([M]). Recall that S^{2} × S^{2} admits a handle body, with one 0 and 4 handles, and two 2 handles. The two 2 handles is expressed by a link, Hopf link with 0 framing.

Let D^{4} be a smooth disk. Then 3(S^{2} × S^{2})\D^{4} has the handle body:

D^{4} ∪ h_{1} ∪ ⋯ ∪ h_{6}

where h_{i} are 2 handles with framing 0. Each pair (h_{2i−1}, h_{2i}) represents a Hopf link with 0 framing. Only h_{2i−1} and h_{2i} links from each other.

Now by the Hurwitz surjection, there is a smooth map:

h : 3(S^{2} × S^{2})\D^{4} ↦ M

which induces an injection π_{2}(3(S^{2} × S^{2})\D^{4}) ↪ π_{2}(M). This corresponds to the decomposition π_{2}(M) ≅ π_{2}(3(S^{2} × S^{2})) ⊕ π_{2}(|2(−E_{8})|). Thus this map preserves the intersection form. By the general position argument, one may assume the following:

(1) h is a smooth immersion, (2) h|D^{4} is an embbeding.

The main construction by Casson and Freedman, is to find h̃_{i} ⊂ h_{i} with the following properties:

(1) h̃_{i} contains ¯ ∂_{0}h_{i}, a small neighbourhood of ∂_{0}h_{i}, where ∂_{0}h_{i} = S^{1} × D^{2} is the attaching region of h_{i} = D^{2} × D^{2},

(2) h̃_{i} is homeomorphic to h_{i} itself,

(3) h̃_{i} ∪ ¯ ∂_{0}h_{i} has a tree like end,

(4) there are regular homotopies of h|h̃_{i} rel h|∂_{0}h_{i} such that the result h′|h̃_{i} is a topological embedding. Moreover h′| inth̃_{i} ∪ ¯ ∂_{0}h_{i} is a smooth embedding,

(5) Let us put D = 3(S^{2} × S^{2})\[D^{4} ∪_{i} h̃_{i}]. Then D has a topological color neighbourhood. Notice that D inherits a smooth structure from the inclusion D ↪ 3(S^{2} × S^{2}).

Proposition 2.1 (F) D is homeomorphic to R^{4}.

Each pair (h̃_{i}, S^{1} × D^{2}) is called a Casson handle. It is denoted by CH. In the Appendix, one will recall some basic construction of the Casson handles. Smooth type of each CH can be expressed by an infinite tree T with a base point, such that every edge is assigned one of ±. Later one calls such a tree as a signed tree.

Let us put Y = D^{4} ∪_{i} h̃_{i} ⊂ M. One may express Y by D^{4} ∪_{i} CH(T_{i}), where each T_{i} is a signed tree. Induced from the embedding, Y admits a smooth structure.

In the above case, one can consider the pair (M, Y = D^{4} ∪_{i} CH(T_{i}) ⊂ M). In part II, one will verify the following:

Proposition 2.2 Suppose Y is constructed from a finite number of signed trees of finite type. Then Y contains another Y′ with the property from (1) to (5) above, and Y′ has an admissible pair (g, w).

2.C Donaldson’s invariant: Let M be an oriented simply connected smooth closed four manifold. Let h be any element in H_{2}(M; Z). One may represent h by a smoothly embedded Riemann surface. In fact by Hurwitz surjection, h has a representative by a smooth immersion f : S^{2} ↦ M. Let p_{1}, p_{2} ∈ S^{2} be a self intersection pair. Remove small neighbourhoods B_{ϵ}(p_{i}), and add a small cylinder S^{1} × [0, 1] along ∂B_{ϵ}(p_{i}) which gives an embedded Riemann surface which represents the same homology class as h (notice that H_{2}(M; Z) is torsion free).

Let Σ_{1}, …, Σ_{m} ⊂ M be embedded surfaces representing homology classes [Σ_{i}] ∈ H_{2}(M; Z). Let ϕ : H_{2}(M; Z) ≅ ⊕Z be a marking. Thus the form of M is isomorphic to (−E_{8}) ⊕ (k − 1)(−E_{8}) ⊕ lH, or (−1) ⊕ (k − 1)(−1) ⊕ l(1), k ≥ 2 and l ≥ 1. Associated to the decomposition of this form, one may express:

⊕Z = ⊕_{n_{1}}Z ⊕_{n_{2}} Z ⊕_{n_{3}} Z, (n_{1}, n_{2}, n_{3}) = (8, 8(k − 1), 2l) or (1, k − 1, l).

Through ϕ and this decomposition, one has decompositions of [Σ_{i}] as [Σ_{i}] = h_{i}^{1} ⊕ h_{i}^{2} ⊕ h_{i}^{3}, where h_{i}^{1} ∈ ⊕_{n_{1}}Z, h_{i}^{2} ∈ ⊕_{n_{2}}Z, and h_{i}^{3} ∈ ⊕_{n_{3}}Z.

Now suppose Y ⊂ M is homeomorphic to l(S^{2} × S^{2})\D^{4}, or lCP^{2}\D^{4} which induces an isomorphism:

i : H^{2}(Y; Z)(↪ H^{2}(M : Z)) ≅ ⊕Z

through ϕ. Moreover suppose the induced smooth structure on Y admits a tree like end of finite type. Thus one has an admissible pair (g, w) on Y.

Under the above assumption, one can construct a Donaldson’s invariant Q(P; (Y, g, w); h_{1}, …, h_{m}). One can construct Q for all oriented closed smooth four manifold. The constructions for Y are parallel to the closed case. Here one reviews for the former.

We follow [FS2]. Let E be an SO(3) bundle over M, and take any embedded Riemann surface Σ ⊂ M. Let 𝔅_{k}^{∗}(Σ; E|Σ) = 𝔄(E|Σ)_{k}^{∗}/𝔊_{k+1}(E|Σ) be the gauge equivalence classes of irreducible Sobolev k connections. Similarly one has 𝔅̃_{k}^{∗}(Σ; E|Σ) = 𝔄(E|Σ)_{k}^{∗}/𝔊_{k+1}(E|Σ)_{0}, where 𝔊_{k+1}(E|Σ)_{0} ⊂ 𝔊_{k+1}(E|Σ) is a subgroup with g(x_{0}) is the identity for a fixed point x_{0} ∈ Σ. Thus one has a fibration:

SO(3) ↪ 𝔅̃_{k}^{∗}(Σ; E|Σ) ↦ 𝔅_{k}^{∗}(Σ; E|Σ)

which gives an SO(3) principal bundle β over 𝔅_{k}^{∗}(Σ; E|Σ). Now one has the sublemma:

Sublemma 2.1 (FS2) (1) If < w_{2}(E), Σ >≠ 0, β lifts to a U(2) bundle β̃.

(2) For Σ, there is another embedded surface Σ′ ⊂ M such that restriction of any irreducible connection A over E to E|Σ is also irreducible.

For our application, it is enough to assume the condition (1) in the above sublemma. Thus let us choose a lift β̃ over 𝔅_{k}^{∗}(Σ; E|Σ). One has c_{1}(β̃) = w_{2}(β) mod 2. Let us choose a complex line bundle Λ over 𝔅_{k}^{∗}(Σ; E|Σ) with c_{1}(Λ) = c_{1}(β̃).

Now by the above sublemma (2), one has the restriction map:

r : 𝔐_{k}^{∗}(P) ↦ 𝔅_{k}^{∗}(Σ; E|Σ).

One chooses a generic section s of Λ. Then by the transversality argument ([DK] p192), one has a smooth manifold 𝔐_{k}^{∗}(P) ∩ r^{−1}( Kers) of codimension 2. One may generalize this construction. Suppose dim𝔐_{∗}(P) = 2m, and choose m embedded surfaces Σ_{1}, …, Σ_{m} ⊂ M. As before one has determinant bundles over 𝔅_{k}^{∗}(Σ_{j}; E|Σ_{j}). Then for generic sections, s_{1}, …, s_{m}, one has a smooth submanifold of dimension 0 as:

𝔐_{∗}(P) ∩ r_{1}^{−1}( Kers_{1}) ∩ ⋯ ∩ r_{m}^{−1}( Kers_{m}).

By choosing these embeddings with general positions, one may assume that this 0 dimensional manifold consists of finite oriented points. Thus one may count the number of points so that one obtains an integer. This number is, by definition, Q(M, P; [Σ_{1}], …, [Σ_{m}]) (modulo positive constant).

2.D The main theorem: Our main theorem is as follows:

Theorem 2.4 Let π_{1}(M) = 1, all h_{i}^{1} = h_{i}^{2} = 0, 1 ≤ i ≤ l and α̂_{i} ≠ 0, i = 1, 2, 3. Then if the Donaldson’s invariant does not vanish Q(M, P; h_{1}, …, h_{m}) ≠ 0, then the end of Y is tree like of infinite type, or L^{2} ASD moduli space over (Y, g) is generically w-separated.

P roof : Let us fix a marking of H^{2}(M; Z), and take a smooth embedding Y = l(S^{2} × S^{2})\D ↪ M, where D is a topologically closed disk, and this embedding is compatible with the forms. On the other hand, there exists a topological disk (D^{3}, S^{2}) ⊂ (M\Y, ∂Y) such that (1) M\Y ∪ D^{3} consists of two components M_{1} ∪ M_{2},

(2) for even case, M_{1} and M_{2} are homeomorphic to |(−E_{8})|\ pt and |(k − 1)(−E_{8})|\ pt respectively,

(3) for odd case, these are homeomorphic to C̄P^{2}\ pt and (k − 1)C̄P^{2}\ pt respectively.

By taking any integral lift α ∈ H^{2}(M; Z), it gives a lift of the SO(3) bundle to another U(2) bundle P̄. Then for any h ∈ H_{2}(M : Z) and its mod 2 reduction [h]_{2}, one has the equality < w_{2}(P), [h]_{2} >=< c_{1}(P̄), h > mod 2.

Now Y is an open manifold on which one can equip a complete Riemannian metric g of bounded geometry. Moreover one has a smooth function w : Y ↦ [0, ∞) so that one gets weighted Sobolev spaces W_{w}^{l}(Y; Λ^{∗}). Let us choose any complete Riemannian metrics of bounded geometry g_{1} and g_{2} on the interiors of M_{1} and M_{2}.

Let E → Y be an SO(3) bundle with \frac{1}{4}p_{1}(E) = −k and w_{2}(E) = α̂, which satisfies the conditions in the theorem. Let us take an L^{2} ASD connection A_{0} over (E, Y, g). Then by a generic choice of g, one sees that for all A_{0}, d_{A_{0}} is an injection and d_{A_{0}}^{+} is a surjection. Thus one has a smooth moduli space 𝔐_{∗}(E) of ASD connections over E. Let Σ_{1}, …, Σ_{m′} ⊂ Y be embedded surfaces as above. Then one has a smooth manifold 𝔐(E; [Σ_{1}], …, [Σ_{m′}]) defined similarly as before.

Now let Σ_{1}, …, Σ_{m} ⊂ M be as the theorem. By the assumption, one may assume that all these surfaces are included in Y. Moreover one may assume the general position of these, for any Σ_{i}, Σ_{j}, Σ_{k}, one has empty intersection Σ_{i} ∩ Σ_{j} ∩ Σ_{k} = ϕ.

Suppose one has Q(P, M; [Σ_{1}], …, [Σ_{m}]) ≠ 0. This shows that for a generic choice of Riemannian metric g on M, and also generic choice of these surfaces, one has non empty 0 dimensional moduli space 𝔐(P, (M, g); [Σ_{1}], …, [Σ_{m}]) whose algebraic number is equal to the above invariant Q.

Recall that one has n signed trees T_{1}, …, T_{n} Then one obtains Y by the tower construction as Y = D^{4} ∪ CH(T_{1}) ∪ ⋯ ∪ CH(T_{n}).

Let us choose an exausting families of compact subsets as:

K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ Y,
K′_{0} ⊂ K′_{1} ⊂ ⋯ ⊂ M_{1},
K′′_{0} ⊂ K′′_{1} ⊂ ⋯ ⊂ M_{2}.

Let us choose a family of metrics g′_{i} on M with g′_{i}|K_{i} = g|K_{i}, g′_{i}|K′_{i} = g_{1}|K′_{i} and g′_{i}|K′′_{i} = g_{2}|K′′_{i}. Let us choose a family of small constants ϵ_{i} > 0, i = 0, 1, …, so that ϵ_{i} → 0. As section 1.E, in order to construct a smooth moduli space 𝔐(E, M; [Σ_{1}], …, [Σ_{m}]), one needs to perturbe g′_{i} slightly. Let us choose a family of generic Riemannian metrics g_{i} on M with |g − g_{i}|W_{w}^{l}(K_{i} ∪ K′_{i} ∪ k′′_{i}, g ∪ g_{1} ∪ g_{2}) ≤ ϵ_{i}. One may assume that the family {g_{i}} consists of uniformly bounded geometry. Then one has a family of smooth manifolds:

{𝔐(P, (M, g_{i}); [Σ_{1}], …, [Σ_{m}])}_{i≥0}.

Let us choose any sequence [A_{i}] ∈ 𝔐(P, (M, g_{i}); [Σ_{1}], …, [Σ_{m}]), i = 0, 1, . . .. Then there is a subsequence which we also denote by [A_{i}], and gauge transformations h_{i} ∈ Aut(P) so that h_{i}^{∗}(A_{i}) converges to [A_{∞}], L^{2} ASD connection with respect to (Y, g) in C^{∞} topology on every compact subset of Y \ pts. More precisely for any compact subset K_{n} ⊂ Y \ pts and any small ϵ > 0, there exists i_{0} >> 0 so that for any i ≥ i_{0}, one has |h_{i}^{∗}(A_{i}) − A_{∞}|C^{l}(K_{n}) < ϵ.

Similarly for l = 1, 2, h_{i}^{∗}(A_{i})|M_{l}\ pts converges to L^{2} ASD connections A_{∞}^{l} over (M_{l}, g_{l}) respectively.

Now let {p_{1}, …, p_{l}} ⊂ Y be the bubbling points. Then at most 2l surfaces can contain these points. Thus by reordering surfaces, one has:

A_{∞} ∈ 𝔐(E, (Y, g); [Σ_{1}], …, [Σ_{m−2l}])

Let us see that A_{∞} defines an SO(3) bundle E′ on Y with strictly smaller −p_{1}(E′) than −p_{1}(E) on M. For this, one will see |F_{A_{∞}^{l}}|L^{2} > 0, l = 1, 2. One has chosen E so that each α̂_{j} ≠ 0 where w_{2}(E) = α̂ = Σ_{j=1}^{3}α̂_{j} ∈ Z_{2}^{8} ⊕_{k−1} Z_{2}^{8} ⊕ ⊕_{2l}Z_{2} ≅ H^{2}(M; Z_{2}).

Let α_{j} ∈ H^{2}(M; Z) be the corresponding lift. Let us take homology classes h = h_{1} + h_{2} + h_{3} ∈ H_{2}(M; Z) such that the mod 2 reduction of < α_{j}, h_{j} >_{2}∈ Z_{2} are all non zero (= 1). First one sees w_{2}(E′) = α̂_{3} ∈ H^{2}(Y; Z_{2}). Let Σ_{j} ⊂ M be embedded surfaces which represent h_{j}. As below one has w_{2}(E′) ≠ 0.

If the family {F_{A_{i}}}_{i} concentrates at some points p ∈ M_{1} and p′ ∈ M_{2}, then clearly one has −p_{1}(E′) ≤ −p_{1}(E) − 4. Now suppose it does not concentrate on M_{1}. Let Σ_{1} ⊂ M_{1} as above. Then one has the equality < w_{2}(E), Σ_{1} >=< c_{1}(P̄), Σ_{1} >=< c_{1}(A_{i}), Σ >≠ 0 mod 2. Since there are no concentrating points, one has also < c_{1}(A_{∞}^{1}), Σ >≠ 0. Then clearly lim sup_{i}|F_{A_{i}}|L^{2}(M_{1}, g_{i}|M_{1}) ≥ 4. Thus in any situation, one has −p_{1}(E′) ≤ −p_{1}(E) − 4.

Then using A_{∞} as the base point, one has non empty moduli space of ASD connections 𝔐(E′, (Y, g); [Σ_{1}], …, [Σ_{m−2l}]). One may assume that this space contains no reducible orbits. Thus one has dim𝔐 = dim Kerd_{A_{∞}}^{+}/ imd_{A_{∞}} − 2(m − 2l). Since α̂_{i} ≠ 0, i = 1, 2, one has −p_{1}(P) > −p_{1}(E). Thus one has dim Kerd_{A_{∞}}^{+}/ imd_{A_{∞}} + 4 − 3 − 2(m − 2l) ≤ 0. In particular the formal dimension of 𝔐 is negative. This is a contradiction.

2.E Asymptotics of ASD connections: Let T be any infinite tree and | | be the norm which is the distance from the base point. Every integer value corresponds to some vertices of T. Let us use notations e to denote some edge, and v to some vertex. If v is one end point of e, then one denotes v ∈ e. One defines a finite subtree E(v) = {e ∈ T |v ∈ e}. Let us define a growth function of T by F (T) : N ↦ N:

F (T)(n) = sup_{v}♯{e ∈ T |v ∈ e, |v| = n}.

For a family of trees {T_{j}}_{j}, one puts F ({T_{j}}_{j})(n) = max_{j}F (T_{j})(n). Let us denote T (n) = {x ∈ T : |x| ≤ n}. Any homogeneous tree T admits a family of homogeneous trees of finite type {T^{n}} such that there are isomorphisms φ_{n} : T (n) ⊂ T ≅ T^{n}(n) ⊂ T^{n}. These are called an asymptitics of T.

Let M be a simply connected oriented smooth closed manifold with an embedding Y ⊂ M as before. Here the end of Y = D^{4} ∪ ∪_{j=1}^{l}CH(T_{j}) may be of infinite type. For this case also, one can admit a complete Riemannian metric g and a weight function w on Y as before. Suppose the defining signed trees T_{1}, …, T_{l} are homogeneous trees of infinite type. By taking asymptotics {T_{j}^{n}} for all T_{j}, one gets a family of open manifolds by Y_{n} = D^{4} ∪ ∪_{j=1}^{l}CH(T_{j}^{n}). Let us put W′_{n} = D^{4} ∪ ∪_{j=1}^{l}CH(T_{j}^{n}(n)). Y_{n} has end Y_{n}([0, ∞)) on vertical direction, where Y_{n}([0, ∞)) is isometric to Y_{n}(0) × [0, ∞) and Y_{n}(0) is a complete Riemannian three manifold (see 5.A). Similarly W′_{n} has vertical end isometric to W_{n}(0)′ × [0, ∞). Let us choose an increasing sequence of large numbers T_{0} << T_{1} << . . .. Then one puts W_{i} = W′_{i}\W′_{i}([T_{i}, ∞)).

Recall that each kinky handle of CH(T) is determined by each E(v) ⊂ T. Thus the family {Y_{n}}_{n} can be regarded as an approximation of CH(T). Let (g_{n}, w_{n}) be a family of admissible pairs over Y_{n}. Let g be a complete Riemannian metric of bounded geometry on Y with g_{n}|W_{n} = g|W_{n}. Then {(Y_{n}, g_{n}, w_{n})} gives an admissible approximation of (Y, g, w) with an increasing sequence W_{0} ⊂ W_{1} ⊂ . . ..

Let us choose a sequence A_{i} ∈ 𝔐((M, g_{i}), [Σ_{1}], …, [Σ_{m}]). Then a subsequence {A_{i}} converges to a L^{2} ASD connection A_{∞} over (Y, g) on each compact subset except finite set of points. Then one can assign an asymptotics of {A_{i}} as a proper function G({A_{i}}) : N × N ↦ (0, C] by:

G({A_{i}})(i, j) = |F_{A_{i}}|L_{w_{i}}^{2}((W_{i}\W_{i−j}}), g).

Let F = {F : N ↦ N}, G = {h : N × N ↦ (0, C(h)]} and H = {h : N ↦ (0, C(h)]} be the sets of functions.

Let P be an SO(3) bundle over M with α̂_{i} ≠ 0, i = 1, 2, 3 as before. Suppose the restriction of the Donaldson’s invariant Q|H_{2}(Y : Z) is non zero. Then there exists a family of ASD connections A_{i} over P which converges to A_{∞} with the following property. Let {λ_{i}} be the family of partial lower spectrums with respect to {W_{i}} (see 1.F). Theorem 2.5 There exists a universally defined map 𝔉 : F × H ↦ G and a constant C such that one has the pointwise estimate at (i, j) with j ≥ Cλ_{i}^{−1}:

𝔉(F ({T_{j}}_{j}), {λ_{i}}) ≤ G({A_{i}}).

P roof : Let us take an exaustion of Y by W_{1} ⊂ ⋯ ⊂ W_{n} ⊂ … Y. Recall the properties g|W_{n} = g_{n}|W_{n}.

We show that for a small ϵ = ϵ(F ({T_{j}}_{j})(n), λ_{n}), if G({A_{i}})(n, n′) < ϵ, then one can find A′_{n} ∈ 𝔐((Y_{n}, g_{n}), [Σ_{1}], …, [Σ_{m′}]), m′ ≤ m, which should be a smooth manifold of negative formal dimension. This is enough for the proof.

Let H_{2}(Y : Z) ≅ ⊕Z be an isomorphism with a generating set a_{1}, …, a_{v}. Since Q is polynomial, there exist some choice of a_{i(1)}, …, a_{i(m)} from the generating set with Q(M, P; a_{i(1)}, …, a_{i(m)}) ≠ 0. Then one chooses the embedded surfaces Σ_{1} = a_{i(1)}, …, Σ_{m} = a_{i(m)} ⊂ Y and A_{i} ∈ 𝔐((M, h_{i}); a_{i(1)}, …, a_{i(m)}) so that the above ϵ is independent of Σ_{i}.

Let us take a subsequence of {A_{i}} which converges to A_{∞} = lim_{i}A_{i} on each compact subsets. If the above condition is satisfied, then one finds an ASD connection A′_{n} over (Y_{n}, g_{n}) such that |F_{A′_{n}}^{+}|L_{w_{n}}^{2}(W_{n}) < C(g_{n})G(A_{∞})(n, n′), where C is determined by g_{n}. ϵ and C are determined by {T_{j}^{n}} and {λ_{i}}. Thus one may assume ϵ and C are determined by F ({T_{j}}_{j}) and {λ_{i}}. Let us take a cut off function φ on M with φ|W_{n−n′} ≡ 1 and φ|(W_{n})^{c} ≡ 0. Then if G({A_{i}})(n, n′) is sufficiently small, one can get a smooth connection A′_{n} over (Y_{n}, g_{n}) such that A′_{n}|W_{n−n′} is ASD and |F_{A′_{n}}^{+}|L_{w_{n}}^{2}((Y_{n}, g_{n}), Λ_{+}^{2}) is sufficiently small. Then by the continuous method, one gets A′′_{n} ∈ 𝔐((Y_{n}, g_{n}), [Σ_{1}], …, [Σ_{m′}]). The argument that this space has negative formal dimension, is the same as the proof of the previous theorem. This completes the proof.

3 The position of Kahler forms and Picard lattice

One has an application of the vanshing result above from some relationship between Kahler forms and Picard lattice over algebraic surfaces. Let M be a simply connected complex algebraic surface with p_{g} > 0. Suppose M is of even type, and take a marking ψ : H^{2}(M; Z) ≅ ⊕_{8k}Z ⊕_{2l} Z with respect to the decomposition of the form as k(−E_{8}) ⊕ lH. Then one has an open manifold S ⊂ M which corresponds to lH. The diffeomorphism type of the end of S is determined by T_{1}, …, T_{2l}, where each T_{j} is an infinite signed tree.

M can be holomorphically embeddable into a projective space CP^{N}. Let us take such embedding i : M ↪ CP^{N}. Then one has a natural homology class h ∈ H_{2}(M; Z), the hyperplane class. This is defined as the intersection of M with the hyperplane CP^{N−1} ⊂ CP^{N}. Let ω be the Fubini Study form over CP^{N}. Then the Poincaré dual to h is the restriction of ω = ω|M. This form gives a Kahler (Hodge) structure on M.

Let h be a hyperplane class of M, and decompose h = h_{1} ⊕ h_{2} with respect to above. Suppose there is an integral (1, 1) form α = α_{1} ⊕ α_{2} ∈ H^{1,1}(M; R) ∩ H^{2}(M; Z), such that the mod 2 reduction α̂_{i} ≠ 0 ∈ H^{2}(M; Z_{2}), i = 1, 2.

Theorem 3.1 Let M be as above and α satisfy the above condition. Then if there exists a hyperplane class h ∈ H^{2}(M; Z) with h_{1} = 0, then one of the following holds; (1) at least one of T_{j} is of infinte type, or (2) L^{2} ASD moduli space over (S, g) is generically w-separated.

This theorem follows from our main theorem before, and the non vanishing result as follows:

Theorem 3.2 (Do) Let M be as before and take any hyperplane class h. Then there is k_{0} = k(S, h) such that for any k ≥ k_{0}, and any α̂ which is mod 2 reduction of an integral (1, 1) class, the Donaldson’s invariant Q_{α̂,k}(h) is positive.

Theorem 3.1 above gives some condition on the smooth structure on the end of S. Recall that the Picard number is the rank of the image of the Picard group Pic(M) in H^{2}(M : Z). Let M be a complex algebraic surface with even type of the form, k(−E_{8}) ⊕ lH. Let us say that the Picard group of M is large if the Picard number is grater than max{8k + 1, 2l + 1}. Thus the Picard numbers of all projective Kummer surfaces are large (more than 17).

Lemma 3.1 Let M be a complex algebraic surface of even type of the form. Let ω be a Hodge form. Then there exists a marking ϕ : H^{2}(M : Z) ≅ ⊕_{8k}Z ⊕_{2l} Z corresponding to the decomposition k(−E_{8}) ⊕ lH, such that ϕ(ω) ∈ ⊕_{2l}Z.

This follows easily from the result in [J] for unimodular forms over Z.

Then one has the following statement; let M be as above, and suppose the Picard group of M is large. Then there exists a marking ϕ as above, such that ϕ(ω) ∈ ⊕_{2l}Z, and for some α = α_{1} + α_{2} + α_{3} ∈ im Pic(M), one has [α_{j}]_{2} ≠ 0 ∈ ⊕Z_{2}.

Corollary 3.1 Let M be as above. Let S be an open sub manifold in M with tree like end as before. Then for the defining signed trees T_{1}, …, T_{2l}, at least one of them is of infinite type with respect to ϕ above, or L^{2} ASD moduli space over (S, g) is generically w-separated.

For K3 surface M, the Kahler cone is described as ℭ = {y ∈ H^{1,1}(M; C)| < y, y >> 0, < y, d >> 0, d ∈ Δ}, where Δ consists of the effective classes with self intersection −2 in the Picard lattice.

For K3 surface M, there is a 1 dimensional holomorphic section w of holomorphic 2 forms. w satisfies (1) < w, w >= 0, < w, w̄ >> 0. Then ω ∈ H^{2}(S; Z) can be a Kahler form on some K3 surface M if and only if the following conditions are satisfied:

(2) < ω, w >= 0, < ω, ω >> 0,

(3) < ω, d >≠ 0 for any d ∈ H^{2}(M; Z) with < d, d >= −2 and < w, d >= 0.

In the case when M is a Kummer surface, then one has the natural marking. With respect to this particular marking, one has the following non vanishing result.

Lemma 3.2 Let M be a projective Kummer surface. Then there exists a particular marking φ such that for any Hodge form ω, φ(ω) = φ(ω)_{1} ⊕ φ(ω)_{2} satisfies φ(ω)_{1} ≠ 0.

P roof : Notice that for this marking, −2 curves d spans ⊕^{16}Z ⊂ ⊕^{16}Z⊕^{6}Z with respect to the decomposition. Then one takes any d ∈ ⊕^{16}Z with < w, d >= 0. For this d, one must have < ω, d >≠ 0 which implies ω_{1} ∈ ⊕^{16}Z does not vanish.

For another K3 surfaces, one knows the following.

Proposition 3.1 Let (M, ϕ) be a marked Kahler K3 surface with Kahler form ω, and w ≠ 0 ∈ H^{2,0}(M; C). Let E(w) ⊂ H^{2}(M; R) be a linear 2 plane spanned by Rew and Imw. Suppose H^{2}(M; Z) ∩ E(w) consists of a primitive rank 2 lattice in H^{2}(M; Z). Then there is an automorphism τ on H^{2}(M; Z) such that for all Hodge form ω, one has (τ ∘ ϕ(ω))_{1} ≠ 0 ∈ ⊕^{16}Z.

Recall that a sublattice L′ ⊂ L is primitive if L/L′ is free as a Z module. Proof of the proposition: Let us put L = H^{2}(M; Z) ∩ E(w). Then there

is an automorphism τ of H^{2}(M; Z) such that τ(L)_{1} = 0 ∈ ⊕^{16}Z (see [BPV] p15 2.9(2)). Since τ(L) is primitive, one has an orthogonal decomposition H^{2}(M; Z) = (τ(L))^{⊥} ⊕τ(L), Let us take d ∈ (τ(L))^{⊥} with < d, d >= −2. Then by the condition of ω above, one has < τ(ω), d >≠ 0. This shows the result.

4 Appendix

Here we recall basic construction of Casson and Freedman’s tower construction.

Casson handles: Before entering the tower construction, one recalls about the spin structure on four manifolds. Let M be a simply connected closed smooth four manifold. Then one has the second Stiffel Whitney class w_{2}(M) ∈ H^{2}(M; Z_{2}).

Lemma 4.1 Let M be as above. Then M admits a spin structure (i.e. w_{2}(M) = 0) if and only if the intersection form of M is of even type.

Suppose M has form λ = λ_{1} ⊕ λ_{2}, such that ||λ_{1}|| is realized by a smooth N. Later when we construct some smooth embedding N\D ≡ Y ↪ M, one assumes one of the following conditions:

(1) w_{2}(M) = 0, or

(2) w_{2}|H_{2}(M : Z_{2}) ∩ (H_{2}(N : Z_{2}))^{⊥} ≠ 0.

For the Kummer surface, (1) is satisfied. For the treatment of topological manifolds M which does not satisfy (1) or (2), we refer to [FQ].

In general, N will contain 1 and 3 handles for a handle body. However one has the following ([FQ] p176):

Lemma 4.2 For any handle body on N, one may change 1 and 3 handles into 2 handles by taking appropreate number of connected sums with S^{2} × S^{2}.

Suppose N admits a handle body with one 0 and 4 handles. All the other handles consist of 2 handles. From the handle body theory, one may assume that all 2 handles are disjoint. Let Y = N\D^{4} ↦ M be a smooth immersion as 2.B:

h : N\D^{4} = D^{4} ∪ h_{1} ∪ ⋯ ∪ h_{k} ↦ M.

Let D_{i} = 0 × D^{2} ⊂ h_{i} be the cores of the 2 handles.

Lemma 4.3 (see FQ) Suppose w_{2}(M) satisfies the above condition. Then:

(1) By changing h by a regular homotopy, one may assume that the (self) intersection numbers of h|D_{i} are all trivial.

(2) For each D_{i}, there are α_{i} ∈ π_{2}(M) so that the intersection numbers are given by λ(h(D_{i}), α_{j}) = δ_{i,j}.

Tower construction: Let D^{2} be a 2 dimensional disk, and consider an immersion f : D^{2} ↦ M into a 4 manifold. When one considers such immersion, it is always assumed that f is an embedding on the boundary, S^{1} = ∂D^{2}. Let us regard D^{2} as the core, D^{2} × {0} ⊂ D^{2} × D^{2}. We say that f is a framed immersion if f has an extension to f|D^{2} × D^{2}, and for every double points p, p′ ∈ D^{2}, f(p) = f(p′), one has the identification (z_{1}, w_{1}) ≅ (w_{2}, z_{2}), where (z_{1}, w_{1}) and (z_{2}, w_{2}) are local charts around p = (0, 0) and p′ = (0, 0) respectively.

Let M be a simply connected 4 manifold, and consider a framed immersion f : S^{2} ↦ M and an immersion h : D^{2} ↦ M. The image f(S^{2}) ≡ (D^{2})^{t} is called a transverse sphere for h if h(D^{2}) ∩ (D^{2})^{t} consists of one point. (D^{2})^{t} is called an algebraically transverse sphere if the intersection number of these is equal to 1. When h is a map from a disjoint union of D^{2}, h : ∪_{i}(D^{2})_{i} ↦ M, then a transverse sphere for h consist of transverse spheres for every component (D^{2})_{i}. The main theorem coming from the tower construction, is as follows:

Theorem 4.1 (F) Let h : ∪D^{2} ↦ M be an immersion whose algebraic intersection numbers are all 0, with algebraic transverse spheres. Then h is regularly homotopic rel ∂ to a topologically union of embedded disks with transverse spheres.

One may replace the condition, that algebraic intersection numbers h((D^{2})_{i}) ∩ h((D^{2})_{j}) are all 0, by that such numbers for (D_{i}^{2})^{t} ∩ (D_{j}^{2})^{t} are all 0.

One finds proofs of this theorem in [F], [FQ]. Roughly speaking, the process of the proof is as follows. Suppose one has an immersion as above. One may extend h, and thick the domain as h|∪D^{2} ×D^{2}. The image of every component h(D^{2} × D^{2}) is diffeomorphic to S^{1} × D^{3}♮_{D^{3}}S^{1} × ♮_{D^{3}} … S^{1} × D^{3} (n times). It is called a kinky handle. The number of times to take end connected sums, is equal to the one of double points. Now one tries to remove intersections in the image, by adding another kinky handles. Let γ_{i} : S^{1} ↪ h(D^{2} × D^{2}), 1 ≤ i ≤ n, be a set of embeddings, representing a set of generator of π_{1}(h(D^{2} ×D^{2})). Then one may construct another immersions, h_{i} : D^{2} ↦ M whose restriction h_{i}|∂D^{2} are identical to γ_{i}. One may extend it to a framed immersion h_{i}|D^{2} ×D^{2}. Then the images of h_{i} produces another kinky handles. One may iterate this process infinitely many times. The above theorem saids that the result of union of kinky handles, are homeomorphic to D^{2} × D^{2} rel boundary.

Here we will recall the process more concretely. Let us denote a 2 handle by (D^{2} × D^{2}, ∂D^{2} × D^{2}), where ∂D^{2} × D^{2} is the attatching region.

A 2 handle is expressed by a knot in S^{3}. A knot corresponds to the attatching circle S^{1} × {0} ⊂ ∂D^{2} × D^{2}. For example the trivial knot implies B^{4} ∪ D^{2} × D^{2}, diffeomorphic to D^{2} × S^{2}. In general one needs to determine a framing for a knot which is an integer. For example a trivial knot with 1 framing implies CP^{2}\D^{4}.

Let h : D^{2} × D^{2} ↦ M be a framed immersion, such that h|∂D^{2} × D^{2} is embedding. Then the image is diffeomorphic to ♮_{n}S^{1} × D^{2}. For each double point of the image, or each generator γ_{i} ∈ π_{1}(♮_{n}(S^{1} × D^{3})), one associates ± sign. We call the pair (♮_{n}S^{1} × D^{3}, (±, …, ±)), a kinky handle with n kinks. Thus there are at most 2^{n} kinky handles with n kinks.

Kinky handles are conveniently described by link diagrams. Usually a 1 handle is denoted a circle with a dot. This is understood as follows. Let us choose a standard embedding i : S^{1} = ∂D^{2} ↪ S^{3} = ∂B^{4}, and extend this to the embedding of pairs, i : (D^{2}, S^{1}) ↪ (B^{4}, S^{3}). We remove a tublar neighbourhood of the image of i in B^{4}. Then the result, B^{4}\D^{2}×D^{2} is diffeomorphic to S^{1}×D^{3}, union of one 0 handle and one 1 handle.

Notice that a generator γ of π_{1}(B^{4}\D^{2} × D^{2}) is given as the following loop in S^{3} ⊂ B^{4}:

Figure

Let k be a kinky handle with 1 kink. Thus it is diffeomorphic to S^{1} × D^{3}. Let us consider ∂k = S^{1} × S^{2}, and regard it as a union of two solid tori, ∂k = ∪_{±}(S^{1} × D^{2})_{±}. Notice that when k is an image of an immersion h : D^{2} × D^{2} ↦ M, It is important to consider the position of the attatching circle, h(S^{1} × {0}) ⊂ (S^{1} × D^{2})_{+} ⊂ ∂k. For 1 kinks, there are two possible positions as the following pictures:

Figure

Thus if one writes a 1 kink with specified attatching region, it is expressed using link picture as follows:

Figure When one has n kinks, where n is more than 1, one has the following link picture, to denote the kinky handle with the attaching region:

Figure

Let k be a n kinky handle. Then one has a disjoint n loops in k, γ_{1}, …, γ_{n}, which represents a generator of π_{1}(k). For every γ_{i}, let us assign another kinky handle k_{i}, with the attatching region γ_{i}. It is clear that γ_{i} is null homotopic in k_{i}. In particular one has the zero map π_{1}(k) → π_{1}(k ∪_{i} k_{i}) induced from the embedding. The link diagram for k ∪_{i=1}^{n} k_{i} is as follows:

Figure Now we look for the atatching regions in more detail. Let k be a n kinky handle. Let μ ⊂ ∂k be the atatching circles. Let us regard ∂k as:

♯_{n}S^{1} × S^{2} ≅ ∪_{±}(♮_{n}S^{1} × D^{2})_{±}.

Then one may assume that μ has the image in (♮_{n}S^{1} × D^{2})_{+}. The image of μ is described as follows. Let k_{+} or k_{−} be 1 kinky handles. Then k_{±} can be denoted by an embedding of a circle μ_{±} into ∂k_{±} = S^{1} × S^{2}, k_{±} = (S^{1} × D^{3}, μ_{±}). Associated to the end connected sum of two S^{1} × S^{2}, one may take a bund connected sum of two μ_{±}’s. The result becomes (♮^{2}S^{1} × D^{3}, μ_{±}♮μ_{±}), or shortly μ_{±}♮μ_{±}. Let μ_{1}, …, μ_{n} be the one of μ_{+} or μ_{−} respectively. Then ♮_{i=1}^{n}μ_{i} is exactly representing the previous μ in (♮_{n}S^{1} × D^{2})_{+} as above, for some choice of ±.

Let us consider the previous situation, k ∪_{i=1}^{n} k_{i}. From the above description, one may represent any kinky handle k with n kink by a tree T (k) with n edges e_{1}, …, e_{n}, and all the edges has the common one end point v(k). Moreover every edge is assigned with a sign ±. Recall that π_{1}(k) has a generator by γ_{1}, …, γ_{n}. Then every edge e_{i} corresponds to γ_{i}. Now suppose another kinky handle k_{i} is attatched along γ_{i}. Let us represent k_{i} by another tree T (k_{i}). Then in order to express k ∪ k_{i}, one connects T (k) and T (k_{i}) by identifying the end point of e_{i} which is different from v(k), and v(k_{i}). The result is another tree T (k ∪ k_{i}). By making this attatching for every i, one gets a tree T (k ∪_{i=1}^{n} k_{i}).

Figure

Now let us consider k ∪k_{i}. Associated to every loop γ_{i,l}, in k_{i} which represent an element in π_{1}(k_{i}) as before, one may attatch another kinky handle k_{i,l}. By adding T (k_{i,l}) with the vertex corresponding to γ_{i,l}, one may similarly construct another tree, T (k∪k_{i}∪k_{i,l}). One may iterate this process infinitely many times. The result, k ∪_{i} k_{i} ∪_{i,l} k_{i,l} ∪_{i,l,m} k_{i,l,m} ∪ …, is called a Casson handle, CH. Thus every CH may be expressed by an infinite signed tree T with one end point. Let S^{1} × D^{2} ⊂ CH be the attatching region of the first kinky handle k. Theorem 4.2 (F) Any Cassson handle pair (CH, S^{1}×D^{2}) is pairwisely homeomorphic to (D^{2} × D^{2}, S^{1} × D^{2}).

II Spectrum on complete Riemannian manifolds

5 A complete Riemannian metric on int W_{0}

Roughly speaking the set of the anti self dual connections modulo equivalence forms a finite dimansional manifold, in the case when the base four manifold is closed. Then the dimension is equal to the Fredholm index of the elliptic complex, the Atiyah Hitchin Singer complex. One may make a Fredholm operator for four manifold with boundary, using suitable complete Riemannian metrics on its interior. In this case one uses the weighted Sobolev spaces on the elliptic operator. For our purpose this is a basic construction, and later we will use infinitely many such open Riemannian manifolds. In fact by glueing out all of them iteratedly, we will get another complete Riemannian manifold such that the Atiyah Hitchin Singer complex becomes Fredholm between the weighted Sobolev spaces.

5.A A specified embedding of W_{0} into R^{N}: Let W_{0} be a compact manifold with boundary of arbitrary dimension n. Let us choose compact submanifolds with boundary M_{1}, N_{0} ⊂ ∂W_{0} of the same dimension, with empty intersection, M_{1} ∩ N_{0} = ϕ. Suppose M_{1} and N_{0} are diffeomorphic each other, and denote W̃_{0} = intW_{0} ∪ M_{1} ∪ N_{0}. Let us take a countable number of W̃_{0} and make indices them by W̃_{0}^{i} = W̃_{0}, i ∈ Z. Then one constructs an open manifold Y by ∪W̃_{0}^{i} identified M_{1}^{i} with N_{0}^{i+1}:

Y = … W̃_{0}^{i} ∪_{M_{1}^{i}∼N_{0}^{i+1}} W̃_{0}^{i+1} ∪ …

There is a natural smooth Z action on Y, and we call Y a periodic open manifold. We have the obvious notion, periodic, for vector bundles over Y, or for differential operators over E.

In order to define (weighted) Sobolev spaces over Y, we need first of all, complete Riemannian metric on int W_{0} which extends to the one of Y. We recall an explicit picture of end connected sums along M_{1} and N_{0}. Let us choose smooth embeddings:

γ(γ′) : M_{1}(N_{0}) × [0, ∞) ↪ ∂W_{0} × [0, ∞) ⊂ int W_{0}.

Then we have the following equality (at this stage it is C^{∞} sense, by smoothing corners):

int W_{0}♮ int W_{0} = int W_{0}\ im γ ∪ int W_{0}\ im γ′

where we identify ∂M_{1}×[0, ∞)∪M_{1}×{0} with ∂N_{0}×[0, ∞)∪N_{0}×{0}. In order to make this construction compatible with the metric on int W_{0}, we choose a particular style of Riemannian metric on int W_{0} as follows. Let us choose an embedding:

I : W_{0} ↪ R^{N+2}

with the following properties:

(1) Let us choose a small neighbourhood of ∂W_{0} in W_{0} and identify it with ∂W_{0} × [0, ϵ] ≡ W_{0}(ϵ), ∂W_{0} = ∂W_{0} × {0}. Then for any τ ∈ [0, ϵ], we have the following:

W_{0} ∩ R^{N+1} × [0, ϵ] = W_{0}(ϵ), W_{0}(ϵ) ∩ ∂W_{0} × τ ⊂ R^{N+1} × τ.

(2) For a smaller ϵ′ < ϵ, pr{W_{0}(ϵ′) ∩ ∂W_{0} × τ} ⊂ R^{N+1} is independent of τ ∈ [0, ϵ′], where pr: R^{N+2} ↦ R^{N+1} is the projection.

Figure

(3) I|W_{0}\W_{0}(ϵ) is C^{∞} embedding (i.e. without corners).

(4) Let M̄_{1} and N̄_{0} be tublar neighbourhoods of M_{1} and N_{0} in ∂W_{0} respectively. As before we identify small neighbourhoods of ∂M̄_{1} and ∂N̄_{0} with ∂M_{1} × (0, δ] ≡ M_{1}(δ) and ∂N_{0} × (0, δ] ≡ N_{0}(δ) respectively. Then we have the following:

(a) M̄_{1} and N̄_{0} are diffeomorphic to M_{1} and N_{0} respectively,

(b) ∂W_{0} ∩ R^{N} × {0} = M_{1}, ∂W_{0} ∩ R^{N} × {1} = N_{0},

(c) for 0 < τ < δ, M_{1}(δ)∩R^{N} ×τ and N_{0}(δ)∩R^{N} ×{1−τ} are diffeomorphic to ∂M_{1} and ∂N_{0} respectively,

Figure

(4) W_{0}(ϵ) ∩ R^{N} × τ × [0, ϵ] and W_{0}(ϵ) ∩ R^{N} × {1 − τ} × [0, ϵ], τ ∈ (0, δ], are diffeomorphic to ∂M_{1} × [0, ϵ] ∪ M_{1} and ∂N_{0} × [0, ϵ] ∪ N_{0} respectively.

Figure

Moreover W_{0}(ϵ) ∩ R^{N} × τ × [0, ϵ] ⊂ R^{N} × {0} × [0, ϵ] and W_{0}(ϵ) ∩ R^{N} × {1 − τ} × [0, ϵ] ⊂ R^{N} × {0} × [0, ϵ] are independent of τ.

(5) Let us denote M̄_{1}(τ) = W_{0}(ϵ) ∩ R^{N} × {τ} × [0, ϵ] = {(a, b, c) ∈ M̄_{0}(δ) ∩ R^{N} × R × R}. We have similar notations for N_{0}. Then we have N̄_{0}(τ) = M̄_{1}^{♯}(τ) ≡ {(a, 1 − b, c)|(a, b, c) ∈ M̄_{1}(τ)}, τ ∈ (0, δ].

Figure

Let us denote W̃′_{0} = W_{0}\W_{0} ∩ R^{N} × {0, 1} × [0, ϵ]. Then I|W̃′_{0} is a smooth embedding with corners. Let us denote by g_{0}, the Riemannian metric on W̃′_{0} induced from the standard one on R^{N+2} through I. Notice important properties of g_{0}; for ϵ′ < ϵ:

g_{0}|W_{0}(ϵ′) = (g_{0}|∂W_{0}) × dy^{2}, g_{0}|M̄_{1}((0, δ]) ∩ W_{0}(ϵ′) = (g_{0}|∂M_{1}) × dx^{2} × dy^{2}

where we have the coordinates (t, x, y) ∈ R^{N} × R × R. We have a similar expression for g_{0}|N̄_{0}(δ).

Now we define the desired Riemannian metric on int W_{0} as follows. Let us put:

W̃_{0} = W̃′_{0} ∪ [∂W_{0}\{M_{1} ∪ N_{0}}] × (−∞, 0]∪

[∂M_{1} × (−∞, ϵ] ∪ M_{1} × {ϵ}] × (−∞, 0] ∪ [∂N_{0} × (−∞, ϵ] ∪ N_{0} × {ϵ}] × [1, ∞).

Figure

Clearly W̃_{0} is diffeomorphic to int W_{0}. On the other hand, by the above remark, there is a natural extension of g_{0} to g|W̃_{0}. This is the desired complete Rimannian metric. Later we will use the following notations. Let us choose large t >> 0 and s >> 0. Then we denote:

\begin{aligned}W̃_{0}(t) = [∂W_{0}\\{M_{1} ∪ N_{0}\}] × \{−t\} & ∪ ∂M_{1} × \{−t\} × (−∞, 0] \\ & ∪ ∂N_{0} × \{−t\} × [1, ∞), \end{aligned}

M_{1}(s) = {∂M_{1} × (−∞, ϵ] ∪ M_{1} × {ϵ}} × {−s}.

We denote M_{1}([S_{0}, S_{1}]) = ∪_{s∈[S_{0},S_{1}]}M_{1}(s). We have a similar notation N_{0}(s).

5.B Casson handles as Riemannian manifolds: Let us generalize the previous construction over the triple (W_{0}, M_{1}, N_{0}) as follows. Suppose we have n+1 submanifolds M_{0}, M_{1}, …, M_{n} in ∂W_{0} such that they are disjoint from each other, M_{i}∩M_{j} = ϕ. Then by choosing an appropreate embedding I : W_{0} ↪ R^{N}, one can construct a complete Riemannian metric on intW_{0}, by the same way as 5.A. Recall the notation W̃′_{0} in 5.A. Then one expresses this as W̃_{n}:

W̃_{n} = W̃′_{0} ∪ [∂W_{0}\ ∪_{i} M_{i}] × (−∞, 0] ∪_{i} [∂M_{i} × (−∞, ϵ] ∪ M_{i} × {ϵ}] × (−∞, 0].

Let Σ be a Riemann surface of genus g ≥ 1, and Σ̄ be its solid Riemann surface. Thus Σ̄ is a three manifold with boundary Σ. Let us put W_{g} = Σ̄×[0, 1], and take g embeddings of S^{1} × D^{2} into ∂W_{g} which represents a generating set of π_{1}(W_{g}). We denote their images by M_{1}, …, M_{g}. One may assume that all M_{i} and M_{j} are disjoint. A kinky handle with k kinks, are expressed as (W_{k}, N_{0}, M_{1}, …, M_{k}), where N_{0} is another embedding of S^{1} × D^{2} into ∂W_{k}. The isotopy type of N_{0} is determined by the sign (±, ⋯ ±) (k times). Thus there are exactly 2^{k} types of kinky handles with k kinks.

Casson handle is constructed by taking end connected sums of kinky handles infinitely many times. It is expressed by an infinite tree (embedded in R^{2}) with the additional information. Let us introduce the following:

Definition 5.1 Let T be an infinite tree with only one end point ∗. T is said to be a signed tree, if every edge v is assigned one of + or −.

Let n(v) + 1 be an integer which is the number of edges of T with common end point v. A signed tree T is said to be of bounded multiplicity if there exists a constant C with C ≥ n(v) for any v ∈ T.

Any tree with one end point ∗, admits a natural distance from ∗. One will denote it by | |. Let T be a signed tree with one end point ∗. Let us take any vertex v ∈ T. Then there exists a unique edge e on T such that one of the end point of e is v, and the other v′ satisfies |v′| < |v|. To specify this e, one denotes it as e(v). Let us denote finite subtrees by T_{v} = ∪_{j=0}^{n(v)}e_{j}, where all the edges e_{j} has v as one end point, and e_{0} = e(v). Recall that each e_{j} is assigned with ±. Since e_{j}, 1 ≤ j ≤ n(v), are all assigned one of ±, the set {e_{1}, …, e_{n(v)}} has one to one correspondence with the set (W_{n(v)}, N_{0}, M_{1}, …, M_{n(v)}), where W_{n(v)} is diffeomorphic to Σ̄_{n(v)} × [0, 1], and all N_{0} and M_{j} are diffeomorphic to S^{1} × D^{2} with specified embeddings into ∂W_{n(v)}. Thus every T_{v} corresponds to a kinky handle with n(v) kinks. In particular one may assign a complete Riemannian manifold W̃_{v} for every vertex v ∈ T.

Let us take an edge e with end points v and v′, and assign a large number S >> 0. Then one can make end connected sums of W̃_{v} with W̃_{v′} as:

W̃_{v′}♮_{e,S}W̃_{v} ≡ W̃_{v′}\M_{e}([S, ∞))♮W̃_{v}\N_{v}([S, ∞)).

One will also denote the set (T, S) by T, and also call it a signed tree. For every e, one may connect W̃_{v} and W̃_{v′} as above. By iterating these end connected sums for every edge, one gets a complete Riemannian four manifold CH(T). The diffeomorphism type of this space, is the Casson handle corresponding to T.

We denote (W_{0}, N_{0}, M_{1}) as the simplest kinky handles. Then Ỹ (S) is a periodic simplest Riemannian Casson handle:

CH(R) = Ỹ (S) = ∪_{j∈Z}W̃_{0}^{j}([−S, S]), CH(R_{+}) = Ỹ (S)_{0} = ∪_{j∈N}W̃_{0}^{j}([−S, S]) where W̃_{0}^{j}([−S, S]) = W̃_{0}^{j}\N_{0}([S, ∞)) ∪ M_{1}([S, ∞)), and one numerates the same W̃_{0} by W̃_{0}^{j}, j ∈ Z.

Let (W_{l}, N_{0}, M_{1}, …, M_{l}) be a kinky handle with l kinks. Then one defines the following another Riemannian manifolds. Let us choose any 1 ≤ j ≤ l, say j = 1. Now one defines:

V (S, l)_{0} ≡ W̃_{l}\ ∪ N_{0}([S, −∞)) ∪_{k=1}^{l} M_{k}([S, ∞)),
Y (S, l, 1)_{0} = V (S, l)_{0} ∪_{j=2}^{l} ∪_{M_{j}(S)∼N_{0}(S)}Ỹ (S)_{0}\N_{0}([S, −∞)),
Y (S, l, 1) = Y (S, l, 1)_{0}/N_{0}(S) ∼ M_{1}(S),
Ỹ (S, l, 1)_{0} = ∪_{j∈N} ∪_{M_{1}^{j}(S)∼N_{0}^{j+1}(S)} Y (S, l, 1)_{0}^{j},
Ỹ (S, l, 1) = ∪_{j∈Z} ∪_{M_{1}^{j}(S)∼N_{0}^{j+1}(S)} Y (S, l, 1)_{0}^{j}.

One expresses the complete Riemannian manifold Ỹ (S, l, 1) by CH(T_{l,1}). Similarly there is an infinite signed tree with a base point (T_{l,1})_{0} such that Ỹ (S, l, 1)_{0} is diffeomorphic to CH((T_{l,1})_{0}). Let us take l half lines R_{+}, … R_{+} (l times). Then one identifies all R_{+} at 0. The result is a connected infinite tree with a base point. One denotes it by T_{+}^{l}. T_{l,1} can be expressed as R ∪_{j∈Z} (T_{+}^{l−1})^{j}, where (T_{+}^{l−1})^{j} are the same (T_{+}^{l−1}) indexed by j ∈ Z. Similarly one has (T_{l,1})_{0} = R_{+} ∪_{j∈N} (T_{+}^{l−1})^{j}.

Let us define:

Y (S, l, m, 1)_{0} ≡ V (S, l)_{0} ∪_{j=2}^{l} ∪_{M_{1}^{j}(S)∼N_{0}^{1}(S)}Ỹ (S, m, 1)_{0},
Y (S, l, m, 1) = Y (S, l, m, 1)_{0}/N_{0}(S) ∼ M_{1}(S),
Ỹ (S, l, m, 1)_{0} = ∪_{j∈N} ∪_{M_{1}^{j}(S)∼N_{0}^{j+1}(S)} Y (S, l, m, 1)_{0}^{j},
Ỹ (S, l, m, 1) = ∪_{j∈Z} ∪_{M_{1}^{j}(S)∼N_{0}^{j+1}(S)} Y (S, l, m, 1)_{0}^{j}.

Similarly one has Ỹ (S, l, m, 1) = CH(T_{l,m,1}), Ỹ (S, l, m, 1)_{0} = CH((T_{l,m,1})_{0}). Let us take (T_{m,1})_{0}, … (T_{m,1})_{0} (l times). Then one identifies all (T_{m,1})_{0} at 0. The result is a connected infinite tree with a base point. One denotes it by (T_{m,1})_{0}^{l}. As above, T_{l,m,1} can be expressed as R ∪_{j∈Z} [(T_{m,1})_{0}^{l−1}]^{j}, where [(T_{m,1})_{0}^{l−1}]^{j} are the same (T_{m,1})_{0}^{l−1} indexed by j ∈ Z. Similarly one uses a notation (T_{l,m,1})_{0}^{l}, which one identifies l (T_{l,m,1})_{0} at 0. Then using this, one can obtain T_{l,m,k,1} by a similar method.

By iteration, one obtains Y (S, l, m, n, 1) and Ỹ (S, l, m, n, 1) = CH(T_{l,m,n,1}) as above. Thus one can obtain CH(T_{n_{1},...,n_{k},1}) by iteration. One calls T_{n_{1},...,n_{k},1} a homogeneous tree of finite type.

Let n̄ = {n_{1}, n_{2}, …, n_{l}, … } be an infinite set of positive integers. One can iterate the previous construction infinitely many times. Then one gets a complete Riemannian manifold:

CH(Tn̄) = CH(T_{n_{1},n_{2},...})

where we call Tn̄ a homogeneous tree. If there is a bound C with n_{j} ≤ C for all j, then we call the homogeneous tree of bounded multiplicity.

6 Some properties of elliptic operators over W̃_{0}

6.A Spectral decomposition: Throughout this paper, one uses the following lemmas. Let P be an order 1 elliptic differential operator over a complete Riemannian manifold W of bounded geometry. Let w be an weight function on W, and weighted Sobolev k norms on W by |u|W_{w}^{k} = (Σ_{l≤k} ∫ exp(w)|∇^{l}u|^{2}vol)^{\frac{1}{2}}. Thus one has a bounded map P : W_{w}^{k+1}(W) ↦ W_{w}^{k}(W). Let us denote P_{w}^{∗}, the formal adjoint operator with respect to the weighted L_{w}^{2} inner product.

Lemma 6.1 Suppose SpecΔ = P_{w}^{∗} ∘ P is contained in [λ^{2}, ∞) for some λ > 0. Then P : W_{w}^{k+1} ↦ W_{w}^{k} has closed range.

Similarly P_{w}^{∗} : W_{w}^{k+1} ↦ W_{w}^{k} also has closed range.

P roof : Let us consider the first statement. One verifies this by induction. Let k = 0. Let {u_{i}} be a sequence in W_{w}^{1} with |P (u_{i})|L_{w}^{2} ↦ 0. Since one has the estimate |P (u)|L_{w}^{2} ≥ λ|u|L_{w}^{2}, one sees |u_{i}|L_{w}^{2} → 0. Moreover by the elliptic estimate:

|u|W_{w}^{k+1} ≤ C_{k}(|P (u)|W_{w}^{k} + |u|W_{w}^{k}), k ≥ 0

for some constants C_{k}, one concludes |u_{i}|W_{w}^{1} → 0. This shows the result for k = 0.

We show by induction that if a sequence {u_{i}} in W_{w}^{k+1} satisfies |P (u_{i})|W_{w}^{k} → 0, then |u_{i}|W_{w}^{k} → 0. Then by the elliptic estimate, one gets the conclusion. Suppose the result holds for k ≤ k_{0}, and take a sequence {u_{i}} in W_{w}^{k_{0}+2} with |P (u_{i})|W_{w}^{k_{0}+1} → 0. Then since |P (u_{i})|W_{w}^{k_{0}} also converges to 0, one knows that |u_{i}|W_{w}^{k_{0}} converges to 0 by the induction hypothesis. By the elliptic estimate, one gets the result for k_{0}+1. Thus one gets the result for k_{0}+1. This completes the proof of the first statement.

Next let us consider the second statement. Let H^{⊥} ⊂ L_{w}^{2} be a closed subspace which consists of the orthogonal complement of H = P (W_{w}^{1}) ⊂ L_{w}^{2}. Notice the following relation:

Δ(W_{w}^{k+2}) = P_{w}^{∗}(H ∩ W_{w}^{k+1}) ⊂ W_{w}^{k}.

Now let {v_{i}} be a sequence in H ∩ W_{w}^{1} with |P_{w}^{∗}(v_{i})|L_{w}^{2} ↦ 0. Since one may express v_{i} = P (u_{i}) for u_{i} ∈ W_{w}^{2}, one has |Δ(u_{i})|L_{w}^{2} → 0. Thus one has |u_{i}|L_{w}^{2} → 0. Then by the Cauchy Schwartz, one has the estimate |P (u_{i})|^{2}L_{w}^{2} ≤ |Δ(u_{i})|L_{w}^{2}|u_{i}|L_{w}^{2}. This shows |v_{i}|L_{w}^{2} → 0. By the elliptic estimate, one sees |v_{i}|W_{w}^{1} → 0. Then by proceeding similarly as above, one verifies the conclusion by induction. This completes the proof.

Let W and w be as above, and consider an elliptic complex:

0 ⟶ E_{0} ⟶^{d_{0}} E_{1} ⟶^{d_{1}} E_{2} ⟶ 0

over vector bundles on W.

Lemma 6.2 (1) Suppose d_{0} : W_{w}^{k+1}(E_{0}) ↦ W_{w}^{k}(E_{1}) has closed range, and Kerd_{0} = 0. Then there exists a constant C such that:

|(d_{0})_{w}^{∗} ∘ d_{0}(u)|W_{w}^{k}(E_{0}) ≥ C|u|W_{w}^{k+2}(E_{0})

for any u. In particular (d_{0})_{w}^{∗} ∘ d_{0} gives an isomorphism.

(2) Suppose the corresponding bounded complex:

0 ⟶ W_{w}^{k+2}(E_{0}) ⟶^{d_{0}} W_{w}^{k+1}(E_{1}) ⟶^{d_{1}} W_{w}^{k}(E_{2}) ⟶ 0

satisfies the following; (1) d_{0} and d_{1} have closed range, (2) 1st cohomology group Kerd_{1}/ imd_{0} = 0, and (3) Kerd_{0} = 0.

Then for P = (d_{0})_{w}^{∗} ⊕ d_{1} and Δ = P_{w}^{∗} ∘ P, one has SpecΔ ⊂ [λ, ∞) for some positive constant λ > 0. Moreover (d_{1})_{w}^{∗} has closed range.

P roof : (1) Suppose contrary, and take a sequence {u_{i}} ⊂ W_{w}^{2} with |(d_{0})_{w}^{∗} ∘ d_{0}(u_{i})|L_{w}^{2} → 0. Then by Cauchy Schwartz, one has a convergent sequence |d_{0}(u_{i})|L_{w}^{2} → 0. Then by the elliptic estimate, one has |d_{0}(u_{i})|W_{w}^{1} → 0. By closedness and the open mapping theorem, this shows existence of non trivial kernel. This is a contradiction. The rest of the proof follows by induction.

(2) follows from (1) and lemma 6.1. This completes the proof.

Let us use the above notations. Let us put H = L_{w}^{2}(E_{1}) ∩ ( Kerd_{1})^{⊥}. Let us consider restriction of Δ = (d_{1})_{w}^{∗} ∘ d_{1}|H.

Corollary 6.1 Suppose d_{0} and d_{1} has closed range as above. Then SpecΔ|H is contained in [λ, ∞) for some positive λ > 0.

6.B Analysis over W̃_{0}: Let us take the complete Riemannian manifold W̃_{0} in 5.A. Let us take a smooth function w : W̃_{0} ↦ [0, ∞). Then as before one has weighted Sobolev k norms on W̃_{0} by |u|W_{w}^{k} = (Σ_{l≤k} ∫ exp(w)|∇^{l}u|^{2}vol)^{\frac{1}{2}}. Then we have the Atiyah Hitchin Singer complex (AHS complex) as:

0 ⟶ W_{w}^{k+2}(W̃_{0}; Λ^{0}) ⟶^{d} W_{w}^{k+1}(W̃_{0}; Λ^{1}) ⟶^{d_{+}} W_{w}^{k}(W̃_{0}; Λ_{+}^{2}) ⟶ 0

where d^{+} = (1 + ∗) ∘ d.

Let M be a complete Riemannian 3 manifold without boundary, but possibly be non compact. We have in mind M = W̃_{0}(t) or N_{0}(s), M_{1}(s). Let g be a Riemannian metric on M, and denote also by g as the product one on M × R. Let Λ^{1}(M × R) and Λ_{+}^{2}(M × R) be the exterior differentials on 1 forms and self dual 2 forms with respect to g. Then we have the natural identification:

Λ^{1}(M × R) = p^{∗}(Λ^{1}(M)) ⊕ p^{∗}(Λ^{0}(M)),
Λ_{+}^{2}(M × R) = p^{∗}(Λ^{1}(M))

where p : M × R ↦ M is the projection. The isomorphisms are given by:

u + vdt ↔ (u, v), ∗_{M}u + u ∧ dt ↔ u.

Let us put X = M × R. Using L^{2} adjoint operator (here we do not use any weight functions), we get the elliptic operator P = d^{∗} ⊕ d^{+} : Λ^{1}(X) ↦ Λ^{0}(X) ⊕ Λ_{+}^{2}(X). Using the above identification, this is expressed as:

P = d^{∗} ⊕ d^{+} : p^{∗}(Λ^{0}(M) ⊕ Λ^{1}(M)) ↦ p^{∗}(Λ^{0}(M) ⊕ Λ^{1}(M))

where p : M × R ↦ M is the projection. Let us use t as the coordinate on R. Then by a straightforward calculation, one has the following expression:

P = −\frac{d}{dt} + (\begin{matrix}∗_{M}d & d \\ d^{∗} & 0 \end{matrix}) ≡ −\frac{d}{dt} + Q.

Notice that Q is an elliptic self adjoint differential operator on L^{2}(M; Λ^{0}(M) ⊕ Λ^{1}(M)). Notice that if M is closed, then Q is not invertible (one has constant functions). When M is non compact, 0 may be contained in the spectrum of Q, even if there are no kernels for Q.

Let us introduce a weight function τ on X as follows. Let M be possibly non compact as above. Let us choose a positive number δ > 0 which will be specified later, and choose a smooth proper function τ′ : R ↦ [0, ∞) with τ′|R\[−1, 1](t) = δ|t|. Then we define τ : M × R ↦ [0, ∞) by τ(m, t) ≡ τ′(t). Let us introduce weighted Sobolev k norms on X by:

|u|W_{τ}^{k} = (Σ_{l≤k} ∫ exp(τ)|∇^{l}u|^{2}vol)^{\frac{1}{2}}.

We denote these Sobolev weighted spaces by W_{τ}^{k} or L_{τ}^{2} (for k = 0). Then one has an isometry:

I : L^{2}(X, p^{∗}(Λ^{0} ⊕ Λ^{1})) ↦ L_{τ}^{2}(X, p^{∗}(Λ^{0} ⊕ Λ^{1}))

by I(u) = exp(−\frac{τ}{2})u. Then one has the equality:

I^{−1}P I = P + \frac{1}{2} \frac{dτ}{dt} : W^{k+1}(X; p^{∗}(Λ^{0} ⊕ Λ^{1})) ↦ W^{k}(X; p^{∗}(Λ^{0} ⊕ Λ^{1})).

Let d_{τ}^{∗} be the L_{τ}^{2} adjoint operator, i.e. < u, d(v) > |L_{τ}^{2} =< d_{τ}^{∗}(u), v > |L_{τ}^{2}, and put P_{τ} = d_{τ}^{∗} ⊕ d^{+}. Then we have the following expression:

I^{−1}P_{τ}I = −\frac{d}{dt} + (\begin{matrix}∗_{M}d & d \\ d^{∗} & −\frac{dτ}{dt} \end{matrix} ) + \frac{1}{2} \frac{dτ}{dt} ≡ −\frac{d}{dt} + Q_{τ}.

We recall the definition of W̃_{0}(t) or M_{1}(s), N_{0}(s) in the last paragraph in 5.A. Practically we will apply the next lemma for M = W̃_{0}(t). Let us consider the operator:

Q_{δ} = (\begin{matrix}∗_{M}d + \frac{δ}{2} & d \\ d^{∗} & −\frac{δ}{2} \end{matrix}) : W^{k+1}(M, Λ^{1} ⊕ Λ^{0}) ↦ W^{k}(M, Λ^{1} ⊕ Λ^{0}).

Notice that Q_{δ} coincides with Q_{τ} above for |t| ≥ 1.

Lemma 6.3 Suppose M is a closed Riemannian manifold. Then for a small δ > 0 and k ≥ 0, Q_{δ} gives an isomorphism.

P roof : Let us take (u, v) ∈ W^{k+1}(M, Λ^{1} ⊕ Λ^{0}), and put Q_{δ}(u, v) = (x, y). Then we have:

(∗d + \frac{δ}{2})u + dv = x, d^{∗}u − \frac{δ}{2}v = y.

For convenience, we recall ∗^{2}u = (−1)^{p(n−p)}u, d^{∗}u = (−1)^{np+n+1} ∗ d ∗ u for p form u on n dimensional manifold. Then from the above, we have:

d^{∗} ∗ du + \frac{δ}{2}d^{∗}u + d^{∗}dv = d^{∗}x = ((\frac{δ}{2})^{2} + d^{∗}d)v + \frac{δ}{2}y.

Thus we can solve v from (x, y) as follows:

v = ((\frac{δ}{2})^{2} + d^{∗}d)^{−1}(d^{∗}x + \frac{δ}{2}y).

Next we have the following equality:

∗^{2}du + \frac{δ}{2} ∗ u + ∗dv = ∗x = du + \frac{δ}{2} ∗ u + ∗dv.

\begin{matrix}d^{∗}du + \frac{δ}{2}d^{∗} ∗ u+ & d^{∗} ∗ dv = d^{∗} ∗ x \\ = d^{∗}du + \frac{δ}{2} ∗ du = d^{∗}du + \frac{δ}{2}(x − \frac{δ}{2}u − dv). \end{matrix}

Then from this, we have the following equation:

\begin{matrix}(d^{∗}d − (\frac{δ}{2})^{2})u = & − \frac{δ}{2}(x − dv) + d^{∗} ∗ x, \\ dd^{∗}u = & \frac{δ}{2}dv + dy. \end{matrix}

Thus one can again solve u as:

u = (Δ − (\frac{δ}{2})^{2})^{−1}(dy + d^{∗} ∗ x − \frac{δ}{2}x + δdv).

It follows that there is a constant C = C(M) such that Spec|Q_{δ}| ≥ C(\frac{δ}{2})^{4}. Notice the following inequalities:

\begin{matrix}|(u, v)|W^{k+1} ≤ & C(\frac{δ}{2})^{−2}|(Δ ± (\frac{δ}{2})^{2})(u, v)|W^{k−1} \\ ≤ & C(\frac{δ}{2})^{−4}(|x|W^{k} + |y|W^{k}) ≤ C(\frac{δ}{2})^{−4}|Q_{δ}(u, v)|W^{k}). \end{matrix}

In fact one may assume that C only depends on Sup_{0≤l≤2} Sup_{x∈M}|∇^{l}R|(x), where R is the curvature form with respect to the Levi Chivita connection.

This completes the proof.

Let (P_{τ})_{τ}^{∗} be the formal adjoint of the Sobolev weighted spaces. Then one considers the same analysis for (P_{τ})_{τ}^{∗}. Notice the equality:

(I^{−1} ∘ P_{τ} ∘ I)^{∗} = I^{−1} ∘ (P_{τ})_{τ}^{∗} ∘ I.

Thus one can analysis an operator of the form, P_{τ}^{∗} = \frac{d}{dt} + Q_{δ}, since Q_{δ} is self adjoint over M.

Let us take a smooth (non proper) function:

τ : W̃_{0} ↦ [0, ∞)

which is horizontalwisely constant, namely, for some positive constant δ > 0, we have τ|W̃_{0}(t) = δt for t ≥ 0. Using τ, one defines weighted Sobolev spaces W_{τ}^{k}((W̃_{0}, g), Λ_{∗}). One has the expression P_{τ}|M_{1}([0, ∞)) = −\frac{d}{ds} + Q_{τ′} where:

Q_{τ′} = (\begin{matrix}∗d & d \\ d_{τ′}^{∗} & 0 \end{matrix}) : W_{τ′}^{k+1}(M_{1}(s); Λ^{1} ⊕ Λ^{0}) ↦ W_{τ′}^{k}(M_{1}(s); Λ^{1} ⊕ Λ^{0})

is an elliptic operator over M_{1}(s), and τ′ = τ|M_{1}(s) is its restriction. It follows that Q_{τ′} has closed range with empty kernel. We do not use this fact later. In this case one can not see whether P_{τ} defined above is Fredholm or not, since one has no simple statement as above for the weighted adjoint operator (Q_{τ′})_{τ′}^{∗}|W_{τ′}^{k}(M_{1}(s); Λ^{1}).

Let us take another weight function μ : W̃_{0} ↦ [0, ∞), with μ|M_{1}(s), N_{0}(s) ≡ δ′s, where we choose a sufficiently small 0 < δ′ << δ. Let us put:

w = τ + μ : W̃_{0} ↦ [0, ∞).

We call w as a weight function with weight (δ, δ′). Let us use w as the weight function. Then one obtains a bounded operator:

P_{w} = d_{w}^{∗} ⊕ d^{+} : W_{w}^{k+1}(W̃_{0}; Λ^{1}) ↦ W_{w}^{k}(W̃_{0}; Λ^{0} ⊕ Λ_{+}^{2}).

Passing through the isometry I : L^{2} → L_{w}^{2}, I(u) = exp(−\frac{w}{2})u, P_{w} has the following expression on M_{1}([0, ∞)) and N_{0}([0, ∞)):

I^{−1} ∘ P_{w} ∘ I = −\frac{∂}{∂s} + (\begin{matrix}∗d & d \\ d_{τ′}^{∗} & 0 \end{matrix}) + \frac{δ′}{2} (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) ≡ −\frac{∂}{∂s} + Q_{w}^{v}.

On the other hand one has another expression on W̃_{0}([0, ∞)):

I^{−1} ∘ P_{w} ∘ I = −\frac{∂}{∂t} + (\begin{matrix}∗d + \frac{δ}{2} & d \\ d_{μ}^{∗} & −\frac{δ}{2} \end{matrix}) = −\frac{∂}{∂t} + Q_{w}^{h}.

6.C Decay estimate: Let M be a possibly non compact n dimensional Riemannian manifold of bounded geometry. In practice M stands for W̃_{0}(t) (or N_{0}(s), M_{1}(s)). Let us take a smooth function μ : M ↦ [0, ∞). Let E and F be vector bundles over M, and take an elliptic operator:

Q : W_{μ}^{k+1}(M; E) ↦ W_{μ}^{k}(M; F).

Let us choose τ : M × R ↦ R_{+} by τ(m, t) = δ|t| for |t| ≥ 1, and put w = τ + μ. Then one considers the following operator:

P = −\frac{d}{dt} + Q : W_{w}^{k+1}(M × R, p^{∗}E) ↦ W_{w}^{k}(M × R; p^{∗}F).

Let P_{w}^{∗} be the formal adjoint operator with respect to the weighted inner product. Then P_{w}^{∗}P admits a spectral decomposition. For u ∈ W_{w}^{k}(M × R, p^{∗}(E)), we denote Specu ∈ [0, ϵ] if u lies in the image of the spectral projection 𝔓([0, ϵ]).

Suppose the above Q is a Fredholm operator, and is self adjoint with respect to the weighted L_{μ}^{2} inner product.

Proposition 6.1 (LM, F,T) For sufficiently small δ > 0, the above P is Fredholm. Moreover if Q is invertible, then one may choose δ = 0. In particular P is also invertible.

Suppose Q : W_{μ}^{k+1}(M; E) ↦ W_{μ}^{k}(M; F) has closed range with KerQ = 0. Let us put Δ = Q_{μ}^{∗}∘Q, and choose a positive constant λ_{0} > 0 with SpecΔ ∈ [λ_{0}^{2}, ∞). Then under the above condition, one has the following; for any small ϵ << λ_{0}, there exist positive constants C, δ_{0} > 0 such that for any u ∈ W^{k+l}(M ×R, p^{∗}E) with Specu ∈ [0, ϵ] for P_{w}^{∗} ∘ P, we have the following decay estimate:

|u( , t)|L_{μ}^{2}(M) ≤ Cexp(−δ_{0}|t|)|u|L_{μ}^{2}(M × [0, ∞)), t ≥ 0.

P roof : The first statement follows from the decay estimate. Next let us take u as above, and denote its slice on M × s by u_{s}. Let us put a smooth function f : [0, ∞) by f(s) = ∫_{M_{s}} exp(μ)|u_{s}|^{2}. Then by differentiating, one has the inequalities:

f′′(s) = 2[∫_{M_{s}} exp(μ) < u′_{s}, u′_{s} > + ∫_{M_{s}} exp(μ) < u′′_{s}, u_{s} >]
≥ 2[∫_{M_{s}} exp(μ) < u′_{s}, u′_{s} > + ∫_{M_{s}} exp(μ) < (Δ − ϵ)u_{s}, u_{s} >]
≥ 2 ∫_{M_{s}} exp(μ) < (Δ − ϵ)u_{s}, u_{s} >≥ 2(λ_{0} − ϵ)f(s).

Thus one has differential inequalities f′′ ≥ λ_{0}f. From this, one can get the desired estimate. This completes the proof.

Let Y be a complete Riemannian four manifold such that except compact subset K ⊂ Y, Y \K is isometric to Y (0) × [0, ∞) where Y (0) is a closed manifold. Let us equip a weight function τ : Y ↦ [0, ∞) by τ|Y (0) × t = δt. Let I : L^{2}(Y) ≅ L_{τ}^{2}(Y) be the isometry.

Corollary 6.2 (LM) The corresponding

P = I^{−1}P_{τ}I : W^{k+1}(Y; Λ^{1}) ↦ W^{k}(Y; Λ^{0} ⊕ Λ_{+}^{2})

gives a Fredholm map.

P roof : This is well known, but for convenience we recall the proof. Let us take a cut off function φ|Y with φ|Y (0) × [1, ∞) ≡ 1, and φ|K ≡ 0. Let us choose any u ∈ W^{k+1}(Y; Λ^{1}). Then one may regard φu ∈ W^{k+1}(Y (0) × R; Λ^{1}). Thus one has the estimate:

|u|W^{k+1} ≤ |φu|W^{k+1} + |(1 − φ)u|W^{k+1}
≤ C{|φP (u)|W^{k} + |[φ, P]u|W^{k} + |(1 − φ)P u|W^{k}
+ |[(1 − φ), P]u|W^{k} + |(1 − φ)u|W^{k}}.

This shows that P has closed range with finite dimensional kernel. One has a similar estimate for P^{∗}. This shows the result.

6.D 1st cohomology: Let w = τ + μ be the last paragraph in 6.B.

Lemma 6.4 For a sufficiently small choice of δ′ << δ, d : W_{w}^{k+1}(W̃_{0}; Λ^{0}) ↦ W_{w}^{k}(W̃_{0}; Λ^{1}) has closed range with Ker d = 0.

P roof : Clearly Ker d = 0.

As lemma 6.3, passing through the isometry I, I^{−1} ∘ d ∘ I|W̃_{0}([0, ∞)) can be expressed as:

\frac{d}{dt} + (d, −\frac{δ}{2}) : W^{k+1}(W̃_{0}([0, ∞)) ↦ W^{k}(W̃_{0}([0, ∞); Λ^{0} ⊕ Λ^{1}).

Clearly this has closed range. One has the same property on N_{0}([0, ∞)) and M_{1}([0, ∞)).

Now let H_{i} ⊂ W^{k+1}(W̃_{0}), i = 0, 1, 2, 3 be closed subsets whose supports lye in N_{0}([0, ∞)), M_{1}([0, ∞)), W̃_{0}([0, ∞)) and K respectively, where K ⊂ W̃_{0} is some compact subset. Then one may assume ∪_{i}H_{i} = W^{k+1}(W̃_{0}). Now by the open mapping theorem and above, every image d(H_{i}) has closed range. This completes the proof.

Let us consider W̃_{0} and the weight function w on it with the weight constants (δ, δ′). Let us recall that we have a bounded complex:

0 ⟶ W_{w}^{k+2}(W̃_{0}) ⟶^{d} W_{w}^{k+1}(W̃_{0}; Λ^{1}) ⟶^{d_{+}} W_{w}^{k}(W̃_{0}; Λ_{+}^{2}) ⟶ 0

where the weight function is w = τ + μ.

It is clear Kerd_{0} = 0. Suppose ω ∈ W_{w}^{1}(W̃_{0}; Λ^{1}) with d_{+}(ω) = 0. Then by integration by parts, one has 0 = ∫W̃_{0} |d_{+}ω|^{2} = ∫W̃_{0} |dω|^{2}. In particular we have dω = 0. The following sublemma shows H^{1}(AHS) = 0.

Let W = W̃_{0} or Ỹ (S) in 5.B. Recall that one has a weight function τ over W̃_{0}. There is a natural extension of τ on Ỹ (S). Let μ : Ỹ (S) ↦ [0, ∞) be another weight function defined as μ|W̃_{0}^{n}([−S, S])(x) = δ′(|n| + t(x)), where t : W̃_{0}^{0}([−S, S]) ↦ [0, 1] satisfies t|N_{0}(S) ≡ 0 and t|M_{1}(S) ≡ 1.

Let w = τ + μ be a weight function on W In order to make explicit the weight constants, sometimes one uses notations τ(δ) and μ(δ′). Later one will show that d : W_{w}^{k+1}(Ỹ (S)) ↦ W_{w}^{k}(Ỹ (S); Λ^{1}) has closed range. Assuming this, one has the following:

Sublemma 6.1 Let f ∈ C^{∞}(W) satisfies |df|L_{w}^{2}(W) < ∞. Then there are constants C and k = k(f) such that:

|f − k|L_{w}^{2} < ∞.

Proof: The situation diffres from [T1] lemma 5.2, in that on the end, we have non compact slices, W̃_{0}(t) and M_{1}(s), N_{0}(s). In order to verify this, one takes two steps. First one sees a weaker version of the sublemma.

Step 1: Let us choose any δ_{0} < δ and δ′_{0} < δ′ so that the pair (δ_{0}, δ′_{0}) is sufficiently near (δ, δ′). Let us put w = τ(δ) + μ(δ′), w′ = τ(δ_{0}) + μ(δ′_{0}). Then we claim that f − k ∈ L_{w′}^{2}.

By lemma 6.2(1) and lemma 6.4, one has an estimate:

∫_{W} exp(w)|u|^{2} ≤ C ∫_{W} exp(w)|du|^{2}.

This shows that one has a Hilbert space, which is a completion of C_{cp}^{∞}(W) by the norm |u|^{2} = ∫_{W} exp(w)|du|^{2}. Let us take any f ∈ C^{∞}(W) with df ∈ L_{w}^{2}. Using a functional s : H → R, by s(v) = ∫W̃_{0} exp(w)(|dv|^{2} − 2 < dv, df >), one gets a unique u = (d_{w}^{∗}d)^{−1}(d_{w}^{∗}df) ∈ H with d_{w}^{∗}d(f − u) = 0. We show that g = f − u is in L_{w′}^{2}. From dg ∈ L_{w}^{2}, one has estimates ∫_{W ([t,t+1])} |dg|^{2} ≤ exp(−δt)ϵ_{t}, where ϵ_{t} → 0. Notice that dg satisfies the elliptic equation, (d_{w}^{∗} ⊕ d)dg = 0. Thus by the local Sobolev embedding and the above estimate, one gets:

Sup_{W̃_{0}(t)}|dg| ≤ Cexp(−\frac{δt}{2})ϵ_{t}.

In particular, one has the next estimate,

|g(m, t)| ≤ C_{1} + ∫_{0}^{t} C_{2}exp(−\frac{δs}{2})ϵ_{s}ds,

where C_{1} = |g(m, 0)|. This shows that there is a constant C_{m} such that lim_{t→∞} Sup_{W (m,t)}|g − C_{m}| = 0. It is clear that C_{m} is independent of m, and we write the same constant by C.

Let us make a similar procedure on s direction along M_{1}(s) and N_{0}(s). Then one finds also another constant C′ (C′′) such that g−C′(C′′)|M_{1}(m, s), (N_{0}(m, s)) vanishes at infinity for every m. Again it is clear that C = C′ = C′′.

Now one finds a constant c with |dg|(x) ≤ cexp(−\frac{w(x)}{2}). Then by integration, one has the inequality:

|g − C|(x) ≤ cexp(−\frac{w(x)}{2}).

Notice that for any α > 0, w^{α} is integrable over W. Combining with this and the above inequality, one gets the first claim.

Step 2: For any sufficiently small pair (δ, δ′), one has a well defined 1st cohomology group H^{1}(δ, δ′) = Kerd ∩ L_{w}^{2}(W; Λ^{1})/d(W_{w}^{1}(W)). Then one has a natural map:

i : H^{1}(δ, δ′) ↦ H^{1}(δ_{0}, δ′_{0}).

We claim that i is injection. This is enough for the proof of the sublemma.

Let us put Δ(δ, δ′) = (d_{w}^{∗} ⊕d)_{w}^{∗}(d_{w}^{∗} ⊕d) on L_{w}^{2}(W; Λ^{1}), w = w(δ, δ′). Then the spectrum of Δ(δ, δ′) is discrete near 0. Thus for (δ_{0}, δ′_{0}) sufficiently near (δ, δ′), one has dimH^{1}(δ, δ′) ≥ dimH^{1}(δ_{0}, δ′_{0}).

Suppose these dimensions are different. Then one chooses another δ_{1} < δ_{0} and δ′_{1} < δ′_{0}. If dimH^{1}(δ_{0}, δ′_{0}) = dimH^{1}(δ_{1}, δ′_{1}), then one puts δ_{0} = δ and δ′ = δ′_{0}, and gets the conclusion. If these dimensions are different, one has dimH^{1}(δ_{0}, δ′_{0}) > dimH^{1}(δ_{1}, δ′_{1}). If these are different, one must have dimH^{1}(δ, δ′) ≥ 2. Let us take another δ_{2} < δ_{1} < δ_{0} and δ′_{2} similarly.

One itarates this process until one gets the equality of dimensions. If this is not the case, one has an infinite sequence (δ_{i}, δ′_{i}). By choice, one may assume that this sequence converges to (δ_{∞}, δ′_{∞}) with 2δ_{0}(′) − δ(′) < δ_{∞}(′) < δ(′). Then Δ(δ_{∞}, δ′_{∞}) must have a continuous spectrum near 0. This is a contradiction. This completes the proof.

Remark 6.1: When δ′ = 0 (no weight in the horizontal direction), one can still find some constant C such that g − C vanishes at infinity. Let us take any (m′, t) ∈ W̃_{0}^{n}([−S, S]) ⊂ Ỹ (S). Then we show that if n is sufficiently large, then |g(m′, t) − C_{m}| is sufficiently small for C_{m} as above. With respect to n, let us choose a sufficiently large T. Then since one has the estimate |dg|Ỹ (S)(T) ≤ Cexp(−\frac{δT}{2} ), one may assume |g(m, T) − g(m′, T)| is sufficiently small. On the other hand, |g(m′, t) − g(m′, T)| is less than ϵ_{n}, where ϵ_{n} → 0 as n → ∞. Thus |g(m′, t) − C_{m}| is sufficiently small.

Lemma 6.5 Let W = Ỹ (S). H^{1}(δ, 0) is naturally isomorphic to H^{1}(δ, δ′).

P roof : Let α ∈ H^{1}(δ, 0) be a harmonic representative. Then from corollary 7.1 below, and by the same method as [T1], proof of lemma 5.3, one can verify α ∈ L_{w(δ,δ′)}^{2}(Ỹ (S); Λ^{1}). This completes the proof.

7 Fourier Laplace transform between open manifolds

Our main application is the analysis over (W_{0}, N_{0}, M_{1}) = kinky handles. Recall that for the simplest kinky handle, W_{0} is diffeomorphic to S^{1} × D^{3}. In this section, one considers (W_{0} = S^{1} × D^{3}, M_{1}, N_{0}), where M_{1}, N_{0} are all diffeomorphic to S^{1} × D^{2}, and the embeddings are given as follows. Let us regard ∂W_{0} = S^{1} × D_{+}^{2} ∪ S^{1} × D_{−}^{2}, and consider two knots μ ⊂ S^{1} × D_{+}^{2}, γ ∈ S^{1} × D_{−}^{2} as follows:

Figure

Notice that μ is nullhomotopic, and γ represents a generator of π_{1}(W_{0}). Let us take tublar neighbourhoods of μ and γ in S^{1} × D_{±}^{2} respectively. We denote them as μ̄ and γ̄. Then we choose γ̄ = M_{1} and μ̄ = N_{0}. Let us denote the quotient space by Y = W_{0}/M_{1} ∼ N_{0}. By a simple calculation, one knows the following:

Sublemma 7.1 H^{1}(Y : R) = R and H^{2}(Y : R) = H^{3}(Y : R) = 0.

In this section, one introduces the Fourier-Laplace transform over Y, which is an open analogue of the one in [T1]. In order to do this, one will only require the above topological conditions. In particular one can apply the method in this section for cylindrical manifolds Y which includes the simplest Casson handles.

Now by the previous construction, one has a complete Riemannian manifold W̃_{0}. Let us take infinitely many W̃_{0}, and for convenience, we numerate the same W̃_{0} as W̃_{0}^{i}, i ∈ Z. Let us choose a sufficiently large S, and introduce notations:

\begin{matrix}W̃_{0}([−S, S]) = W̃_{0}\\{M_{1}([S, ∞)) ∪ N_{0}([S, ∞))\}, \\ Y (S) = W̃_{0}([−S, S])/N_{0}(S) ∼ M_{1}(S). \end{matrix}

One has a natural extension of notation Y (S)(t) = W̃_{0}(t)/N_{0}(S, t) ∼ M_{1}(S, t), t ∈ [0, ∞). Recall that one has a weight function τ : W̃_{0} ↦ [0, ∞) by τ|W̃_{0}(t) = δt. One can make a natural extnsion of τ on Y (S). We will use the same τ for this.

Now one has a periodic Riemannian maifold as:

\begin{aligned}Ỹ (S) = & ⋯ ∪_{M_{1}^{−n}(S)∼N_{0}^{−n+1}(S)} W̃_{0}^{−n+1}([−S, S]) \\ & ∪_{M_{1}^{−n+1}(S)∼N_{0}^{−n+2}(S)} W̃_{0}^{−n+2}([−S, S]) … \end{aligned}

There is a free isometric Z action on Ỹ (S), and we denote the action 1 ∈ Z by T.

Let us take vector bundles E_{i} → Y (S), i = 1, 2 and a differential operator D between them. Then it lifts to the following one:

D : C_{cp}^{∞}(Ẽ_{1}) ↦ C_{cp}^{∞}(Ẽ_{2})

where Ẽ_{i} → Ỹ (S) are the natural lifts.

Let us take any ψ ∈ C_{cp}^{∞}(Ẽ_{1}) and z ∈ C^{∗}. Then we define the Fourier Laplace transform of ψ by:

ψ̂_{z}( ) = Σ_{n=−∞}^{∞}z^{n}(T^{n}ψ)( ).

Let us define another vector bundle over Y (S) × C^{∗} as:

E′_{1} ≡ [Ẽ_{1} ⊗_{R} C]/Z ↦ Y (S) × C^{∗}

where 1 ∈ Z sends (ρ, λ) ∈ Ẽ_{1} ⊗_{R} C to (T ρ, zλ). One may regard E′_{1} as a family of vector bundles {E′_{1}(z)} over Y (S), parametrized by z ∈ C^{∗}.

Now by restriction ψ̂|W̃_{0}^{0}([−S, S]), ψ̂_{z} defines a smooth section over Y (S) of E′_{1}(z), where W̃_{0}^{0}([−S, S]) ⊂ Ỹ (S). Thus one may regard ψ̂_{z} ∈ C_{cp}^{∞}(E′_{1}(z)).

The Fourier Laplace inversion formula is as follows. Let us take a smooth section η̂ ∈ C_{cp}^{∞}(E′_{1}) with η̂_{z} ∈ C_{cp}^{∞}(E′_{1}(z)). Then for any s ∈ (0, ∞), the following:

T^{n}η(x) ≡ \frac{1}{2πi} ∫_{|z|=s} z^{−n}η̂_{z}(π(x))\frac{dz}{z}

defines a smooth section over Ẽ_{1} → Ỹ (S), where x ∈ W̃_{0}^{0}([−S, S]) and π : Ỹ (S) → Y (S) is the projection. By the Cauchy’s formula, these are inverses each other, independently of s.

Let D be as above. Passing through the Fourier Laplace transform, one has another differential operator between (E_{0})′ and (E_{1})′ by:

D̂_{z}ψ̂_{z} ≡ (D̂ψ)_{z}.

Notice that at z = 1, (E_{i})′, i = 0, 1, are isomorphic to E_{i} → Y (S) respectively. In fact every (E_{i})′(z) is isomorphic to E_{i} as follows. Let us take t : W̃_{0}([−S, S]) ↦ [0, 1], a smooth map such that t ≡ 0 near N_{0}(S), and t ≡ 1 near M_{1}(S). By taking [0, ∞) ⊂ C as a branched line, one may define logz. Let us put z^{t} ≡ exp(tlogz), and consider z^{t}ψ̂_{z}. This gives a C valued section over E′_{0}. One calculates D̂_{z} in terms of this identification. The result is as:

D̂(z) = D + z^{t}[D, z^{−t}] : C_{cp}^{∞}(E_{0}) ↦ C_{cp}^{∞}(E_{1}).

Let {E_{i}, D_{i}}_{i=0,1,2} be an elliptic complex over Y (S). It is clear that it gives a bounded one as:

0 ⟶ W_{τ}^{k+2}(Ẽ_{0}) ⟶^{D_{0}} W_{τ}^{k+1}(Ẽ_{1}) ⟶^{D_{1}} W_{τ}^{k}(Ẽ_{2}) ⟶ 0.

Proposition 7.1 The above D_{i} has closed range, if for all z ∈ C = {w ∈ C^{∗}; |w| = 1}, {D̂_{z}^{i} : W_{τ}^{k+i} → W_{τ}^{k+i−1}} is acyclic Fredholm complex, i.e. all cohomologies H^{i} vanish.

Proof: For any u ∈ W_{τ}^{k}(Ỹ (S); Ẽ_{i}), û_{z} has the following property; it lies in W_{τ}^{k}((E_{i})_{z}; Y (S)_{z}) for all z with 0 < |z| ≤ 1.

When |z| = 1, the inner product on E_{i} has a natural extension of a hermitian metric on E_{i}(z). Recall that D_{i} : W_{τ}^{k+1}(Y (S); E_{i}(z)) → W_{τ}^{k}(Y (S); E_{i+1}(z)) has closed range. Let (D_{i})_{τ}^{∗} be the formal adjoint operator. This differential operator is defined independently of z, with |z| = 1 over E_{i}(z). By the assumption, one has an isomorphism:

D = D_{0} ⊕ (D_{1})_{τ}^{∗} : W_{τ}^{k+1}(Y (S); E_{0}(z) ⊕ E_{2}(z)) ↦ W_{τ}^{k}(Y (S); E_{1}(z)).

We denote by D^{−1} as its inverse. On the other hand one has a natural lift of D as:

D : W_{w}^{k+1}(Ỹ (S); Ẽ_{0} ⊕ Ẽ_{2}) ↦ W_{w}^{k}(Ỹ (S); Ẽ_{1}).

Here we show that there is another bounded operator:

R : W_{w}^{k}(Ỹ (S); Ẽ_{1}) ↦ W_{w}^{k+1}(Ỹ (S); Ẽ_{0} ⊕ Ẽ_{2})

with R ∘ D is the identity. From this, one knows that D has closed range. In particular D_{0} has closed range.

Next one proceeds similarly for D_{τ}^{∗} = (D_{0})_{τ}^{∗} ⊕ D_{1}. Then one also knows that its lift D_{τ}^{∗} has closed range. Clearly it follows that D_{1} has also closed range.

Now let us take ψ ∈ W_{w}^{k+1}(Ỹ (S); Ẽ_{0} ⊕ Ẽ_{2}). Then one considers b(ψ) = D^{−1}(ψ̂) ∈ W_{τ}^{k+1}(Y (S); E_{1}(z)). This element also has the property in the first paragraph of the proof.

Then one defines R(ψ) ∈ W_{w}^{k+1}(Ỹ (S); Ẽ_{1}) by:

T^{n}R(ψ)|W̃_{0}^{0}([−S, S])(x) = \frac{1}{2πi} ∫_{C} z^{−n}(D^{−1}(ψ̂)(π(x)))\frac{dz}{z}.

We claim that this assignment:

R : W_{w}^{k}(Ỹ (S); Ẽ_{1}) → W_{w}^{k+1}(Ỹ (S); Ẽ_{0} ⊕ Ẽ_{2})

gives a bounded operator. Then by definition of D_{i} over Y (S), it satisfies R ∘ D = 1.

Let us put C = {z = exp(iθ) ∈ C : |z| = 1}, ψ_{n}|W̃_{0}^{0}([−S, S]) = T^{n}(ψ). By regarging b(ψ)_{z} ∈ W_{τ}^{k+1}(W̃_{0}^{0}([−S, S])), one has a Fourier expansion b(ψ) = Σ_{n}z^{n}φ_{n}, φ_{n+1}|N_{0}(S) = φ_{n}|M_{1}(S). Notice the equality D(φ_{n}) = ψ_{n}. One can regard the value of the inner product < b(ψ), b(ψ) > |W_{τ}^{k}(Y (S); E_{∗}(z)) as a real valued function of θ ∈ [0, 2π]. Then one has the equality:

\begin{matrix}\frac{1}{2π} ∫_{0}^{2π} dθ|b(ψ)|^{2}W_{τ}^{k+1}(Y (S); E_{∗}(z)) = \\ \frac{1}{2π} ∫_{0}^{2π} dθΣ_{n,m}exp(iθ(n − m)) < φ_{n}, φ_{m} > |W_{τ}^{k+1}(W̃_{0}^{0}([−S, S])) = Σ_{n}|φ_{n}|^{2}W_{τ}^{k+1}. \end{matrix}

Combining with the inequality (D^{−1} are bounded operators for all z ∈ C) |b(ψ)|W_{τ}^{k+1}(Y (S); E_{∗}(z)) ≤ C|ψ̂|W_{τ}^{k} for every θ, one gets the estimate:

Σ_{n}|φ_{n}|W_{τ}^{k+1}(W̃_{0}^{0}([−S, S])) ≤ CΣ_{n}|ψ_{n}|W_{τ}^{k}(W̃_{0}^{0}([−S, S])).

From the equality, T^{n}(ψ) = ψ_{n}, one has the desired estimate:

\begin{aligned}|R(ψ)|^{2} & W_{τ}^{k+1}(Ỹ (S); Ẽ_{∗}) = Σ_{n}|φ_{n}|^{2}W_{τ}^{k+1}(W̃_{0}^{0}([−S, S])) ≤ CΣ_{n}|D(φ_{n})|^{2}W_{τ}^{k} \\ & = CΣ_{n}|T^{n}(ψ)|^{2}W_{τ}^{k}(W̃_{0}^{0}([−S, S])) = C|ψ|^{2}W_{τ}^{k}. \end{aligned}

This completes the proof.

One needs to have some circle C ⊂ C^{∗} which satisfies the above conditions. For this, one needs the following:

Sublemma 7.2 D_{i} : W_{τ}^{k+1}(Y (S); E_{i}(z)) ↦ W_{τ}^{k}(Y (S); E_{i+1}(z)) has closed range.

P roof : It is true for z = 1. Recall Y (S) = W̃_{0}([−S, S])/N_{0}(S) ∼ M_{1}(S). Let H_{i} ⊂ W_{τ}^{k+1}(Y (S); E_{i}(z)) be closed subsets satisfying

(1) H_{1} ∪ H_{2} = W_{τ}^{k+1}(Y (S); E_{i}(z)) and

(2) SuppH_{1} ⊂ W̃_{0}([−(S − 1), S − 1)]) and SuppH_{2} ⊂ N_{0}([−S, −(S − 1)]) ∪ M_{1}([S − 1, S]).

By regarding H_{1} ⊂ W_{τ}^{k+1}(Y (S); E_{i}(1)), it follows that D_{i}(H_{1}) is a closed subspace of W_{τ}^{k}(Y (S); E_{i+1}(z)). Let u ∈ H_{2}. Then one may associate ũ ∈ W_{τ}^{k+1}(Y (S); E_{i}(1)) by ũ|N_{0}([−S, −(S − 1)]) = u and ũ|M_{1}([S − 1, S]) = z^{−1}u. Clearly D̃_{i}(H_{2}) = D_{i}(H̃_{2}) is a closed subspace of W_{τ}^{k}(Y (S); E_{i+1}(1)). Since this assignment is isometric, it follows that H_{2} is also a closed subspace of W_{τ}^{k}(Y (S); E_{i+1}(z)).

This completes the proof.

Proposition 7.2 Let {D_{∗}} be the AHS complex. Suppose it gives a Fredholm complex and all the cohomology groups vanish, H^{i}(W_{τ}^{k}(Y (S); Λ^{∗}); D_{∗}) = 0, i = 0, 1, 2. Then for C(1) = {z : |z| = 1}, one has H^{∗}(W_{τ}^{k}(Y (S); Λ^{∗}); (D_{∗})_{z}) = 0 for all z ∈ C(1) and i = 0, 1, 2.

P roof : The first part of the proof of is essentially [T1] p390. Let u ∈ W_{τ}^{k}(Y (S); Λ_{z}^{∗}) with d_{0}(u) = 0. Then passing through the identification z^{t} : Λ_{z}^{0} ≅ Λ^{0}, one has d(z^{−t}u) = 0 where z^{−t}u ∈ W_{τ}^{k}(Y (S); Λ^{0}). Clearly this shows u = 0.

Next let us take α ∈ W_{τ}^{k}(Y (S); Λ_{z}^{1}) with d_{+}(α) = 0, d_{τ}^{∗}(α) = 0. Then by integration by parts, one has d(α) = 0. α is zero in H^{1}(W̃_{0} : R) (see the proof of lemma 8.1). Thus one may express α = d(f) on W̃_{0} for f ∈ C^{∞}(W̃_{0}([−S, S])) and d_{τ}^{∗}d(f) = 0. Let i : N_{0}(S) ≅ M_{1}(S) be the identification. Then one has i^{∗}(f) = zf +const. If one takes f so that it vanishes at infinity, then const = 0. Thus one has the equality:

0 = ∫_{W̃_{0}([−S,S])} exp(τ) < f, d_{τ}^{∗}d(f) > vol = − ∫ exp(τ)|df|^{2}.

This shows df = 0. Thus H^{∗}(W_{τ}^{k}(Y (S); Λ^{∗}); (D_{∗})_{z}) = 0, ∗ = 0, 1 for all z ∈ C(1).

Now let us put D = (D_{0})_{τ}^{∗} ⊕ D_{1}, and consider Δ = D_{τ}^{∗}D on L_{τ}^{2}(Y (S); Λ^{1}(z)). Then there is a constant λ > 0 with λ_{z} > λ, where λ_{z} is the bottom of Δ. Suppose H^{2} ≠ 0 at z = exp(iθ_{0}), but H^{2} = 0 for all z = exp(iθ), 0 ≤ θ < θ_{0}.

Then by choosing some θ, it follows that there exists u ∈ W_{τ}^{1}(Y (S); Λ_{z}^{2}), z = exp(iθ) with (1) |u|W_{τ}^{1} = 1 and (2) |(D_{1})_{τ}^{∗}(u)|L_{τ}^{2} < ϵ.

Since D_{1} is surjective, one finds α ∈ W_{τ}^{2}(Y (S); Λ_{z}^{1}) with D_{1}(α) = u and (D_{0})_{τ}^{∗}(α) = 0. Then there is a constant C independent of z such that:

|u|W_{τ}^{1} ≤ C|α|W_{τ}^{2} ≤ C|(D_{1})_{τ}^{∗}D_{1}(α)|L_{τ}^{2}.

This is a contradiction. This shows that the second cohomology also vanishes for all z ∈ C(1). This completes the proof.

Now we compute the cohomology H^{i}(W_{τ}^{k}(Y (S); Λ^{∗}); D_{∗}) of the AHS complex for the simplest kinky handles (W_{0}, N_{0}, M_{1}). Notice that the end of Y (S) is isometric to M × [0, ∞) for a compact Riemannian three manifold M. We will denote M_{t} = M × {t} ⊂ Y (S). Proposition 7.3 Let Y (S) be as above. Then for all i = 0, 1, 2, the cohomologies H^{i}(W_{τ}^{k}(Y (S); Λ^{∗}); D_{∗}) vanish. P roof : Clearly H^{0}(AHS) = 0. Let us consider H^{1}(AHS). Let us take any representative u. By integration by parts, one finds du = 0. One may express u = w + fdt around the end Y (S) ≅ M × [0, ∞), where w does not contain dt component. Then through this isometry, one considers w_{t} ∈ Λ^{1}(M) for t ∈ [0, ∞). Let d_{3} be the differential over M. Then for every t, one has d_{3}w_{t} = 0. Thus one may express w_{t} = Σ_{i}g_{i}(t)α_{i} + dμ_{t}, where {α_{i}}_{i} consists of the orthogonal basis of H^{1}(M; R). Here one may assume d_{3}^{∗}(α_{i}) = 0. By taking into account of dt component of du, one has the following equation:

Σ_{i}g_{i}(t)′α_{i} + d_{3}μ(t)′ = d_{3}f_{t}.

This shows g_{i}(t)′ ≡ 0. Since u ∈ L_{τ}^{2}, one concludes g_{i}(t) ≡ 0. Then one has another equality, f_{t} = μ(t)′ + c(t), where c(t) are constants depending on t. Thus one gets u = d_{3}μ_{t} + (μ′_{t} + c(t))dt. Sublemma 7.3 If d_{3}μ_{t} ∈ W_{τ}^{k}, then for a smooth family of constants d(t), one has μ_{t} − d(t) ∈ W_{τ}^{k}. P roof : Let us put d(t) = (vol_{M})^{−1} ∫_{M} μ_{t}vol. Then we show that this family is the desired one. Notice that one has the following bound: |d_{3}μ_{t}|L_{τ}^{2} ≥ C|μ_{t} − d(t)|L_{τ}^{2} for some positive constant C. In particular one has μ_{t} − d(t) ∈ L_{τ}^{2}. Notice that one has the equality μ_{t} − d(t) = (Δ_{0})^{−1}d^{∗}d(μ_{t} − d(t)), and (Δ_{0})^{−1}d^{∗} is a translation invariant bounded operator. Thus we get:

\frac{μ_{t} − d(t)}{dt} = \frac{d}{dt}[(Δ_{0})^{−1}d_{3}^{∗}d_{3}(μ_{t} − d(t))] = (Δ_{0})^{−1}d^{∗} \frac{d}{dt}[d_{3}μ_{t}].

From the last term, one sees that \frac{μ_{t}−d(t)}{dt} ∈ L_{τ}^{2}. By a similar consideration, one gets the result. This completes the proof.

Proof of Proposition: Then by replacing μ_{t} by μ_{t} − d(t), one may assume μ_{t} ∈ W_{τ}^{k}. Thus dμ_{t} = d_{3}μ_{t} + μ′_{t}dt ∈ W_{τ}^{k−1}. On the other hand, let us consider the equality f_{t} = μ′_{t} + c(t). Then it follows c(t) ∈ W_{τ}^{k−1}. Let us put C(t) = ∫_{0}^{t} c(s)ds − ∫_{0}^{∞} c(t). Then by [T1] lemma 5.2 (see sublemma 6.1), one finds that C(t) ∈ W_{τ}^{k}, and dC(t) = c(t). In particular we have u| endY (S) = df, f ∈ W_{τ}^{k}. Then using a cut off function, one may represent u by a compactly supported smooth 1 form, which is itself exact by a compactly supported smooth function, since H_{cp}^{1}(Y (S); R) = 0. This shows H^{1}(AHS) = 0.

Next we consider H^{2}(AHS). Let us take a representative u ∈ H^{2}(AHS). One may choose u so that it satisfies d(e^{τ}u) = e^{−τ}d(ue^{τ}) = 0. Since H^{2}(Y (S); R) = 0, one may express e^{τ}u = dμ, μ ∈ C^{∞}(Λ^{1}). Let us denote μ| endY (S) = β+fdt, where β does not contain dt component. Then we have the following relation:

dμ| endY (S) = d_{3}β_{t} + (d_{3}f − β′_{t})dt = d_{3}β_{t} + ∗_{3}d_{3}β_{t} ∧ dt

where ∗_{3} is the star operator over M. Let us decompose β_{t} = β_{t}^{1} + β_{t}^{2}, where β_{t}^{1}(β_{t}^{2}) does (not) consists of a closed form over M. Then from the last two terms, one finds d_{3}f_{t} = (β_{t}^{1})′. In particular one may represent:

e^{w}u| endY (S) = dβ_{t}^{2} = d_{3}β_{t}^{2} − (β_{t}^{2})′ ∧ dt = d_{3}β_{t}^{2} + ∗_{3}d_{3}β_{t}^{2} ∧ dt.

By the decomposition, one finds a positive constant C such that:

|d_{3}β_{t}^{2}|W^{k−1}(M_{t}) ≥ C|β_{t}^{2}|W^{k}(M_{t}).

Now we have the next relations (put μ = β_{t}^{2} on the end):

e^{τ}u = dμ, |μ_{t}|W^{k}(M_{t}) ≤ C|e^{τ}u|W^{k−1}(M_{t}).

Sublemma 7.4 If δ > 0 is sufficiently small with respect to S, then e^{τ}u ∈ L^{2}(Y (S); Λ_{+}^{2}).

P roof : Let us consider e^{τ}u|end = dμ. One may assume that for every t, μ_{t} ∈ C^{∞}(Λ^{1}(M_{t})) lies on the orthogonal complement of Kerd. Then ∗_{3}d is invertible on (Kerd)^{⊥} by corollary 6.1. Moreover it is self adjoint with respect to L^{2} inner product. Since μ satisfies the equation, (\frac{∂}{∂t} + ∗_{3}d_{3})μ = 0, one has the exponential decay estimate for μ. More precisely there exist constants C > 0, λ_{0} > 0 which are independent of δ, such that:

|μ|L^{2}(M_{t}) ≤ exp(−λ_{0}t) sup{|μ|L^{2}(M_{s}); 0 ≤ s ≤ 2t}.

Notice that a priori, μ satisfies the following bound of its growth |μ|L^{2}(M_{t}) ≤ Cexp(t\frac{δ}{2})|u|L^{2}(M_{t}). Combining these estimates, one gets the exponential estimate for μ. Then one has also the exponential decay estimate for dμ on the end. This completes the proof.

Proof of proposition: Let φ_{t} : Y (S) ↦ [0, 1] be a cut off function such that φ|M × [t + 1, ∞) ≡ 0, φ|(M × [t, ∞))^{c} ≡ 1. Let us fix a large S, and let δ > 0 satisfy the above condition. By the above estimate on W^{k} norm, one finds φ_{t}μ ∈ L^{2}(Y (S); Λ_{+}^{2}) for all t. Then for any small ϵ > 0, one finds a large t so that:

∫ |e^{w}u|^{2} = | ∫ d(φ_{t}μ) ∧ d(φ_{t}μ) + ∫ |e^{w}u|^{2}| < ϵ.

This shows u ≡ 0, and we have shown H^{2}(AHS) = 0 for this pair (δ, S). This completes the proof.

Remark 7.1 Suppose for all δ′ ∈ [δ, δ_{0}], the differential of AHS complex has closed range, where δ is sufficiently small. Then H^{∗}(AHS) also vanish for all δ′. This is seen as follows. Notice that for any choice of the constants, one knows H^{0}(AHS) = H^{1}(AHS) = 0. We want to calculate H^{2}(AHS) when we vary δ′ > 0. Let us take an isomorphism:

I : W_{τ(δ)}^{∗}(Y (S); Λ^{∗}) ≅ W_{τ(δ′)}^{∗}(Y (S); Λ^{∗})

where we denote τ(δ) to express the weight constant. Passing through I, one has {W_{τ(δ)}^{∗}(Λ^{∗} : Y (S)); I^{−1} ∘d_{∗} ∘I}, a continuous family of Fredholm complexes. In particular, the indices of these complexes are invariant. Since for all cases, one has H^{0}(AHS) = H^{1}(AHS) = 0, one concludes H^{2}(AHS) = 0.

Remark 7.2 Suppose a cylindrical four manifold Y has non zero H^{2}(Y : R). Then in this case, one has a bound:

dimH^{2}(AHS) ≤ 2 dimH^{2}(Y : R).

This directly follows from the proof of proposition 7.3.

Corollary 7.1 Let us choose a sufficiently small δ > 0. Then one has an invertible operator:

P_{w} : W_{w}^{k+1}(Ỹ (S); Λ^{1}) ≅ W_{w}^{k}(Ỹ (S); Λ^{0} ⊕ Λ_{+}^{2}).

8 An asymptotic method to compute cohomology

8.A P_{w} over Y (S, 2): Let (W_{0}, N_{0}, M_{1}) be a simplest kinky handle, and W̃_{0} be the Riemannian manifold constructed before. In the previous section, one has the invertible operator:

P_{τ} : W_{τ}^{k+1}(Y (S); Λ^{1}) ↦ W_{τ}^{k}(Y (S); Λ^{0} ⊕ Λ_{+}^{2}).

By the Fourier Laplace transform, one gets an invertible Fredholm P_{τ} over Ỹ (S) where:

Ỹ (S) = ⋯ ∪ W̃_{0}^{i−1}([−S, S]) ∪ W̃_{0}^{i}([−S, S]) ∪ … i ∈ Z.

Let T_{2} be the periodic tree as R ∪_{n∈Z} R_{0}, where R_{0} is the half real line. The aim here is to show that the corresponding P_{τ} over CH(T_{2}) is invertible. Let (W_{2}, N_{0}, M_{1}, M_{2}) be a kinky handle with 2 kinks, and W̃_{2} be the corresponding complete Riemannian manifold. Thus one has three ends in the horizontal direction, N_{0}([0, ∞)), M_{j}([0, ∞)), j = 1, 2. Let us choose a large S, and put:

Ỹ (S)_{0} = W̃_{0}^{1}((−∞, S]) ∪ W̃_{0}^{2}([−S, S]) ∪ …,
Y (S, 2)′ = W̃_{2}\N_{0}([S, ∞)) ∪ M_{1}([S, ∞))/N_{0}(S) ∼ M_{1}(S),
Y (S, 2) = Y (S, 2)′\M_{2}([S, ∞)) ∪ Ỹ (S)_{0}\N_{0}([S, ∞)).

One may equip a weight function τ on Y (S, 2) as before.

Lemma 8.1 d_{∗} : W_{τ}^{k+1}(Y (S, 2); Λ^{∗}) ↦ W_{τ}^{k}(Y (S, 2); Λ^{∗+1}) has closed range with H^{0}(AHS) = H^{1}(AHS) = 0.

P roof : Closedness follows from the excision method used before. Let us take (W_{s} = S^{1} × D^{3}, N_{0} = S^{1} × D^{2}), where N_{0} represents a generator of π_{1}(W_{s}). By the construction in 5.A, one can get a complete Riemannian manifold W̃_{s} with one end N_{0}([0, ∞)) along the horizontal direction. Let us put:

Y (S, 2)(0) = Y (S, 2)′\M_{2}([S, ∞)) ∪ W̃_{s}\N_{0}([S, ∞)).

Notice that Y (S, 2)(0) is diffeomorphic to Y (S, 2)′, but Riemannian metrics are different. (Y (S, 2)(0) has cylindrical end.) By corollary 6.2, the differential of the AHS complex has closed range.

Let H_{i} ⊂ W_{τ}^{k+1}(Y (S, 2); Λ^{∗}) be closed subsets with

(1) H_{1} ∪ H_{2} = W_{τ}^{k+1}(Y (S, 2); Λ^{∗}),

(2) SuppH_{1} ⊂ Y (S, 2)′\M_{2}([S, ∞)) and SuppH_{2} ⊂ Ỹ (S)_{0}\N_{0}([S − 1, ∞)). One regards H_{1} ⊂ W_{τ}^{k+1}(Y (S, 2)(0); Λ^{∗}) and H_{2} ⊂ W_{τ}^{k+1}(Ỹ (S); Λ^{∗}). From

this, it follows that d_{∗}(H_{i}) are both closed subspace of W_{τ}^{k}(Y (S, 2); Λ^{∗+1}).

Let us see KerP_{w} = 0. KerP_{w} is isomorphic to 1st cohomology H^{1}(AHS). One can easily check H^{1}(Y (S, 2) : R) = R. Let us take a non zero element w. Then this has the property < w, C >≠ 0 for any loop C representing a generator of π_{1}(Y (S, 2)). One may choose C sufficiently near to infinity while having a bounded length.

Let us take u ∈ H^{1}(AHS), and consider its class [u] ∈ H^{1}(Y (S, 2) : R). it follows [u] = 0 , since one can make < u, C > arbitrarily small by choosing C as above. Thus one may express u = df for f ∈ C^{∞}(Y (S, 2)). One may assume f ∈ L_{w}^{2}(Y (S, 2)), by subtracting some constant. This shows u = 0. Thus we have shown the result.

8.B Some continuity of H^{1}(AHS): Let us introduce another weight function:

μ : Ỹ (S) ↦ [0, ∞)

by μ|W̃_{n}(s ≤ S)(x) = |n| + t(x). We choose another small constant δ′ (we take it so that δ′ << δ, where δ is the weight constant for τ). Then we introduce another weight w = δ′μ + τ and weighted Sobolev k spaces W_{w}^{k}, where:

(|u|W_{w}^{k})^{2} = ∫Ỹ exp(δ′μ + τ)(Σ_{l=0}^{k}|∇^{l}u|^{2}).

Since one knows H^{∗}(AHS) = 0 over Ỹ (S), it follows that there exists a positive constant C > 0 such that |P_{w}|, |(P_{w})_{w}^{∗}| ≥ C, for any small δ′ ∈ [0, δ′_{0}].

Let Y be a complete Riemannian manifold of bounded geometry. Let us take a family of smooth maps:

w(δ′) : Y ↦ [0, ∞), w(δ′′)(x) ≤ w(δ′)(x), δ′′ ≤ δ′ ∈ [0, δ], x ∈ Y.

Suppose that for all w(δ′), the corresponding AHS complex between weighted Sobolev spaces are Fredholm. In any case it is immediate to see H^{0}(AHS) = 0. Here one has some continuity property as:

Lemma 8.2 H^{1}(AHS) = 0 for δ′ = 0 when H^{1}(AHS) = 0 for δ′ > 0.

P roof : Let us denote Ker(d_{+})_{w(δ′)}^{∗} = H^{2}(δ′) (the second cohomology H^{2}(AHS) when it is defined). Let us choose sufficiently small constants:

δ >> δ′ > δ′′ ≥ 0.

Then one has H^{2}(δ′′) ≅ H^{2}(δ′). This follows from that by varing these weights from w(δ′) to w(δ′′), one gets a family of Fredholm complexes between the weighted Sobolev spaces (these spaces also vary with respect to the deformation of the weights). For every value of weights, one has H^{0}(AHS) = H^{1}(AHS) = 0. Thus one gets the above statement.

Sublemma 8.1 If H^{2}(δ′′) ≠ 0, then one has also H^{2}(δ′) ≠ 0.

P roof : Let us denote w = w(δ′) and w′ = w(δ′′). Let us take v ∈ L_{w′}^{2} with d^{∗}(exp(w′)v) = 0. Then one puts u = exp(−w+w′)v. One may assume u ∈ L_{w}^{2}. Since u satisfies d^{∗}(exp(w)u) = 0, one gets the result.

Proof of lemma: Recall that by putting w = w(δ′) one also has a Fredholm complex {W_{w}^{∗}(Y (S, 2); Λ^{∗}), d^{∗}}. Then by variying a parameter δ′ ∈ [0, δ], one has a family of Fredholm complex:

{W_{w(δ′)}^{∗}(Y (S, 2); Λ^{∗}), d^{∗}}.

From the above proof, one has the inclusion H^{2}(δ′′) ⊂ H^{2}(δ′) for all 0 ≤ δ′′ ≤ δ′. Now let us see H^{2}(δ′) = H^{2}(0) and H^{1}(0) = 0. Suppose H^{1}(0) ≠ 0. Then by invariance of Fredholm indices, one must have dimH^{2}(0) > dimH^{2}(δ′). This contradicts to the above. This completes the proof.

8.C Computation of H^{2}(AHS): To show that P_{τ} : W_{τ}^{k+1}(Y (S, 2); Λ^{1}) ↦ W_{τ}^{k}(Y (S, 2); Λ^{0} ⊕ Λ_{+}^{2}) is invertible, one uses an asymptotic method. Roughly speaking one approximates Y (S, 2) by a family of Rimannian manifolds with cylindrical ends. Then the spectrum of P_{τ} over each cylindrical manifold has a uniform lower bound on 1 forms. From this one gets a lower bound of (d_{+})_{τ}^{∗} over Y (S, 2). In the presence of H^{2}(Y : R), one can get a uniform bound of dimH^{2}(AHS). The approximation of spaces corresponds to the one of infinite tree by its finite subtrees. This method completely works for higher stages in 8.D.

Computation of dimH^{2}(AHS) uses information of H^{1}(AHS) on both approximation spaces and the limit space. The former is obtained by proposition 7.3 and the latter by sublemma 6.1. On the other hand for all cases H^{0}(AHS) = 0. Then using this and proposition 7.3, one can apply the asymptotic method in this section to verify H^{1}(AHS) = 0 for the limit space. This will be another method to compute H^{1}(AHS) without using sublemma 6.1.

Let us take (W_{s} = S^{1} ×D^{3}, N_{0} = S^{1} ×D^{2}) where N_{0} represents a generator of π_{1}(W_{s}). Then by the construction in 5.A, one gets a complete Riemannian manifold W̃_{s} with one end N_{0}([0, ∞)) along horizontal direction. Let (W_{0}, N_{0}, M_{1}) be the simplest kinky handle as before. Let us put:

Y (S)_{0}(n)′ = W̃_{0}^{1}([−S, S]) ∪_{M_{1}^{1}(S)∼N_{0}^{2}(S)} W̃_{0}^{2}([−S, S])
⋯ ∪_{M_{1}^{n−1}(S)∼N_{0}^{n}(S)} W̃_{0}^{n}([−S, S]),
Y (S)_{0}(n) = Y (S)_{0}(n)′ ∪_{M_{1}^{n}(S)∼N_{0}(S)} W̃_{s}\N_{0}([S, ∞)),
Y (S)′′_{0} = … W̃_{0}^{−n}([−S, S]) ∪_{M_{1}^{−n}(S)∼N_{0}^{−n+1}(S)} W̃_{0}^{−n+1}([−S, S]) …
∪_{M_{1}^{n−1}(S)∼N_{0}^{n}(S)} W̃_{0}^{n}([−S, S]) ∪_{M_{1}^{n}(S)∼N_{0}(S)} W̃_{s}\N_{0}([S, ∞)).

Notice Y (S)_{0}(∞)′ = Y (S)_{0}. Let us put Y (S, 2)(n)′ = Y (S, 2)′ ∪_{M_{2}(S)∼N_{0}^{0}(S)} Y (S)_{0}(n)′ and:

Y (S, 2)(n) = Y (S, 2)′ ∪_{M_{2}(S)∼N_{0}^{0}(S)} Y (S)_{0}(n).

Y (S, 2)(n) is a complete Riemannian manifold with cylindrical end. Let τ : Y (S, 2)(n) ↦ [0, ∞) be the weight function with weight δ as before.

Proposition 8.1 One gets isomorphisms:

P_{τ} : W_{τ}^{k+1}(Y (S, 2)(n); Λ^{1}) ≅ W_{τ}^{k}(Y (S, 2)(n); Λ^{0} ⊕ Λ_{+}^{2}).

P roof : By proposition 7.3, the conclusion is true when one uses a sufficiently small weight constant δ′. On the other hand, by the first part of the proof of lemma 8.1, the differentials of AHS complex have all closed range for all δ′ ∈ (0, δ]. From this, one gets H^{0}(AHS) = H^{1}(AHS) = 0 for all δ′. To see H^{2}(AHS) = 0, one can use the same method as proposition 7.2. This completes the proof.

In particular there are constants C_{n} such that for Δ = (P_{τ})_{τ}^{∗} ∘P_{τ}, one has the bounds |u|W_{τ}^{k+2}(Y (S, 2)(n)) ≤ C_{n}|Δ(u)|W_{τ}^{k}(Y (S, 2)(n)). Also one has bounds:

|u|W_{τ}^{k+2}(Y (S, 2)) ≤ C|Δ(u)|W_{τ}^{k}(Y (S, 2)),
|u|W_{τ}^{k+2}(Y (S)′′_{0}) ≤ C|Δ(u)|W_{τ}^{k}(Y (S)′′_{0}).

Sublemma 8.2 There is an upper bound C_{n} ≤ C for all n.

P roof : For any small ϵ > 0, there exists a large n with the following property; for any u ∈ W_{τ}^{k+2}(Y (S, 2)(n)) with |u|W_{τ}^{k+2} = 1, there is at least one n′, 0 ≤ n′ ≤ n with |u|W_{τ}^{k+2}(W̃_{0}^{n′}([−S, S])) < ϵ. Let φ(n′) be a cut off function on Y (S, 2)(n) with φ(n′)|(W̃_{0}^{n′}([−S, S]))^{c} ≡ 1 and Y (S, 2)(n)\ Suppdφ(n′) has two components. Then one may express φ(n′)u = u_{1} + u_{2}. One may regard u_{1} ∈ W_{τ}^{k+2}(Y (S, 2)), u_{2} ∈ W_{τ}^{k+2}(Y (S)′′_{0}). In particular one has the estimates:

\begin{aligned}|u|W_{τ}^{k+2} & ≤ |u_{1}|W_{τ}^{k+2} + |u_{2}|W_{τ}^{k+2} + |(1 − φ(n′)u|W_{τ}^{k+2} \\ & ≤ C\{|Δ(u_{1})|W_{τ}^{k} + |Δ(u_{2})|W_{τ}^{k}\} + ϵ ≤ C|Δ(u)|W_{τ}^{k} + Cϵ \end{aligned}

where C is independent of ϵ and u. This gives the result.

Now suppose H^{2}(AHS) is non zero over Y (S, 2), and take u ∈ Ker(d_{+})_{τ}^{∗} ∩ W_{τ}^{k+1}(Y (S, 2); Λ_{+}^{2}) with |u|W_{τ}^{k+1} = 1. For each n, let us take a cut off function φ(n) on Y (S, 2) with φ(n)|Y (S, 2)(n − 1)′ ≡ 1, φ(n)|(Y (S, 2)(n)′)^{c} ≡ 0. Then one may regard φ(n)u ∈ W_{τ}^{k+1}(Y (S, 2)(n); Λ_{+}^{2}). Thus there exists v_{n} ∈ W_{τ}^{k+2}(Y (S, 2)(n); Λ^{1}) with d_{τ}^{∗}(v_{n}) = 0, d_{+}(v_{n}) = φ(n)u. By the above sublemma, one has the uniform estimates:

\begin{matrix}1 − ϵ ≤ |φ(n)u|W_{τ}^{k+1} = |d_{+}(v_{n})|W_{τ}^{k+1} \\ ≤ C|v_{n}|W_{τ}^{k+2} ≤ C|d_{+}(v_{n})|W_{τ}^{k+1} ≤ C|φ(n)u|W_{τ}^{k+1}. (∗) \end{matrix}

By the above (∗), one has the uniform estimates C ≤ |v_{n}|W_{τ}^{k+2} ≤ C′. Let us take a sequence 1 >> ϵ_{1} >> ϵ_{2} >> ⋯ → 0. Then for each n, there exists a large N = N(n) and L(n) >> 0 such that at least one of |v_{N}|W_{τ}^{k+2}(W̃_{0}^{n′}([−S, S])) is less than ϵ_{n} for N − L(n) ≤ n′ ≤ N. Let us take a subsequence {v_{N(n)}}_{n} and denote it by {v_{n}}_{n}. For simplicity of the notation, one may assume n′ = n − 1.

Using this, one can verify Ker(d_{+})_{τ}^{∗} = 0 over Y (S, 2). Let us put Δ = dd_{τ}^{∗} ⊕ (d_{+})_{τ}^{∗}d_{+} on W_{τ}^{k+2}(Y (S, 2); Λ^{1}). From above, one has the estimate:

\begin{aligned}|dd_{τ}^{∗} & (φ(n − 1)v_{n})|W_{τ}^{k} \\ & ≤ C\{|dv_{n}|W_{τ}^{k}(W̃_{0}^{n−1}([−S, S])) + |v_{n}|W_{τ}^{k}(W̃_{0}^{n−1}([−S, S]))\} ≤ Cϵ_{n} \end{aligned}

where C is independent of n. Then one has the estimates:

|Δ(φ(n − 1)v_{n})|W_{τ}^{k} ≤ C[|(d_{+})_{τ}^{∗}(φ(n − 1)φ(n)u)|W_{τ}^{k}
+ |(d_{+})_{τ}^{∗}[d_{+}, φ(n − 1)]v_{n}|W_{τ}^{k}( Suppdφ(n − 1))] + Cϵ_{n}
≤ C[|φ(n − 1)φ(n)(d_{+})_{τ}^{∗}(u)|W_{τ}^{k}
+ |[(d_{+})_{τ}^{∗}, φ(n − 1)φ(n)]u|W_{τ}^{k}(∪_{j=n−1,n}W̃_{0}^{j}([−S, S]))
+ |(d_{+})_{τ}^{∗}[d_{+}, φ(n − 1)]v_{n}|W_{τ}^{k}(W̃_{0}^{n−1}([−S, S])) + Cϵ_{n}
≤ |[(d_{+})_{τ}^{∗}, φ(n − 1)φ(n)]u|W_{τ}^{k} + |(d_{+})_{τ}^{∗}[d_{+}, φ(n − 1)]v_{n}|W_{τ}^{k} + Cϵ_{n}.

One may assume that the last term is arbitrarily small. This shows that there is a sequence {w_{n} ∈ W_{τ}^{k+2}(Y (S, 2); Λ^{1})} with C ≤ |w_{n}|W_{τ}^{k+2} ≤ C′ and |P (w_{n})|W_{τ}^{k+1} converges to zero. This is a contradiction. This shows Ker(d_{+})_{τ}^{∗} = 0 over Y (S, 2).

Let (W_{t}, N_{0}) = (D^{4}, S^{1}×D^{2}) be the standard disk. Let Y = W̃_{t}\N_{0}([S, ∞))∪ Ỹ (S)_{0}\N_{0}([S, ∞)). In this case, one has H^{0}(AHS) = H^{1}(AHS) = 0 (notice Y is simply connected). But H^{2}(Y : R) = R. In this case, one has the following: Corollary 8.1 Suppose Y has non zero H^{2}(Y : R). Then one has a bound dimH^{2}(AHS) ≤ 2 dimH^{2}(Y : R). P roof : Suppose dimH^{2}(AHS) ≥2 dimH^{2}(Y : R) + 1 = b_{2} + 1. Let us take L_{τ}^{2} orthogonal vectors u_{1}, …, u_{b_{2}+1}, ⋯ ∈ L_{τ}^{2}(Y : Λ_{+}^{2})∩ Ker(d_{+})_{τ}^{∗}, with |u_{i}|L_{τ}^{2} = 1.

One can make a family of cylindrical manifolds Y (n) by the above method. By a straightforward calculation, one has an upper bound of dimH^{2}(Y (n) : R) by b_{2}. By remark 7.2, it follows that dimH^{2}(AHS) has also an upper bound over Y (n). Now let φ(n) be as above. Then for any small ϵ > 0, there exists a large n_{0} such that for all n ≥ n_{0}, one has:

\begin{matrix}|(d_{+})_{τ}^{∗}(φ(n)u_{i})|L_{τ}^{2} ≤ ϵ, |φ(n)u_{i}|L_{τ}^{2} ≥ 1 − ϵ, \\ < φ(n)u_{i}, φ(n)u_{j} > |L_{τ}^{2} ≤ ϵ, i = 1, …, b_{2} + 1. \end{matrix}

Let v_{1}, …, v_{l} be an orthonormal basis of H^{2}(AHS) over Y (n), l ≤ b_{2}. Then one may express φ(n)u_{i} = Σ_{j}a_{j}^{i}v_{j} + d_{+}(α_{i}), a_{j}^{i} ∈ R. Let ā_{i} = (a_{1}^{i}, …, a_{l}^{i}) ∈ R^{l}. Then, the set of vectors ā_{1}, …, ā_{b_{2}+1} would satisfy |ā_{i}| ≥ 1−ϵ and < ā_{i}, ā_{i′} >≤ ϵ with respect to the standard norm in R^{l}. Since l < b_{2} + 1, it is impossible to find such set. This completes the proof.

8.D Fourier Laplace transform on higher stages: Let us put: W̃_{2}([−S, S]) = W̃_{2}\N_{0}([S, ∞)) ∪_{j=1,2} M_{j}([S, ∞)),

Y (S, 2)_{0} = W̃_{2} ∪_{M_{2}(S)∼N_{0}(S)} Ỹ (S)_{0}\N_{0}([S, ∞)),
Ỹ (S, 2) = … Y (S, 2)_{0}^{−n} ∪_{M_{1}^{−n}(S)∼N_{0}^{−n+1}(S)} Y (S, 2)_{0}^{−n+1} …
Ỹ (S, 2)_{0} = Y (S, 2)_{0}^{1} ∪_{M_{1}^{1}(S)∼N_{0}^{2}(S)} Y (S, 2)_{0}^{2} ∪_{M_{1}^{2}(S)∼N_{0}^{3}(S)} Y (S, 2)_{0}^{3} …

where (Y (S, 2)_{0}^{n}, M_{j}^{n}(S), N_{0}^{n}(S)) are all the same triple (Y (S, 2)_{0}, M_{j}(S), N_{0}(S)) as before. Notice that Ỹ (S, 2) is diffeomorphic to CH(T_{2}), where T_{2} = R ∪_{n∈Z} R_{+}. Now by the Fourier Laplace transform, one gets the following: Corollary 8.2 P_{τ} : W_{τ}^{k+1}(Ỹ (S, 2); Λ^{1}) ↦ W_{τ}^{k}(Ỹ (S, 2); Λ^{0} ⊕ Λ_{+}^{2}) is invertible.

Let us put T_{2}^{0} = R_{+} ∪_{n∈N} R_{+}. Then one defines:

T_{3} = R ∪_{n∈Z} T_{2}^{0}.

Similarly one puts T_{3}^{0} = R_{+} ∪_{n∈N} T_{2}^{0}, and T_{4} = R ∪_{n∈Z} T_{3}^{0}. One inductively defines T_{j}, j = 1, 2, … as:

T_{j+1} = R ∪_{n∈Z} T_{j}^{0}, T_{j+1}^{0} = R_{+} ∪_{n∈N} T_{j}^{0}

where one puts T_{1} = R, T_{1}^{0} = R_{+}.

One can construct the corresponding spaces. Let (W_{2}, N_{0}, M_{1}, M_{2}) be a kinky handle with two kinks. One has already defined Y (S), Y (S, 2). Let us define inductively Y (S, j) as follows:

Y (S, j)_{0} = W̃_{2}([−S, S]) ∪_{M_{2}(S)∼N_{0}(S)} Ỹ (S, j − 1)_{0},
Y (S, j) = Y (S, j)_{0}/N_{0}(S) ∼ M_{1}(S),
Ỹ (S, j) = … Y (S, j)_{0}^{−n} ∪_{M_{1}^{−n}(S)∼N_{0}^{−n+1}(S)} Y (S, j)_{0}^{−n+1} …
Ỹ (S, j)_{0} = Y (S, j)^{1} ∪_{M_{1}^{1}(S)∼N_{0}^{2}(S)} Y (S, j)^{2} ∪_{M_{1}^{2}(S)∼N_{0}^{3}(S)} Y (S, j)^{3} …

One may express CH(T_{j}) = Ỹ (S, j) and CH(T_{j}^{0}) = Ỹ (S, j)_{0}. The previous method works for all Y (S, j) iteratedly.

Proposition 8.2 P_{τ} over Y (S, j) are all Fredholm with H^{∗}(AHS) = 0 for ∗ = 0, 1, 2 and j = 1, 2, . . ..

In the notation of 5.B, one expresses T_{j} = T_{2,2,...,2,1} ((2, …, 2): j − 1 times). Let n_{1}, …, n_{k} ∈ {1, 2, … } be a set of positive integers. Then using kinky handles with n_{j} kinks, one has an natural extension, and gets T_{n_{1},...,n_{k},1} which is an infinite tree. For Riemannian metric on CH(T_{n_{1},...,n_{k},1}), see 5.B.

Let n̄ = {n_{1}, …, n_{l−1}, 1} be a set of positive integers, and denote the corresponding homogeneous tree of finite type by Tn̄. By the previous method, one gets complete Riemannian metrics and weight functions on every CH(Tn̄). Let us take n̄ = {n_{1}, …, n_{l−1}, 1}. Recall that one has constructed complete Riemannian metrics and weight functions on Y (S, n̄).

Proposition 8.3 P_{τ} : W_{τ}^{k+1}(Y (S, n̄); Λ^{1}) ↦ W_{τ}^{k}(Y (S, n̄); Λ^{0} ⊕ Λ_{+}^{2}) gives an isomorphism for any n̄ = {n_{1}, …, n_{l−1}, 1}.

P roof : One has shown the result for l = 2. Suppose the result is true for l ≤ l_{0}. Let us put n̄ = {n_{1}, …, n_{l_{0}}, 1}. Then by Fourier Laplace transform and excision method used before, one knows P_{τ} : W_{τ}^{k+1}(Y (S, Tn̄); Λ^{1}) ↦ W_{τ}^{k+1}(Y (S, Tn̄); Λ^{0}⊕ Λ_{+}^{2}) gives a closed operator with H^{∗}(AHS) = 0 for ∗ = 0, 1. One may follow the same process to see H^{2}(AHS) = 0 as 8.C. Thus one has shown the result for l_{0} + 1. This completes the induction step.

For a practical application, one considers open four manifolds with 0 handles attatched with Casson handles. Recall that S ≡ k(S^{2} × S^{2})\ pt is homotopy equivalent to some wedges of S^{2}. In particular H_{cp}^{1}(S : R) = H^{3}(S : R) = 0. Recall also that S has a link picture by k disjoint union of Hopf links with 0 framings. Thus (W_{t}, M_{1}, …, M_{2k}) = (D^{4}, S^{1} × D^{2}, …, S^{1} × D^{2}) expresses the link diagram of k(S^{2} × S^{2}) (see section 4).

Let T_{1}, …, T_{2k} be signed homogeneous trees of finite type. Let us take (D^{4}, S^{1} × D^{2}, …, S^{1} × D^{2}) as above, which represents k(S^{2} × S^{2})\ pt. Then as before, one may construct a complete Riemannian manifold W̃_{t}. Then one considers:

S ≡ W̃_{t}♮ ∪_{l} CH(T_{l}).

As before one can equip a weight function τ, and the AHS complex over S.

Corollary 8.3 The differential of AHS complex has closed range over S with H^{0}(AHS) = H^{1}(AHS) = 0 and dimH^{2}(AHS) ≤ 2 dimH^{2}(S : R).

The diffeomorphism types of S as above, appears very often as closed four manifolds minus a topological disk (see [K]).

Lemma 8.3 (F) The end of S admits a topological color, ≅ S^{3} × [0, ∞).

Now we have completed to verify that any open four manifold with tree-like end of finite type can admit an admissible pair (g, τ) on it.

References

[AHS] M.Atiyah, N.Hitchin and I.Singer, Self-duality in four dimensional Riemannian geometry, Proc. R.Soc.London, 362 pp. 425 - 461 (1978).

[APS] M.Atiyah, V.Patodi and I.Singer, Spectral asymmetry in Riemannian geometry, I, II and III, Math. Proc. Cambridge Philos. Soc. 77 pp. 43 - 69 (1975a), 78 pp. 405 - 432 (1975b) and 79 pp. 71 - 99 (1976).

[BPV] W.Barth, C.Peters and A.Van de ven, Compact complex surfaces, Springer (1984).

[B] Ž. BiŽaca, A reimbedding algorithm for Casson handles, Trans. A.M.S. 345 - 2 pp. 435 - 510 (1994).

[C] A.Casson, Three lectures on new infinite constructions in 4-dimensional manifodls, [Co] A.Connes, Non commutative geometry, Academic press (1994).

[DF] S.Demichelis and M.H.Freedman, Uncountably many exotic R^{4}’s in standard 4-space, Journal of Differential Geometry 35 pp. 219 - 254 (1992).

[Do1] S.K.Donaldson, An application of gauge theory to four dimensional topology, Journal of Differential Geometry 18 pp. 279 - 315 (1983).

[Do2] S.K.Donaldson, Poynomial invariants for smooth 4-manifolds, Topology 29 pp. 257 - 315 (1990).

[FS1] R.Fintushel and R.J.Stern, SO(3)-connections and the topology of 4-manifolds, Journal of Gifferential Geometry 20 pp. 523-539 (1984).

[FS2] R.Fintushel and R.J.Stern, Definite 4-manifolds, Journal of Differential Geometry 28 pp. 133-141 (1988).

[Fl] A.Floer, An instanton invariant for 3 manifolds, Comm.Math.Phy. 118 pp. 215 - 240 (1988).

[FU] S.Freed and K.Uhlenbeck Instantons and four manifolds, Springer 2nd edition (1991).

[Fr] M.H.Freedman, The topology of four-dimensional manifolds, Journal of Differential Geometry 17 pp. 357 - 454 (1982).

[FQ] M.Freedman and F.Quinn, The topology of four dimensional manifolds, Princeton Univ. press (1990).

[GL] M.Gromov and H.B.Lawson, Positive scalar curvature on complete Riemannian manifolds, I.H.E.S Publ. Math. 58 pp. 295-408 (1983).

[J] I.D.James , On Witt’s theorem for unimodular quadratic froms, Pacific journal of Math. 26 pp. 303-316 (1968).

[K] R.C.Kirby, The topology of 4 manifolds, Springer L.N. 1374.

[Ko] V.A.Kondrat’ev Boundary value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc. 16 (1967).

[LM] R.B.Lockhhart and R.C.McOwen, Elliptic differential operators on non compact manifolds, Ann. Sci. Ec. Norm. Sup. Pisa 12 pp. 409 - 446 (1985).

[M] J.Milnor On simply connected 4-manifolds, Symp. Int. top. alg., Mexico pp. 122 - 128 (1958).

[Q] F.Quinn, Ends of maps, III: dimension 4 and 5, Journal of Differential Geometry 17 pp. 503-521 (1982).

[T1] H.Taubes, Gauge theory on asymptotically periodic 4 manifolds, Journal of Differential Geoetry 25 pp. 363 - 430 (1987).

[T2] H.Taubes, Casson’s invariant and gauge theory, Journal of Differential Geoetry 31 pp. 547 - 599 (1990).

[T3] H.Taubes, Self-dual Yang-Mills connections over non-self-dual 4manifolds, Journal of Differential Geoetry 17 pp. 139 - 170 (1982).

[T4] H.Taubes, L^{2} moduli spaces on 4-manifolds with cylindrical ends, Monographs in Geometry and Topology 1, International Press (1993).

[U1] K.Uhlenbeck Connections with L^{p} bounds on curvature, Comm. Math. Phys. 83 pp. 31 - 42 (1982).

[U2] K.Uhlenbeck Removable singularities in Yang-Mills fields, C.M.P. 83 pp. 11 - 29 (1982).

[W] F.Warner Foundations of differentiable manifolds and Lie groups, Springer G.T.M. (1971).

Tsuyoshi Kato

Department of Mthematics Faculty of Science Kyoto University Kyoto 606-01 Japan