MAPS BETWEEN 2N-MANIFOLDS OF NONZERO DEGREE

Haibao Duan^{1} and Shicheng Wang^{1,2}

Peking University^{1} and Max-Planck-Institut f. Math.^{2}

§1. Introduction and Results.

In this paper all 2n-manifolds are closed and orientable, and all matrices are integer matrices.

Let M be a 2n-dimensional closed oriented manifold and let H̄^{n}(M, Z) be the free part of H^{n}(M; Z). Then the cup product operator

(1.1)  H̄^{n}(M; Z) ⊗ H̄^{n}(M; Z) → H^{2n}(M; Z),

defines a bilinear unimodule form over H̄^{n}(M, Z), which is denoted by X_{M}, where a ∪ a′ = X(a, a′)[M] and [M] is the fundamental class of H^{2n}(M, Z).

Let α = (a_{1}, ..., a_{m}) be a basis for H̄^{n}(M). Then X_{M} determines an m by m matrix A = (a_{ij}), where a_{ij} is given by a_{i} ∪ a_{j} = a_{ij}[M]. We can also present this formula as

(1.2)  α^{⊥} ∪ α = A[M],

where α^{⊥} is the transpose matrix of α.

Let α′ = (a′_{1}, ..., a′_{m}) ⊂ H̄_{n}(M, Z) be the Poincare dual basis of α, then

(1.3)  α′^{⊥} ∩ α′ = A,

Below we will call X_{M} the intersection form of M, and A the intersection matrix of M (under basis α). Note that those forms (matrices) is symmetry if n is even and is anti-symmetry if n is odd. When n is odd, the Betti number of middle dimensional cohomology must be even.

Say the intersection form X(M) is even if for any x ∈ H̄^{2}(M; Z), x ∪ x = c[M], where c is an even integer. Equivalently, X(M) is even if under the basis α = (a_{1}, ..., a_{m}) of H̄^{2}(M; Z), a_{i} ∪ a_{i} = c_{i}[M], c_{i} is even for all i.

Preliminary version, comments are welcome. The work is partly done during the visit of the second author in MPIM. The work is also partly supported by outstanding Young Grantof NSFC.

2 HAIBAO DUAN^{1} AND SHICHENG WANG^{1,2}

Let H be a subgroup of H̄^{n}(M, Z), then X is defined on H, simply denoted as X_{H}. Say X_{M} is a direct sum of X_{H} and X_{H′}, denoted as X_{M} = X_{H} ⊕ X_{H′}, if H̄^{n}(M, Z) = H ⊕ H′ and X(a, a′) = 0 for any a ∈ H and a′ ∈ H′.

For a closed oriented 2n-manifold L with basis β = {b_{1}, ..., b_{l}} ⊂ H̄^{n}(L; Z), similarly we have the intersection form X_{L} and the intersection matrix B.

For any map f : M → L, there is an m by l integer matrix P such that f^{∗}(β) = αP. Let α_{∗} ∈ H̄_{n}(M, Z) and β_{∗} ∈ H̄_{n}(L, Z) be the algebraic dual basis of α and β

respectively. Then

(1.4)  f^{∗}(β) = αP if and only if f_{∗}(α_{∗}^{⊥}) = P β_{∗}^{⊥}.

where f^{∗} and f_{∗} are be the induced homomorphism on H^{n} and H_{n} respectively, and f_{∗}(α_{∗}^{⊥}) = P β_{∗}^{⊥} is the abbreviation of

(1.5)  ( f_{∗}(a_{∗1}) ; ⋮ ; f_{∗}(b_{∗n}) ) = P ( b_{∗1} ; ⋮ ; b_{∗l} )

and so on.

When f^{∗}(β) = αP (or f_{∗}(α_{∗}^{⊥}) = P β_{∗}^{⊥}), we say P is realized by a map f : M → L. Below we assume that the basis α for H̄^{n}(M, Z) and β for H̄^{n}(L, Z) have been

chosen, therefore the intersection matrices of M and L are given.

Our main result is Theorem 1, which gives a necessary and sufficient conditions for the existence of a degree k map between (n − 1)-connected 2n-manifolds, n > 1, will be presented in Section 3.

Some part of the necessary condition of Theorem 1 is true for all maps between any 2n-manifolds. This is the following

Theorem 2. Suppose M and L are two closed oriented 2n-manifolds with intersection matrices A and B respectively, n > 1. If P is realized by a map of f : M → L of degree k, then

(1.6)  P^{⊥}AP = kB.

Moreover if k = 1, then

(1.7)  X_{M} = X_{H} ⊕ X_{H′},

where X_{H} is isomorphic to X_{N}.

In dimension 4, the necessary any sufficient condition for the existence of degree k map in Theorem 1 can be extend to the maps from any 4-manifolds to simply connected 4-manifolds. The criterion for degree one maps is very simple.

Theorem 3. Suppose M and L are two closed oriented 4-manifolds with the intersection matrix of A and B respectively. If L is simply connected, then there is a map f : M → L of degree k such that f^{∗}(β) = αP if and only if

(1.8)  P^{⊥}AP = kB

Moreover there is a map f : M → L of degree 1 if and only if

(1.9)  X_{M} = X_{H} ⊕ X_{H′},

where X_{H} is isomorphic to X_{N}.

Note that −X_{M} = X_{−M}. In the statements above and below, if we replace ”degree 1,” ”X_{H}” and ”L” by ”degree −1”, ”−X_{H}” and ”−L” simultaneously, then those statement are still true and the proofs are just routine replacements.

Now we will give sequences of corollaries and examples of Theorems 1, 2 and 3. It will be interseting to compare those results with the situation of n = 1, the case

of surfaces (see [Ed] for example).

First we have a look of maps between simply connected 4-manifolds.

Corollary 1. Let f : M → L be a map between two simply connected oriented 4-manifolds.

(1) If f is of degree one, then M has the homotopy type of L#Q, where Q is a simply connected 4-manifold; moreover Q is unique and M is homeomrphic to L#Q when X(M) is even.

(2) If f is of non-zero degree, then χ(M) ≥ χ(L); if f is of degree one and χ(M) = χ(L), then M and L are homotopic (or homeomorphc if X(M) is even).

The first part of (2) of Corollary 1 can be easily generalized to the following

Corollary 1^{∗}. Suppose there is a map f : M → L between two closed orientable 2n-manifolds of vanishing odd dimension Betti number, n > 1. Then χ(M) ≥ χ(L).

Before to state the next corollary, we introduce the following definition.

For any two closed oriented n-manifolds M and L, define

(1.10)  D(M, L) = {degf|f : M → L},

i.e., D(M, L) is all degrees of maps from M to L. Let Z(2) = {0, 1, 4, 9, 16, ...}, the union of all perfect squares, and 2Z be the all even numbers.

Corollary 2. Suppose L be a (n − 1)-connected 2n-manifold, n > 1.

(1) D(L, L) is an infinite set, in particular M admits a self map of degree larger than 1. Moreover If L is a closed oriented simply connected 4-manifold, then Z(2) ⊂ D(L, L).

(2) If L is a closed oriented (n-1)-connected 2n-manifold and M is an closed oriented 2n-manifold, Then D(M, L) is either {0}, or an infinite set.

Corollary 3. Suppose M is a closed oriented 2n-manifold and the Betti number β_{n}(M) = m is odd, Then for any map f : M → M, the degree k of f is a perfect square. In particular there is no orientation reversing homeomorphism on M.

As a corollary of Corollaries 3 and 4, we have that if M is a simply connected 4manifold with odd β_{2}, then D(M, M) are exactly all perfect squares. We noticed that in [Ba], it was proved D(M, M) are eaxcly all perfect cubes for certain 6-manifolds which have algebraic topology close to CP^{3}.

Corollary 4. Let f : M → L be a map of degree k between closed oriented 4manifolds. If A is even and B is not, then k must be even.

Suppose M and L are smooth. Let w_{i} be the i-th Stiefel-Whitney class of M. Since M is orientable, w_{1} = 0. Then w_{2} = 0 implies X_{M} is an even form (or M is spin) (see [p. 27, K]. Then the following proposition generalizes Corollary 4.

Proposition 4^{∗}. If f : M → L is a map of odd degree between smooth manifolds of dimension n. Let w_{i} and w′_{i} be the i−th Stiefel-Whitney classes of M and L respectively. Then w_{i} = 0, i = 1, 2, ..., k ≤ n, implies that w′_{i} = 0, i = 1, 2, ..., k ≤ n.

The following definitions for 3-manifolds are given in [BW] and [LWZ]. They can be defined for any dimension n.

Definition: Let M, L be two n-dimensional oriented manifolds. Say M (k-)domi -nates L if there is a map f : M → L of non-zero degree (degree k), denoted by M > N (M >_{k} L). Say M is minimal, if M >_{1} L implies that either M = S^{4} or M = N.

Corollary 5. The homotopy type of (n−1)-connected 2n-manifold domimated by any given closed orientable 2n-manifold M is finite. In particular each closed orientable 4-manifolds dominates at most finitely simply connected 4-manifolds.

Now we will apply the results above to some simple examples.

Examples.

Let F_{g}, CP^{2}, T^{4} and K be the surface of genus g, complex projective plane, 4-torus and Kummer surface respectively.

(1) Any closed orientable 4-manifold with non-trivial intersection form dominates CP^{2}.

(2) If P_{m×1} is realized by a map f : M → CP^{2}, then any map realizes P is homotopic to f. If P_{m×2} is realized by a map f : M → S^{2} × S^{2} and M is simply connected, then all maps realize P have 4 homotopy classes. There are infinitely many homotopy classes of maps S^{2} × S^{2}#S^{3} × S^{1} → S^{2} × S^{2} realizing I_{2}.

(3) D(CP^{2}#CP^{2}, S^{2} ×S^{2}) = D(S^{2} ×S^{2}, CP^{2}#CP^{2}) = D(CP^{2}#(−CP^{2}), CP^{2}# CP^{2}) = D(CP^{2}#CP^{2}, CP^{2}#(−CP^{2})) = {0} and D(CP^{2}#(−CP^{2}), S^{2} × S^{2}) = D(S^{2} × S^{2}, CP^{2}#(−CP^{2})) = 2Z.

(4) D(T^{4}, #_{3}S^{2} × S^{2}) = Z. That is for any k, there is a map of degree k from 4torus to the connected sum of 3 copies of S^{2}×S^{2}. In general D(F_{s}×F_{r}, #_{q}S^{2}×S^{2}) = Z if q ≤ 2rs + 1. D(F_{s} × F_{r}, (#_{p}CP^{2}))##_{q}(−CP^{2}) = 2Z if p, q ≤ 2rs + 1.

(5) The degree of any self-map on K#CP^{2} must be a perfect square. CP^{2}, S^{2}×S^{2}, M(E_{8}) are minimal.

The paper is organized as follow: In section 2 we will discuss related facts about Thom-Pontrjagin constructions and the prepresentations of pre-n-spaces, which will be uesd to present and to prove Theorem 1 in Section 3. Theorem 2 and Theorem 3 are proved in Section 4 and Section 5 respectively, the corollaries are proved in Section 6 and the examples are verified in Section 7.

§2. Thom-Pontrjagin Construction and presentation of pre-n-space. Let f : S^{2n−1} → S^{n} be any map with y ∈ S^{n} a regular value. Then W = f^{−1}(y) ⊂

S^{2n−1} is a (n-1)-manifold and τ = f^{∗}(T_{y}S^{n}), the pull-back of the tangent bundle of S^{n} at y, gives framed normal bundle of W in S^{2n−1}. Call the pair (W, τ) a manifold in S^{2n−1} with given framed normal bundle.

If both S^{2n−1} and S^{n} are oriented, then both W and the normal framed bundle are naturally oriented as follow: The normal space at each point of W has the orientation lifted from the tangent space at y ∈ S^{n}, and W is oriented so that the orientations of W it tangent bundle give the orientation of S^{2n−1}.

On the other hand, for any (W, τ), the ϵ-neighborhood W_{ϵ} of W in S^{2n−1} has a product structure D^{n} × W induced by the given framing. Let p : D^{n} × W → D^{n} be the projection. q : D → D/∂D = S^{n} be the quotient map, then we can extend q ∘ p : W_{ϵ} → S^{n} to a map f : S^{2n−1} → S^{n} by sending the remaining part of S^{2n−1} to the point q(∂D) ∈ S^{n}.

Define

F(n) = { (n − 1)-manifold W^{n−1} ⊂ S^{2n−1} with framed normal bundle}

and

F^{∗}(n) = { (n − 1)-spheres S^{n−1} ⊂ S^{2n−1} with framed normal bundle}

Note the underlying space of a element in F(n) or F^{∗}(N) are not necessary to be connected.

Let Ω(n) and Ω^{∗}(n) be the bordism classes of F(n) and F^{∗}(n) respectively. For each s ∈ F(n), its bordism class is denoted by [s].

Theorem 2.1. The construction (Thom-Ponjaging) above gives an 1-1 and onto map T : Ω(n) → π_{2n−1}(S^{n}).

For a detailed proof of this Theorem, see [p.209 DFN].

In such a codimension, Ω(n) is not a group. For any two disjoint s_{1}, s_{2} ∈ F(n), let lk(s_{1}, s_{2}) be the linking number of those two (n − 1)-manifolds in S^{2n−1} carrying s_{1} and s_{2}, and s_{1} ⊔ s_{2} be the disjoint union of s_{1} and s_{2}, which is again in F(n). One can verify directly T ([s_{1} ⊔ s_{2}]) = T ([s_{1}]) + T ([s_{2}]) if lk(s_{1}, s_{2}) = 0. In general we have

Theorem 2.2. For s_{1}, s_{2} ∈ F(n), we have

(2.1)  T ([s_{1} ⊔ s_{2}]) = T ([s_{1}]) + T ([s_{2}]) + lk(s_{1}, s_{2})[s^{n}, s^{n}],

where s^{n} is a generator of π_{n}(S^{n}) and [s^{n}, s^{n}] ⊂ π_{2n−1}(S^{n}) is the Whitehead product.

Proof. Let Δ(n) be the diffeotopy classes of F^{∗}(n). For each s ∈ F^{∗}(n), its diffeotopy class is denoted by [s]′. Then there is a natural surjection S : Δ(n) → Ω^{∗}(n) defined by S([s]′) = [s].

For s_{1}, s_{2} ∈ F^{∗}(n), by [Lemma 2 (6), Wa]. there is a map α : Δ(n) → π_{n−1}(SO_{n}) and we have

(2.2)  α([s_{1} ⊔ s_{2}]′) = α([s_{1}]′) + α([s_{2}]′) + lk(s_{1}, s_{2})(∂s^{n}).

where s^{n} is the generator of π_{n}(S^{n}) and ∂ : π_{n}(S^{n}) → π_{n−1}(SO_{n}) is the boundary homomorphism for the homotopy exact sequence of the fibration S^{n} = SO_{n+1}/SO_{n}. (Note in [Wa] the domain of α is H_{n}, but it is defined more directly on F^{∗}(n), and the intersection number if (6) of [Wa] is precisely the linking number in (2.2). Compare with [pp 164-165, Wa]).

Now apply a classical homomorphism J : π_{n−1}(SO_{n}) → π_{2n−1}(S^{2n−1}) to the both side of (2.2). Since J∂s_{n} = [s_{n}, s_{n}] ([p.167, Wa]) We have

(2.4)  J ∘ α([s_{1} ⊔ s′_{2}]) = J ∘ α([s_{1}]′) + J ∘ α([s_{2}]′) + lk(s_{1}, s_{2})[s^{n}, s^{n}].

Since J ∘ α = T ∘ S and S([s_{i}]′) = [s_{i}], we have

(2.5)  T ([s_{1} ⊔ s_{2}]) = T ([s_{1}]) + T ([s_{2}]) + lk(s_{1}, s_{2})[s^{n}, s^{n}].

and the theorem is proved for elements in F^{∗}(n).

Now for any s_{1} and s_{2} in F(n), it is not hard to see that we can find s′_{1}, s_{1}”, s′_{2}, s_{2}” ∈ F(n) such that

(2.6) s_{1}”, s_{2}” ∈ F^{∗}(n).

(2.7) [s_{1}] = [s′_{1} ⊔ s_{1}”] and [s_{2}] = [s′_{2} ⊔ s_{2}”].

(2.8) [s_{1} ⊔ s_{2}] = [s′_{1} ⊔ s_{1}” ⊔ s′_{2} ⊔ s_{2}”].

(2.9) lk(s′_{i}, s′_{j}) = lk(s′_{i}, s_{j}”) = lk(s′_{i}, s_{i}”) = 0 and lk(s_{i}”, s_{j}”) = lk(s_{i}, s_{j}).

Then

T ([s_{1} ⊔ s_{2}]) = T ([s′_{1} ⊔ s_{1}” ⊔ s′_{2} ⊔ s_{2}”]) = T ([s′_{1}]) + T ([s_{1}” ⊔ s_{2}”]) + T ([s′_{2}])

= T ([s′_{1}]) + T ([s_{1}”]) + T ([s_{2}”]) + T ([s′_{2}]) + lk(s_{1}”, s_{2}”)[s^{n}, s^{n}]
= T ([s′_{1} ⊔ s_{1}”]) + T ([s′_{2} ⊔ s_{2}”]) + lk(s_{1}, s_{2})[s^{n}, s^{n}]
= T ([s_{1}]) + T ([s_{2}]) + lk(s_{1}, s_{2})[s^{n}, s^{n}]

Those five equalities follow by (2.8), the first half of (2.9), (2.6) and (2.5), (2.9), (2.7) in order. □

Call M is a pre-n-space, if M has the homotopy type of a space obtained from V_{i=1}^{m}S_{i}^{n}, the m-fold wedge of n-spheres with one point y in common, by attaching 2n-cell D^{2n} via attaching map

(2.10)  g : S^{2n−1} = ∂D^{2n} → V_{i=1}^{m}S_{i}^{n}.

So the homotopy type of M is determined by g ∈ π_{2n−1}(V_{i=1}^{m}S_{i}^{n}). For short denote this space by V_{i=1}^{m}S_{i}^{n} ∪_{g} D^{2n}.

Suppose both S_{i}^{n} and D^{2n} are oriented. Let y_{i} ∈ S_{i}^{n} be a point (other than base point y). Make g transverse to y_{i} for each i = 1, ..., m so that K_{i} = g^{−1}(y_{i}) is an oriented (n-1)-manifold with framed normal bundle in S^{2n−1}. The orientation of K_{i} is given by the orientation of ∂D^{2n} = S^{2n−1} and S_{i}^{n}, and framed normal bundle of K_{i} is given by Thom-Pontragin construction, i.e., the pull back of the tangent bundle at y_{i} in S_{i}^{n}. The oriented framed (n-1)-manifolds K = {K_{1}, ..., K_{m}} gives a linking matrix A = (a_{vw}), where a_{vw} = lk(K_{v}, K_{w}), the linking number of K_{v} and K_{w}. The manifold K_{i} with the given framed normal bundle gives an element t_{i} ∈ π_{2n−1}(S^{n}).

So we have a set of invariants (A; t_{1}, ..., t_{m}) of M. Denote

(2.11)  I(m, n) = {(A_{m×m}; t_{1}, ..., t_{m} ∈ π_{2n−1}(S^{n})},

where A is a symmetric matrix if n is even and is anti-symmetric matrix if n is odd.

Note when a pre-n-space has the homotopy type of a closed orientable manifold, its linking matrix is A unimodule.

Theorem 2.3 ([Wall p. 182] and [p.26, Ki]). There is a 1-1 correspondence between H : π_{2n−1}(V_{i=1}^{m}S_{i}^{n}) → I(m, n) Hence (A; t_{1}, ..., t_{m}) is a complete set of homotopy type invariant of the pre-n-space M.

Proof. The follwoing classsical facts, will be also used later, explain that Theorem 2.3 is the same as the statement of fact in [p. 128, Wall], and coinside with fact in [p.26, Ki] when n − 2. □

Lemma 2.4.

(1) π_{2n−1}(S^{n}) is isomorphic to < ν > ⊕G if n is even and isomorphic to G if n is odd, where G is a finite abelian group, and ν is an element of infinite order. Moreover π_{3}(S^{2}) = Z.

(2) Suppose n is even and t ∈ π_{2n−1}(S^{n}), then t = eH(t)ν + μ, where H : π_{2n−1}(S^{n}) →< ν >= Z is the Hopf map, which is the self-linking number defined by the map t : S^{2n−1} → S^{n} and e = \frac{1}{H(ν)}. In particular, for each t_{v} in Theorem 2.3, H(t_{v}) = a_{vv}.

(3) Also H([s^{n}, s^{n}]) = 2 for [s^{n}, s^{n}] with suitable chosen orientations in the proof of Theorem 2.2.

Proof. (1) and (2) can be found in the Chapter XI of [Hu], and (3) ha been indicated in [p.167, Wall]. □

§3. Theorem 1 and its proof.

Suppose M and L are two (n − 1)-connected 2n-manifolds with representation

M = V_{i=1}^{m}S_{i}^{n} ∪_{g} D^{2n} L = V_{j=1}^{l}S_{j}^{n} ∪_{h} D^{2n}

where g ∈ π_{2n−1}(V_{i=1}^{m}S_{i}^{n}) is corresponding to (A; t_{1}, ..., t_{m}) ∈ I(m, n) and h ∈ π_{2n−1}(V_{j=1}^{l}S_{i}^{n}) corresponding to (B; u_{1}, ..., u_{l}) ∈ I(l, n) by Theorem 2.3.

Let a_{i∗} = [S_{i}^{n}] ∈ H_{n}(M, Z) and b_{j∗} = [S_{j}^{n}] ∈ H_{n}(L, Z). Then α_{∗} = (a_{∗1}, ..., a_{∗m}) and β_{∗} = {b_{1∗}, ..., b_{l∗}} form basis of H_{n}(M, Z) and H_{n}(L, Z) respectively.

Theorem 1. Suppose M and L are two closed oriented (n − 1)-connected 2n-mani -folds with detas as above and P = (p_{ij}) ∈ M_{m×l}. Then there is a map f : M → L of degree k such that f_{∗}(α_{∗}^{⊥}) = P β_{∗}^{⊥} if and only if P satisfies the following equations

(3.1)  P^{⊥}AP = kB

and

(3.2)  ku_{r} = ∑_{v} p_{vr}t_{v} + (∑_{v} \frac{1}{2}p_{vr}(p_{vr} − 1)a_{vv} + ∑_{v<w} p_{vr}p_{wr}a_{vw})[s_{n}, s_{n}]

Moreover the homotopy classes of maps f : M → L of such that f^{∗}(β) = αP is 1-1 corresponding to the elements of π_{2n}(L). In particular there are only finitely many homotopy classes of maps realizes P when n > 2.

Proposition 3.1. Let f : V_{i=1}^{m}S_{i}^{n} → V_{j=1}^{n}S_{j}^{n} be map such that f_{∗}α_{∗}^{⊥} = P β_{∗}^{⊥} on H_{n} = π_{n}. Then for any element g ∈ π_{2n−1}(V_{i=1}^{m}S_{i}^{n}) corresponding to (A; t_{1}, ..., t_{m}) ∈ I(m, n), the element f_{∗}(g) ∈ π_{2n−1}(V_{j=1}^{n}S_{j}^{n}) is corresponding to (A′; t′_{1}, ..., t′_{l}) ∈ I(l, n), where

(3.3)  A′ = P^{⊥}AP

and

(3.4)  t′_{r} = ∑_{v} p_{vr}t_{v} + (∑_{v} \frac{1}{2}p_{vr}(p_{vr} − 1)a_{vv} + ∑_{v<w} p_{vr}p_{wr}a_{vw})[s_{n}, s_{n}]

Proof. Call a map from disc D^{n} to a n-sphere S^{n} with base point z is a boundary pinch if the interior of D is mapped onto S^{n}−z homeomorphically and ∂D is mapped onto z. If both D and S^{n} are oriented, We can say a boundary pinch is degree 1 or degree −1.

Recall P = (p_{ij})_{m×l}, then P^{⊥} = (p_{ji})_{l×m}.

For each i, take ∑_{j} |p_{ji}| disjoint n-disc in S_{i}^{n} away from the base point y_{i}. Each n-disk has the induced orientation of S_{j}^{n}. Then map |p_{ji}| discs to S_{j}^{n} via boundary pinches of degree 1 if p_{ji} is positive and −1 otherwise. Doing this for all i, any finally map the complement of all those ∑_{j,i} |p_{ji}| discs to the base point z, we get a map f′ : V_{i=1}^{m}S_{i}^{n} → V_{j=1}^{n}S_{j}^{n} and clearly f_{∗}α_{∗}^{⊥} = P β_{∗}^{⊥} on H_{n} = π_{n}. Up to homotopy, we may assume that f = f′.

Note f^{−1}(z_{j}) consists ∑_{i} |p_{ji}| points, and |p_{ji}| points of them lie in S_{i}^{n} with sign 1 if p_{ji} is positive and sign −1 otherwise. Suppose g is tranversal to V_{i}S_{i}^{n} − y so that the oriented (n-1)-manifolds K_{i} = g^{−1}(y_{i}) and its lifted framing normal bundle provides an element s_{i} ∈ Ω(n) such that A = (lk(K_{i}, K_{j})) and t_{i} = T (s_{i}) provide the set of invariant (A, t_{1}, ..., t_{m}). Than f ∘ g is transversal to V_{j}S_{j}^{n} − z. Now

(3.5)  (f ∘ g)^{−1}(z_{r}) = g^{−1}(f^{−1}(z_{r})) = ∑_{p=1}^{m} ∑_{e=1}^{p_{vr}} K_{v}^{e},

where K_{v}^{e} is the preimage of a point in f^{−1}(z_{r}) ∩ S_{v} under g. Let s_{v}^{e} ∈ F(n) be the K_{v}^{e} with its lifted framing normal bundle.

By the definition we have

(3.6)  A′ = (a_{rs}) = lk((f ∘ g)^{−1}(z_{r}), (f ∘ g)^{−1}(z_{s}))

and

(3.7)  t′_{r} = T (⊔_{p=1}^{m} ⊔_{e=1}^{p_{vr}} s_{v}^{e})

Then

(3.8) lk(s_{v}^{e}, s_{w}^{e′}) = lk(K_{v}^{e}, K_{w}^{e′}) = lk(K_{v}, K_{w}) = a_{vw} and T ([s_{v}^{e}]) = T ([s_{v}]) = t_{v}

For simplicity we will denote ∑_{e=1}^{p_{vr}} s_{v}^{e} by p_{vr}s_{v} and ∑_{e=1}^{p_{vr}} K_{v}^{e} by p_{vr}K_{v}. Then

lk((f ∘ g)^{−1}(z_{r}), (f ∘ g)^{−1}(z_{s})) = lk(∑_{v} p_{vr}K_{v}, ∑_{w} p_{ws}K_{w})

(3.9)  = ∑_{p,q} p_{vr}p_{ws}lk(K_{v}, K_{w}) = ∑_{p,q} p_{vr}p_{ws}a_{vw}

Combine (3.6) and (3.9), we get (3.3). Applying Theorem 2.2 to T (p_{vr}s_{v}) we have

(3.10)  T (p_{vr}s_{v}) = p_{vr}T (s_{v}) + \frac{1}{2}p_{vr}(p_{vr} − 1)lk(s_{v}, s_{v})[s^{n}, s^{n}]

Apply Theorem 2.2 to T (∑_{v} p_{vr}s_{v}) , and by (3.10) then We have

T (∑_{v} p_{vr}s_{v}) = ∑_{v} T (p_{vr}s_{v}) + ∑_{p<q} p_{vr}p_{qr}lk(s_{v}, s_{w})[s^{n}, s^{n}]

(3.11)  = ∑_{v} p_{vr}T (s_{v}) + (∑_{v} \frac{1}{2}p_{vr}(p_{vr} − 1)a_{vv} + ∑_{v<w} p_{vr}p_{wr}a_{vw})[s^{n}, s^{n}]

(3.7) and (3.11) gives (3.4). Lemma 3.3 is proved. □

Proof of Theorem 1.

(1) The Sufficiant Part. Let J = (J_{1}, ..., J_{l}) ⊂ B ⊂ S^{2n−1} be the (n-1)-manifold provided by h : S^{2n−1} →

V_{j=1}^{l}S_{j}^{n} and B is a (2n-1)-ball. Let p_{k} : S^{2n−1} → S^{2n−1} is a branched covering over a standard sphere S^{2n−3} ⊂ S^{2n−1} which misses B. Then p_{k}^{−1}(J) = (J^{1}, ..., J^{k}), each J^{i} lies in a homeomorphic lift B_{i} of B. Then

(3.12)  lk(J_{v}^{i}, J_{w}^{i}) = lk(J_{v}, J_{w}) and lk(J_{v}^{i}, J_{w}^{j}) = 0 if i ≠ j

Now consider the composition

(3.13)  S^{2n−1} ⟶^{p_{k}} S^{2n−1} ⟶^{h} V_{j}S_{j}^{n},

by (3.12), the discussion above and Theorem 2.2, it is easy to verify that the element in I(l, n) corresponding to [h ∘ p_{k}] = k[h] ∈ π_{2n−1}(V_{j=1}^{l}S_{j}^{n}) is (kB; ku_{1}, ..., ku_{l}).

Then by (3.3) and (3.4) in Lemma 3.3, we have

(3.14)  [f ∘ g] = k[h] = [h ∘ p_{k}].

Since and the composition

(3.15)  S^{2n−1} ⟶^{g} V_{i}S_{i}^{n} ⟶^{f} V_{j}S_{j}^{n} ⟶^{i} V_{j}S_{j}^{n} ∪_{h} D^{2n}

is homotopic to the composition

(3.16)  S^{2n−1} ⟶^{p_{k}} S^{2n−1} ⟶^{h} V_{j}S_{j}^{n} ⟶^{i} V_{j}S_{j}^{n} ∪_{h} D^{2n}

by the (3.14), where i is the inclusion. Now extending the map in (3.16) by coning p_{k} : S^{2n−1} → S^{2n−1} to D^{2n} → D^{2n}, to a map, stiil denited by f,

(3.17)  M = V_{i=1}^{m}S_{i}^{n} ∪_{g} D^{2n} → N = V_{j=1}^{n}S_{j}^{n} ∪_{h} D^{2n}.

Clearly f is degree k. and f_{∗}(α^{⊥}) = P β^{⊥}. □

(2) The 1-1 correspondence:

Suppose P is realized by a map f′ from M to L, that is f′_{∗}(α_{∗}^{⊥}) = P β_{∗}^{⊥}. Since M and L are (n-1)-connected 2-manifolds, we may assume that up to homotopy that

f′, f : M = V_{i=1}^{m}S_{i}^{n} ∪_{g} D^{4} → N = V_{j=1}^{n}S_{j}^{n} ∪_{h} D^{4}.

only differ only the interior of D^{2n}, where f is the map of degree k constructed in (3.17).

Now restriction of f and f′ on D^{2n} give a map S^{2n} → L, which is an element in π_{2n}(L), denote as C(f, f′). Clearly f and f′ are homotopic if and only if C(f, f′) = 0, any for any τ ∈ π_{2n}(L), we can construct an f′ such that C(f, f′) = amstexdw3τ (present D^{2n} as S^{2n}#D^{2n} then present f′ as τ#f). This prove that the homotopy classes of maps realizing P is 1-1 corresponding to the elements in π_{2n}(L).

The finiteness of π_{2n}(L), n > 2, is known (see [p. 45, Ka] for example) It follows from the Hilton’s formular

(3.18)  π_{2n}(V_{j}S_{j}^{n}) = lπ_{2n}(S^{n}) ⊕ \frac{1}{2}l(l − 1)π_{2n}(S^{2n−1})

and the finiteness of groups in the rightside of (3.18). □

(3) The necessary Part.

Suppose there is a map f′ : M → L such that f′_{∗}(α_{∗}^{⊥}) = P β_{∗}^{⊥}. Then up to homotopy, we may assume that f′(V_{i}S_{i}^{n}) ⊂ V_{j}S_{j}^{n}) and the restriction on V_{i}S_{i}^{n} is the same as the map f given in the proof of Lemma 3.1. Then by Lemma 3.1, the complete set of homotopy invariants < A′; t′_{1}, ..., t_{l} > assocaiated with the attaching map g ∘ f′ is given by (3.3) and (3.4).

Since the kernal of

(3.19)  i_{∗} : π_{2n−1}(V_{j}S_{j}^{n}) → π_{2n−1}(V_{j}S_{j}^{n} ∪_{h} D^{2n}) = π_{2n−1}(V_{j}S_{j}^{n})/[h]

induced by the inclusion i : V_{j}S_{j}^{n} → V_{j}S_{j}^{n} ∪_{h} D^{2n} is the cyclic group < [h] >, and [i ∘ f ∘ g] = 0, we have [f ∘ g] = [dh]

Then one can verify d must be k. □

Since Theorem 2 and 3 is stated in terms of cohomology, we need translate the sufficient part of Theorem 1 as below:

Theorem 1*. Suppose M and L are two oriented (n − 1)-connected (2n)-manifolds and P ∈ M_{m×l} satisfying the equation (3.1) and (3.2) in Proposition 3.2, where A and B are the intersection matrix of M and L under some basis α for H̄^{n}(M, Z) and β for H̄^{n}(L, Z). Then there is a map f : M → L of degree k such that f^{∗}(β) = αP.

Proof. Let α_{∗} ∈ H_{n}(M, Z) and β_{∗} ∈ H^{n}(L, Z) be the algebraic dual basis of α and β respectively.

For α_{∗} = (a_{1∗}, ..., a_{n∗}) ∈ H_{n}(M, Z), there is a presentation V_{i=1}^{m}S_{i}^{n} ∪_{g} D^{2n} for M (contained in [p.166, Wa]), where g ∈ π_{2n−1}(V_{i=1}^{m}S_{i}^{n}) such that a_{i∗} = [S_{i}^{n}] ∈ H_{n}(M, Z) and the linking matrix A′ provided by g.

The following Lemma is known.

Lemma 3.2. A′ = A, (and similarly B′ = B).

Proof. Now we can further homotope g so that K_{i} = g^{−1}(y^{i}) ⊂ S^{2n−1} is connected. Let F′_{i} ⊂ D^{2n} be an oriented n-chain bounded by K_{i}. Let F_{i} be the cycle in M = V_{i=1}^{m}S_{i}^{n} ∪_{g} D^{2n}, defined by F′_{i}/K_{i}, where g(K_{i}) = y_{i}. Then

(a) S_{i} ∩ F_{j} is 1 if i = j and is 0 if i ≠ j,

(b) the intersection matrix of H_{n}(M, Z) under F_{1}, ..., F_{m} is the linking matrix A′ provided by g.

The fact (a) implies that α is the Poincare dual basis of (F_{1}, ..., F_{m}). Therefore the intersection matrix of H^{n}(M, Z) under the basis α is the same as the intersection matrix of H_{n}(M, Z) under the basis (F_{1}, ..., F_{m}), Hence A′ = A. □

Now by Theorem 1, P can be realied by f : M → N such that f_{∗}(α_{∗}^{⊥}) = P β_{∗}^{⊥}. By (1.4), this is equivalent to f^{∗}(β) = αP □

§4. Proof of Theorem 2.

Recall that α = (a_{1}, ..., a_{m}) and β = (b_{1}, ..., b_{l}) are basis of H̄^{2}(M, Z) and H̄^{2}(L, Z) respectively. We also have

(4.1) α^{⊥} ∪ α = A[M] = (a_{ij})[M],  β^{⊥} ∪ β = B[N] = (b_{ij})[N],  f^{∗}(β) = αP.

Suppose f : M → N is of degree k, then f^{∗}([N]) = k[M] and

kB_{l}[M] = (b_{ij}k[M]) = (f^{∗}(b_{ij,l}[N])) = (f^{∗}(b_{i} ∪ b_{j,}))
(4.2)  = (f^{∗}(b_{i}) ∪ f^{∗}(b_{j})) = (f^{∗}(β))^{⊥}(f^{∗}β) = (αP)^{⊥}αP
= P^{⊥}α^{⊥}αP = P^{⊥}AP [M].

That is

(4.3)  P^{⊥}AP = kB.

Suppose f : M → L is of degree one. Then for f^{∗} : H^{2}(L, Z) → H^{2}(M, Z) by [Theorem I.2.5, B], there is μ : H^{2}(M, Z) → H^{2}(L, Z) such that μ ∘ f^{∗} : H^{2}(L, Z) → H^{2}(L, Z) is the identity on H^{2}(L, Z). Hence μ ∘ f^{∗}| : H̄^{2}(L, Z) → H̄^{2}(L, Z) is the identity on H̄^{2}(L, Z). In particular H̄^{2}(M, Z) = f^{∗}(H̄^{2}(L, Z) ⊕ kerμ|.

For any b_{i}, b_{j} ∈ β ⊂ H̄^{2}(L, Z), Since

X_{M}(f^{∗}(b_{i}), f^{∗}(b_{j}))[M] = f^{∗}(b_{i}) ∪ f^{∗}(b_{j})
(4.4)  = f^{∗}(b_{i} ∪ b_{j}) = f^{∗}(b_{ij}[N]) = b_{ij}f^{∗}(N) = b_{ij}[M]

Hence the restriction of X_{M} on the subgroup f^{∗}(H̄^{2}(L, Z)) ⊂ H̄^{2}(M, Z) is isomorphic to X_{N}. In particular it is still unimodular. By orthogonal decomposition lemma [p.5 MH],

X_{M} = X_{f^{∗}H̄^{2}(L,Z)} ⊕ X_{ker(μ|)} = X_{N} ⊕ X_{kerμ|}

□

§5. Proof of Theorem 3.

Proposition 5.1. Let M and L be two closed oriented 4-manifolds satisfying

(5.1)  X_{M} = X_{H} ⊕ X_{H′},

where X_{H} is isomorphic to X_{L}. If L is simply connected, then there is map g : M → L of degree 1.

The proof divided into three steps:

(1) First to prove Proposition 5.1 is true in the case M is also simply connected. This is a corollary of Freedmann’s work [Fr]. Suppose X(M) = X(L) ⊕ X(H′).

Clearly X(H′) is a symmetric unimodule, therefore C represents the intersection form of a simply connected 4-manifold Q. That is we have X(M) = X(L)#X(Q) = X(L#Q). By Whitehead’s theorem [Theorem 2.1, K], M and L#Q are homotopic. then there is a homotopy equivalence h : M → L#Q, which is degree one. There is also an obvious degree one pinch p : L#Q → L. Then p ∘ h : M → L is a map of degree one.

(2) Then to prove that if X(M) = X(L) and L is simply connected, then there is a degree one map f : M → L.

Let F = {F_{j}, j = 1, ..., m} be a set of oriented embedded surfaces which provides a basis of H̄_{n}(M, Z).

If M is simply connected, then M and L have the same homotopy type, hence there is a degree one map M → L.

Suppose M is not simply connected. Let C_{i}, i = 1, ..., l be a set of disjoint simple closed curves which generate π_{1}(M) and

(5.2)  (∪_{i}N(C_{i})) ∩ (∪_{j}F_{j}) = ∅.

where N(C_{i}) = C_{i} × D^{3} is the regular neighborhood of C_{i} in M. Let M^{∗} be the simply connected 4-manifolds obtained from M by surgery on {c_{i}}. Precisely

(5.3)  M^{∗} = (M − int ∑_{i} C_{i} × D^{3}) ∪_{{h_{i}}} D_{i}^{2} × S^{2},

where the gluing map

(5.4)  h_{i} : S_{i}^{1} × S^{2} = ∂D_{i}^{2} × D^{2} → ∂(C_{i} × D^{3}) = C_{i} × S^{2}

is an orientation reversing homeomorphism.

Since F is linear independent in H_{2}(M, Z), it is easy to see that F is still linear independent in H_{2}(M^{∗}, Z). Hence F a basis of a submodule H of H_{2}(M^{∗}, Z) and the restriction of the intersection form X(M^{∗}) on F is still X(M). In particular the restriction of the form of M^{∗} on F is still unimodular. By the famous orthogonal decomposition lemma [p.5 MH], We have proved that

(5.5)  X_{M^{∗}} = X_{H} ⊕ X_{H′} = X_{L} ⊕ X_{H′}

In particular,

(5.6)  H_{2}(M^{∗}, Z) = H ⊕ H′,

By the part (1) of Proposition 5.1 and its proof, we have that there is a degree one map

(5.7)  M^{∗} ⟶^{h} L#Q ⟶^{p} L

where X(L) = X(H), h is a homotopy equivalence and p pinches Q. Let f = p ∘ h, in particular

(5.8)  H′ = Kerf_{∗} : H_{2}(M^{∗}, Z) = π_{2}(M^{∗}) → H_{2}(L, Z) = π_{2}(L)

Now we will transfer the degree one map in (5.7) to a degree one from M to N. Pick a point ∗_{i} ∈ D_{i}^{2} for each i. Claim: ∗_{i} × S^{2} ∈ H′ for each i: For simplicity, let S = ∗_{i} × S^{2}. Then we have S = ∑ x_{i}F_{i} + y, where y ∈ H′. By

(5.5),

(5.9)  < S, F_{j} >=< ∑ x_{i}F_{i}, F_{j} >= a_{ij}x_{i}.

On the otherhand since S is disjoint from all F_{j}, we have

(5.10)  ∑ a_{ij}x_{i} = 0,  i = i, ...m,

Since A = (a_{ij}) is unimodule, which is in particular is non-singular, (5.10) has only zero sultion x_{1} = ... = x_{m} = 0. This prove the claim.

By (5.8) and the Claim above, f(∗_{i}×S^{2}) is null homology, therefore null-homotopy. Now we can homotopy f so that f(D_{i}^{2} × S^{2}) = z for each i by first shrinking D_{i}^{2} × S^{2} to its core ∗_{i} × S^{2}, then shrinking ∗_{i} × S^{2} to Z.

Note M^{∗} − int ∪_{i} D_{i}^{2} × S^{2} = M − int ∑_{i} C_{i} × D^{3}. Since f : M^{∗} → L is degree one, f(D_{i}^{i} × S^{2}) = z for all i, we have the restriction

(5.11)  f| : (M − int ∪_{i} C_{i} × D^{3}, ∪_{i}C_{i} × S^{2}) → (L, z)

is map between pairs of degree 1.

Since f(D_{i}^{2} × S^{2}) = z ∈ L, we have f(C_{i} × S^{2}) = z ∈ L. Now we can extend f| to g : M → L by sending C_{i} × D^{3} to z. Since f sent C_{i} × D^{3} to z, the extension g : M → N is of degree 1.

(3) Prove Proposition 5.1 from (1) and (2).

Let Q be a simply connected 4-manifold with X(Q) = X(M) = X(L) + C. Then there is a degree one map from M to Q by (2) and a degree one map from Q to L by (1), hence there is a degree one map form M to L. □

Proof of Theorem 3 from Propositions 5.1 and Theorem 1^{∗}. Let M and L given as in Theorem 3 and let Q be a simply connected 4-manifold with X(Q) = X(M). By proposition 5.1, there is a degree one map f′ : M → L which induces isomorphism f_{∗} : H̄_{2}(M, Z) → H_{2}(Q, Z).

Under the basis f′_{∗}(α_{∗}), the intersection matrix of X(Q) is A the matrix for f′_{∗} is the unit matrix of order n.

For the case of simply connected 4-manifolds, since π_{3}(S^{2}) = Z, the equation (3.2) is covered by (3.1). Then by Theorem 1*, there is a map f” : Q → L of degree k realizes P. Finally f = f” ∘ f′ : M → L is a map of degree k realizes P. □

§6. Proof of the corollaries and examples.

Proof of Corollary 1.

For (1). Suppose there is a degree one map f : M → L. By Theorem 2,

X_{M} = X_{L} ⊕ X_{H′}.

By Freedmann’s Classification Theorem [F], X_{H′} represents the intersection form of a simply connected 4-manifold L. That is we have

X_{M} = X_{L}#X_{Q} = X_{L#Q}.

By Whitehead’s theorem [Theorem 2.1, K], M and L#Q are homotopic. If the intersection form X(M) is even, then both B and C are even, and therefore the simply connected 4-manifold L with intersection form C is unique up to homeomorphism, and M and L#L are homeomorphic by Freedmann’s Classification Theorem.

For (2). If there is a map f : M → L of non-zero degree, then β_{2}(M) ≥ β_{2}(L) by Theorem 2. Since β_{0}(M) = β_{0}(L) = β_{4}(M) = β_{4}(L) = 0, and the odd dimension cohomology groups vanish for simply connected 4-manifolds, we have immediately that χ(M) > 0 and χ(L) > 0 and χ(M) ≥ χ(L).

In the later case, χ(M) = χ(L) implies that β_{2}(M) = β_{2}(L). Since f is of degree one, by (1.6) in Theorem 1, the intersection form of M and L must be isomorphic. Then (2) follows by applying Freedmann’s work, as we did in the proof of Corollary 1. □

Proof of Corollary 1^{∗}. By the proof of Corollary 2, we need only to show β_{i}(M) ≥ β_{i}(L) for all i. A proof of this fact is as follow.

Recall that Poincare duality P_{M} : H^{p}(M, Z) → H_{n−p}(M, Z) is given by z^{p} → z^{p} ∩ [M], where [M] be the fundamental class class of H_{n}(M, Z) and we also have the formula

f_{#}(f^{#}z^{p} ∩ z_{w}) = z^{p} ∩ f_{#}Z_{w}

for any map f : M → L.

To show β_{i}(M) ≥ β_{i}(L), we need only to show that the kernal of f^{#} : H^{#}(L, Z) → H^{#}(M, Z) must be in the torsion part. Suppose that f^{#}(z^{q}) = 0 for some 0 ≠ z^{q} ∈ H^{q}(L, Z). Then by (1) we have

0 = f_{#}(f^{#}z^{p} ∩ [M]) = z^{p} ∩ d[N] = dz^{p} ∩ [N],

where d ≠ 0 is the degree of f. That is dz^{p} = 0. □

Proof of Corollary 2. (1) Let (A; t_{1}, ..., t_{m}) be the complete homotopy invariant of M. Recall that π_{2n−1}(S^{n}) =< ν > ⊕G, where ν is infinite order if n is even and is 0 if n is odd, and G is a finite group. Now t_{v} = a_{vv}ν + μ_{v}, [s^{n}, s^{n}] = 2ν + μ and where μ_{v}, μ ∈ T orπ_{2n−1}(S^{n}), and H : π_{2n−1}(S^{n}) → Z is the Hopf invariant.

Let I_{m} be the m by m unit matrix. Let P = kI_{m} for some integer k, we first have

(6.1)  P^{⊥}AP = k^{2}A.

Now the leftside of (3.1) in Theorem 1 can be deduced to

∑_{v} p_{vr}t_{v} + (∑_{v} \frac{1}{2}p_{vr}(p_{vr} − 1)a_{vv} + ∑_{v<w} p_{vr}p_{wr}a_{vw})[s_{n}, s_{n}]

= kt_{v} + \frac{1}{2}k(k − 1)a_{vv}[s_{n}, s_{n}]

(6.2)  = k(a_{vv}λν + μ_{v}) + \frac{1}{2}k(k − 1)a_{vv}(2λν + μ).

The first equality of (6.2) is due to the fact p_{vv} = k and p_{vw} = 0 if v ≠ w. The second equality is by Lemma 2.4.

When k be a multiple of |T orπ_{2n−1}(S^{n})|, the last term of (6.2) can be further reduced into

k(a_{vv}λν + μ_{v}) + \frac{1}{2}k(k − 1)a_{vv}(2λν + μ)

(6.3)  = ka_{vv}λν + k(k − 1)a_{vv}λν = k^{2}a_{vv}λν

On the otherhand clearly

(6.4)  k^{2}t_{v} = k^{2}(a_{vv}λν + μ_{v}) = k^{2}a_{vv}λν.

(6.1) implies that (3.1) of Theorem 1 hold, and (6.2), (6.3) and (6.4) implies that (3.2) of Theorem 1 hold. By Theorem 1 there is a map of degree k^{2} from M to itself realizing P. In the case n = 2, we have π_{3}(S^{2}) =< ν >, and |T orπ_{2n−1}(S^{n})| = 1. We have proved (1).

(2) If D(M, N) is not the {0}, then there is a map f : M → L of degree d ≠ 0. By (1), for some integer k, there is a map g_{k} : L → L of degree k^{2}, then g_{k} ∘ f : M → L is of degree dk^{2}. That is D(M, N) is infinite set. □

Proof of Corollary 3. Let P be the matrix realized by f, by (1.6) in Theorem 2 we have

(6.5)  P^{⊥}AP = kA,

where both A, P are square matrices of order β_{2}(M). By taking the determinants we have |P |^{2}|A| = k^{β_{2}(M)}|A|. That is |P |^{2} = k^{β_{2}(M)}. Since β_{2}(M) is odd, k itself must be a perfect square. In particular k ≠ −1. □

Proof of Corollary 4. From the equation P AP^{⊥} = kB (1.6) in Theorem 2 we have

(6.6)  kb_{ii} = ∑ a_{rs}p_{ir}p_{is} = ∑ a_{rr}p_{ir}^{2} + ∑_{i<s}(a_{rs}p_{ir}p_{is} + a_{sr}p_{is}p_{ir})

Since X(M) is even, each a_{rr} is even. Since and a_{rs} = a_{sr}, the right side of the equation is even. Since B is not even, b_{ii} is odd for some i. It follows k is even. □

Proof of Proposition 4^{∗}. Let v_{i} and v′_{i} be the i-th Wu class of M and L respectively, then v_{i} is characterized by [p. 132, MS]

(6.7)  < v_{i} ∪ x, [M] >=< sq^{i}x, [M] >

Suppose v′_{i} ≠ 0. Then there is x′ ∈ H_{∗}(M, Z_{2}) such that < v′_{i} ∪ x′, [N] >≠ 0. Let the degree be 2k + 1 Then we have

0 ≠< v′_{i} ∪ x′, [N] >=< sq^{r}x′, [N] >=< sq^{r}x′, f_{#}[M] >
=< f^{#}sq^{r}x′, (2k + 1)[M] >=< f^{#}sq^{r}x′, [M] >
(6.8)  =< sq^{r}f^{#}(x′), [M] >=< v_{i} ∪ f^{#}(x′), [M] >

It follows v_{i} ≠ 0. That is if v_{i} = 0 then v′_{i} = 0.

According to Wu’s formula

w_{i} = ∑_{j+l=i} sq^{l}v_{j} i = 1, ..., n.

it is easy to see that w_{1} = w_{2} = ... = w_{k} = 0 if and only if v_{1} = v_{2} = ... = v_{k} = 0, k ≤ n. Then the conclusion follows. □

Proof of Corollary 5. If M dominates L, then the rank of the form of L is at most the rank of the form of M by Theorem 2. For the given positive integer m, there are only finitely many distinct isomorphic forms of rank m [p.18 and p.25, MH]. Since the torsion part of π_{2n−1}(S^{n}) is finite, so the homotopy type of L is finite. In the case of 4-manifolds, there are at most two simply connected 4-manifolds with the given form X [Fr]. So the conclusion follows. □

§. 7. Verification of Examples.

Let

A_{l} = ⊕_{l} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ,  P_{l,k} = ⊕_{l} ( \begin{matrix}0 & k \\ 1 & 0 \end{matrix} ) ,  B = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ,  I_{l} = ⊕_{l}(1)

(1) Suppose M has intersection matrix A = (a_{ij})_{m×m}. For any a_{ij} ≠ 0, let c_{i} = c_{j} = 1 and c_{r} = 0 for r ≠ i, j. Let C = (c_{i})_{m×1}. Then

C^{⊥}AC = k(1)

Then k = 2a_{ij} if i ≠ j and k = a_{ii} otherwise. By Theorem 1, there is a map f : M → CP^{2} of degree k ≠ 0.

(2) By exact sequence of fibration S^{5} → CP^{2}, we have π_{4}(CP^{2}) = π_{4}(S^{5}) = 0. Moreover π_{4}(S^{2} × S^{2}) = π_{4}(S^{2}) ⊕ π_{4}(S^{2}) = Z_{2} ⊕ Z_{2}. Then the conclusion follows from Theorem 1.

For the remaining Consider the composition

S^{2} × S^{2}#S^{3} × S^{1} ⟶^{p_{1}} S^{2} × S^{2}V S^{3} × S^{1} ⟶^{p_{2}} S^{2} × S^{2}V S^{3} ⟶^{g_{k}} S^{2} × S^{2}

where p_{1} pinches the 3-sphere separating S^{2} × S^{2} and S^{3} × S^{1} to a point to get the the one point union of S^{2} × S^{2} and S^{3} × S^{1}, p_{2}| : S_{3} × S^{1} → S^{3} is the projection; and p_{2}|S^{2} × S^{2} is the identity; g_{k}|S^{3} sends S^{3} to the first factor of S^{2} × S^{2} via with Hopf invariant k and g|S^{2} × S^{2} → S^{2} × S^{2} realizes I_{2}. Let f_{k} = g_{k} ∘ p_{2} ∘ p_{1}, then {f_{k}} forms infinitely many homotopy classes realizing I_{2}.

(3) Let P = ( \begin{matrix}a & b \\ c & d \end{matrix} ). Since

P^{⊥}I_{2}P = ( \begin{matrix}a^{2} + b^{2} & ac + bd \\ ac + bd & c^{2} + d^{2} \end{matrix} )

Hence there is no solution P for P^{⊥}I_{2}P = A_{1} or P^{⊥}I_{2}P = B. Since

P^{⊥}BP = ( \begin{matrix}a^{2} − b^{2} & ac − bd \\ ac − bd & c^{2} − d^{2} \end{matrix} )

Hence there is no solution P for P^{⊥}BP = kI_{2} and P = ( \begin{matrix}1 & 1 \\ k & −k \end{matrix} ) is a solution for

P^{⊥}BP = 2kA_{1}. Since

P^{⊥}A_{1}P = ( \begin{matrix}2ab & ad + bc \\ ad + bc & 2cd \end{matrix} )

Hence there is no solution P such that P^{⊥}A_{1}P = kI_{2} and P = ( \begin{matrix}1 & k \\ −1 & k \end{matrix} ) is a

solution P^{⊥}A_{1}P = 2kB

Since I_{2}, B and A_{1} are the intersection form of CP^{2}#CP^{2}, CP^{2}#(−CP^{2}) and S^{2} × S^{2} respectively, the conclusion for in Example (2) follows from Theorem 3.

(4) Note T^{4} = S_{1}^{1} × S_{2}^{1} × S_{3}^{1} × S_{4}^{1}, Let a_{i} ∈ H^{1}(T^{4}, Z) be the Poincare dual of S_{i}^{1}, Let c_{ij} = a_{i} ∪ a_{j}. Then under the basis (c_{12}, c_{34}, c_{13}, c_{24}, c_{14}, c_{23}) of H^{2}(T^{4}, Z), the intersection matrix is A_{3}, which is the same as the matrix of #_{3}S^{2} × S^{2} under the obvious basis. Then it is easy to verify

P_{l,k}^{⊥}A_{3}P_{,k} = kA_{3}.

There is a map f_{k} : T^{4} → #_{3}S^{2} × S^{2} of degree k by Theorem 3.

Let a_{1}, b_{1}, ..., a_{s}, b_{s} be the standard basis for H_{1}(F_{s}, Z) and c_{1}, d_{1}, ..., c_{r}, d_{r} be the standard basis for H_{1}(F_{r}, Z). Then we use the same symbol to denote their Poincare dual in H^{1}(F_{s} × F_{g}, Z). Let under the basis (a_{1} ∪ b_{1}, c_{1} ∪ d_{1}, a_{1} ∪ c_{1}, b_{1} ∪ d_{1}, a_{1} ∪ d_{1}, b_{1} ∪ c_{1}, ..., a_{s} ∪ c_{r}, b_{s} ∪ d_{r}, a_{s} ∪ d_{r}, b_{s} ∪ c_{r}, ..., a_{s} ∪ c_{1}, b_{s} ∪ d_{1}, a_{s} ∪ d_{1}, b_{s} ∪ c_{1}, ..., a_{s} ∪ c_{r}, b_{s} ∪ d_{r}, a_{s} ∪ d_{r}, b_{s} ∪ c_{r}). The matrix of F_{s} × F_{g} is A_{2sr_{1}}. So for any integer k there is a map f_{k} : F_{s} × F_{r} → #_{2sr+1}S^{2} × S^{2} of degree k. Since there is degree one pinch g : #_{2sr+1}S^{2} × S^{2} → #_{q}S^{2} × S^{2} for any q ≤ 2rs + 1. The conclusion follows.

By last paragraph there is a map f_{k} : F_{r} × F_{s} → #_{2rs+1}CP^{2}##_{2rs+1}(−CP^{2}) of 2k for any k, and clearly there is a degree one pinch #_{2rs+1}CP^{2}##_{2rs+1}(−CP^{2}) → (#_{v}CP^{2})##_{w}(−CP^{2}) for any p, q ≤ 2rs + 1.

(5) The first claim follows from Corollary 3 and its proof. The second claim follows that those forms do not split as direct sum [p.26, Ki].

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