EQUIVARIANT SURGERY WITH MIDDLE DIMENSIONAL SINGULAR SETS. II

Masaharu Morimoto Department of Environmental and Mathematical Sciences Faculty of Environmental Science and Technology Okayama University Tsushimanaka 2-1-1, Okayama, 700 Japan

Abstract. Let G be a finite group. Let f : X → Y be a degree 1, G-framed map of simply connected, closed, oriented, smooth manifolds X and Y of dimension 2k ≧ 6. Assuming that the dimension of the singular set of the G-space X is at most k and further that f is k-connected, we obtained the G-equivariant surgery obstruction σ(f) in the previous article. In the present article, we show that the σ(f) is a G-framed cobordism invariant. Hence σ(f) can be uniquely defined even if f is not k-connected.

1. Introduction

This article is a continuation of [2] and presents an explanation of a part of [1]. Let G be a finite group and X a smooth G-manifold. The term G-equivariant surgery or simply G-surgery will refer to G-surgery on the free part of the G-space X, namely where every nontrivial element of G acts without fixed points. Thus our G-surgery does not change the G-singular set

Sing(G, X) = ⋃_{g∈G\{1}} X^{g},

where X^{g} = {x ∈ X | gx = x}.

Typeset by AMS-TEX 2 MASAHARU MORIMOTO

1991 Mathematics Subject Classification. 57R67, 57R91, 19G24.

Key words and phrases. Equivariant surgery, surgery obstruction, cobordism invariant, quadratic module.

Research partially supported by Max-Plank-Institut für Mathematik in Bonn and also by Grant-in-Aid for Scientific Research Japan

Throughout the paper, R denote the ring of integers ℤ, its localization ℤ_{(p)} at a prime p, or the ring of rational numbers ℚ. Associated to a degree 1, k-connected G-framed map f : X^{n} → Y^{n} with n = 2k such that the dimension of the G-singular set of X is at most k, we [2] defined an element σ(f) in an abelian group W (G, Y; R) such that if σ(f) = 0 then f can be converted to a k-connected, R-homology equivalence f′ : X′ → Y by G-surgery on the free part of X. The group W (G, Y; R) is a generalization of Wall’s group L_{n}^{h}(R[G]) and ingredients for the generalization are quadratic and symmetric form parameters and positioning maps. Here R[G] is the group ring of G with coefficients in R. In the current paper we prove that the obstruction σ(f) is a G-framed cobordism invariant. Thus, one can define a unique element σ(f) in W (G, Y; R) for a degree 1, G-framed map f even if f is not k-connected.

Theorem 1.1. Let G be a finite group and Y a closed, 1-connected, oriented, smooth G-manifold of even dimension n = 2k ≧ 6. Suppose that (1.1.1)–(1.1.3) below hold.

(1.1.1) dim Y^{g} ≦ k for all g ∈ G \ {1}.

(1.1.2) If dim Y^{H} = k for some subgroup H ≦ G then |H| = 2 and Y^{H} is connected and oriented such that each g : Y^{H} → Y^{gHg^{−1}} (g ∈ G) is orientation preserving.

(1.1.3) dim(Y^{H} ∩ Y^{K}) ≦ k − 2 whenever dim Y^{H} = k and dim Y^{K} = k − 1 (H, K ≦ G).

Let X be a closed, connected, oriented, smooth G-manifold satisfying (1.1.1)–(1.1.3) above, ξ and η real G-vector bundles over Y and fff = (f : X → Y, b : T (X) ⊕ f^{∗}η → f^{∗}(ξ ⊕ η)) a degree 1, G-framed map. Suppose that

(1.1.4) if H ≦ G has order 2 and if dim X^{H} ≧ k − 1 or dim Y^{H} ≧ k − 1 then dim X^{H} = dim Y^{H};

(1.1.5) if R = ℤ then f^{H} : X^{H} → Y^{H} is a ℤ-homology equivalence for any hyplerelementary subgroup H ≠ {1} of G;

(1.1.6) if R = ℤ_{(p)} (p a prime) then f^{P} : X^{P} → Y^{P} is a ℤ_{(p)}-homology equivalence for any p-subgroup P ≠ {1} of G; and

(1.1.7) if R = ℚ then χ(X^{g}) = χ(Y^{g}) for any g ∈ G \ {1}, where χ(−) denotes the Euler characteristic.

Then one can associate to fff an element σ(fff) in the abelian group W (G, Y; R) of [2, Theorem 1.1] such that σ(fff) = 0 if and only if fff is G-framed cobordant rel. ⋃_{g∈G\{1}} X^{g} to a degree 1, k-connected, G-framed map fff′ = (f′ : X′ → Y, b′ : T (X′) ⊕ f′^{∗}η → f′^{∗}(ξ ⊕ η)) such that f′ : X′ → Y is an R-homology equivalence.

Theorem 1.1 will be deduced from a slight generalization Theorem 3.3 of it.

Remark 1.2. (1) If σ(fff) = 0 then a G-framed map fff′ satisfying the properties in Theorem 1.1 and a G-framed cobordism between fff and fff′ can be obtained by G-surgery of dimension ≦ k on the free part of X .

(2) If f : X → Y is k-connected then the σ(fff) in the theorem is same as in [2, Theorem 1.1].

The reader can refer to [3] for the G-framed cobordism invariance of G-surgery obstructions under the hypothesis that Sing(G, X) ≦ k − 1.

The remainder of the paper is organized as follows. Section 2 is devoted to preparation for Sections 3 and 4. The G-framed cobordism invariance of the G-surgery obstruction is presented as Theorem 3.2. The main body of its proof is stated as Lemma 3.1. The generalization Theorem 3.3 mentioned above is obtained as a corollary to Theorem 3.2. Section 3 consists of these three results. We prove Lemma 3.1 in Section 4 and a key method of the proof is equivariant handle subtraction.

Acknowledgement

This paper was written during author’s stay in Max-Planck-Institut für Mathematik in Bonn. He would like to express his gratitude to the institute for its warm hospitality.

2. Preliminary

In the current article, n = 2k is an even integer ≧ 6 and a manifold (resp. an action on a manifold) means a smooth manifold (resp. a smooth action) unless otherwise specified. Let ((proj)) (resp. ((s-free)), ((free))) denote the category of all finitely generated projective (resp. stably free, free) R[G]-modules. Let C denote one of the three categories.

Let X be a G-CW complex (e.g. G-manifold, G-simplicial complex). For a point x ∈ X and a subgroup H of G, let H_{x} denote the isotropy subgroup at x in the H-space res_{H}^{G}X. Set

X^{H} = Fix(H, X) = {x ∈ X | G_{x} ⫆ H},
X^{>H} = Fix_{G}(> H, X) = {x ∈ X | G_{x} ⫌ H},
X^{=H} = Fix_{G}(= H, X) = {x ∈ X | G_{x} = H},
Free(H, X) = {x ∈ X | H_{x} = 1} and
Sing(H, X) = {x ∈ X | H_{x} ≠ {1}}.

If g ∈ G then set X^{g} = X^{⟨g⟩}. For a subspace Z ⫅ X, we define the subgroup

ρ_{X}^{G}(Z) := ⋂_{x∈Z} G_{x}.

Let π_{0}(X^{H}) denote the set of all connected components γ of X^{H}. The underlying space of γ is denoted by X_{γ}. If we like to emphasize the group H then we use X_{γ}^{H} instead of X_{γ}. For a nonnegative integer ℓ, let π_{0}(X^{H}, ℓ) denote the subset of π_{0}(X^{H}) consisting of all γ with dim X_{γ} = ℓ. We set

\begin{matrix}Π(G, X) = ∐_{H≦G} π_{0}(X^{H}) and \\ Π(G, X, ℓ) = ∐_{H≦G} π_{0}(X^{H}, ℓ). \end{matrix}

For γ ∈ Π = Π(G, X) such that γ ∈ π_{0}(X^{H}), define ρ(γ) = ρ_{Π}(γ) := H. Clearly, ρ_{Π}(γ) ≦ ρ_{X}^{G}(X_{γ}) holds. The set Π has the canonical G-action such that X_{gγ} = gX_{γ} and ρ(gγ) = gρ(γ)g^{−1}, and Π(G, X, ℓ) is a G-invariant subset of Π. Let Φ : Π → Π denotes the map γ ↦ β such that ρ(β) = ρ_{X}^{G}(X_{γ}) and X_{β} = X_{γ}. Define

Θ(G, X) := Im(Φ) and
Θ(G, X, ℓ) := Θ(G, X) ∩ Π(G, X, ℓ).

Let Mnf Mnf Mnf_{cp}^{n}(G)_{ℓ} (resp. Mnf Mnf Mnf_{cl}^{n}(G)_{ℓ}) denotes the family of all compact (resp. closed), ℓ-connected, oriented G-manifolds of dimension n. Each G-manifold X ∈ Mnf Mnf Mnf_{cp}^{n}(G)_{0} has the orientation homomorphism w_{X} : G → {±1} such that w_{X}(g) = 1 (resp. −1) for orientation preserving (resp. reversing) g ∈ G.

Let G(2) denote the set of all elements of order 2 in G. (The identity element does not belong to G(2).) In the case n = 2k, let Mnf Mnf Mnf_{cp,sg}^{n}(G)_{ℓ} (resp. Mnf Mnf Mnf_{cl,sg}^{n}(G)_{ℓ}), denote the family of all X ∈ Mnf Mnf Mnf_{cp}^{n}(G)_{ℓ} (resp. Mnf Mnf Mnf_{cl}^{n}(G)_{ℓ}) satisfying the following properties.

EQUIVARIANT SURGERY 5

(2.1.1) dim Sing(G, X) ≦ k. (2.1.2) |π_{0}(X^{H}, k)| ≦ 1 for any H ≦ G. (2.1.3) If γ ∈ Θ(G, X, k) then |ρ(γ) ∩ G(2)| = 1. (2.1.4) If γ ∈ Θ(G, X, k) and δ ∈ Θ(G, X, k − 1) then dim(X_{γ} ∩ X_{δ}) ≦ k − 2. (2.1.5) All submanifolds X_{γ}^{H} (γ ∈ Θ(G, X, k) and H = ρ(γ)) are oriented such that each g ∈ G acts as an orientation preserving diffeomorphism X_{γ}^{H} → X_{gγ}^{gHg^{−1}}. If X ∈ Mnf Mnf Mnf_{cp,sg}^{n}(G)_{0} with n = 2k then set

λ = λ_{X} := (−1)^{k},

Q = Q(G, X) := {g ∈ G(2) | dim X^{g} = k − 1} and
S = S(G, X) := {g ∈ G(2) | dim X^{g} = k}.

The set S is identified with Θ(G, X, k) via the map

Θ(G, X, k) → S(G, X); γ ⟼ g such that g ∈ ρ(γ).

Depending on X and the choice of R = ℤ, ℤ_{(p)} (p a prime) or ℚ, let A A A denote the tupple

(R, G, Q, S, λ, w_{X}).

The group ring R[G] is equipped with the anti-involution − such that

\overline{∑_{g∈G} c_{g}g} = ∑_{g∈G} w_{X}(g)c_{g}g^{−1} (c_{g} ∈ R).

If a ∈ G then let ε_{a} denote the map R[G] → R or R/2R[G] → R/2R defined by

ε_{a}(∑_{g∈G} c_{g}g) = c_{a} (c_{g} ∈ R or R/2R).

The choice of R or R/2R will be clear form the context. For a subset T of G, R[T] denotes the R-submodule of R[G] which is generated by all elements in T. Let Λ = Λ(G, X; R) denote the quadratic form parameter defined by

Λ = min^{λ}(R[G]) + R[Q], where min^{λ}(R[G]) = ⟨x − λ\overline{x} | x ∈ R[G]⟩_{R}.

A special quadratic (A A A, Θ(G, X, k))-module M M M is a tupple (M, B, q, α) of

6 MASAHARU MORIMOTO

M a finitely generated, projective R[G]-module,

B : M × M → R[G] a nonsingular λ-Hermitian form,

q : M → R[G \ S]/Λ an A A A-quadratic form and

α : Θ(G, X, [k]) → M a G-map satisfying the conditions: (2.2.1) B is biadditive, (2.2.2) B(ax, by) = bB(x, y)\overline{a}, (2.2.3) B(x, y) = λ\overline{B(y, x)}, (2.2.4) q(gx) = gq(x)\overline{g} and q(rx) = r^{2}q(x) in R[G \ S]/Λ = R[G]/(Λ + R[S]), (2.2.5) q(x + y) − q(x) − q(y) = B(x, y) in R[G \ S]/Λ = R[G]/(Λ + R[S]), (2.2.6) q̃(x) + λ\overline{q̃(x)} = B(x, x) in R[G \ S] = R[G]/R[S] and (2.2.7) [ε_{1}(B(α(s) − x, sx))] = 0 in R/2R, for all x, y ∈ M, a, b ∈ R[G], g ∈ G, r ∈ R and s ∈ S (see [2, pp.278–279]), where q̃(x) is a lifting of q(x).

If M M M = (M, B, q, α) is a special quadratic (A A A, Θ(G, X, k))-module then let −M M M stand for the module (M, −B, −q, −α). Let SQ SQ SQ(A A A, Θ(G, X, k))_{C}, denote the category of all special quadratic (A A A, Θ(G, X, k))-modules whose underlying modules are in C.

For a G-map f : X → Y, we set

K_{ℓ}(X; R) = K_{ℓ}(f; R) := Ker[f_{∗} : H_{ℓ}(X; R) → H_{ℓ}(Y; R)].

It is quite important to see whether the R[G]-module K_{ℓ}(X; R) is stably free or projective. At this end, let us define the following conditions on a G-map f : X → Y.

F_{1}-Condition: χ(X^{g}) = χ(Y^{g}) for any G \ {1}. (Here χ(−) denotes the Euler characteristic.)

F_{p}-Condition: f^{P} : X^{P} → Y^{P} is a ℤ_{(p)}-homology equivalence for any p-subgroup P ≠ {1} of G.

P_{0}-Condition: f^{P} : X^{P} → Y^{P} is a ℤ_{(p)}-homology equivalence for any prime p dividing |G| and any p-subgroup P ≠ {1} of G.

F_{0}-Condition: f^{H} : X^{H} → Y^{H} is a ℤ-homology equivalence for any hyperelementary subgroup H ≠ {1} of G.

Remark 2.3. Let R = ℤ_{(m)} for m = 0, 1 or a prime, where ℤ_{(0)} = ℤ and ℤ_{(1)} = ℚ. Let X and Y be finite G-CW complexes (e.g. compact G-manifolds), let ℓ be a positive integer, and let f : X → Y be an ℓ-connected G-map such that f_{∗} : H_{s}(X; R) → H_{s}(Y; R) is an isomorphism for every s ≠ ℓ and K_{ℓ}(X; R) is ℤ-torsion free. Then the following hold.

(2.3.1) If f satisfies F_{m}-Condition then K_{ℓ}(X; R) is stably free over R[G].

(2.3.2) If R = ℤ and if f satisfies P_{0}-Condition then K_{ℓ}(X; ℤ) is projective over ℤ[G]. If furthermore K̃_{0}(ℤ[G]) = 0 then K_{ℓ}(X; ℤ) is stably free over ℤ[G].

Let X, X′ ∈ Mnf Mnf Mnf_{cl,sg}^{n}(G)_{0} and Y ∈ Mnf Mnf Mnf_{cl}^{n}(G)_{1}. Let fff = (f : X → Y, b : T (X)⊕f^{∗}η → f^{∗}(ξ ⊕ η)) and fff′ = (f′ : X′ → Y, b′ : T (X′) ⊕ f′^{∗}η → f′^{∗}(ξ ⊕ η)) be degree 1, G-framed maps for G-vector bundles ξ and η over Y such that η ⫆ ε_{Y} (ℝ^{n+1}) (cf. [2, p.286]). Let I denote the closed unit interval [0, 1] and p_{Y} : I × Y → Y the canonical projection. Let F F F = (F : (W, ∂W) → (I × Y, ∂I × Y), B : T (W) ⊕ (p_{Y} ∘ F)^{∗}η → (p_{Y} ∘ F)^{∗}(ε_{Y} (ℝ) ⊕ ξ ⊕ η)) be a G-framed cobordism between fff and fff′. In particular, W is an oriented cobordism between X and X′, i.e. ∂W = (−X) ⨿ X′.

Definition 2.4. A G-framed cobordism F F F between fff and fff′ is said to be admissible if the following conditions (2.4.1)–(2.4.3) are fulfilled.

(2.4.1) dim Sing(G, W) ≦ k + 1.

(2.4.2) The inclusion j : X → W (resp. j′ : X′ → W) induces bijections Π(G, X, k) → Π(G, W, k + 1) and Π(G, X, k − 1) → Π(G, W, k) (resp. Π(G, X′, k) → Π(G, W, k + 1), and Π(G, X′, k − 1) → Π(G, W, k)).

We use W_{γ} for W_{j_{#}γ} (resp. also for W_{j′_{#}γ}) if ℓ = k − 1 or k and γ ∈ Π(G, X, ℓ) (resp. Π(G, X′, ℓ)). Moreover we usually identify Θ(G, X′, ℓ) with Θ(G, X, ℓ) via the induced maps when ℓ = k − 1 or k.

(2.4.3) For each γ ∈ Θ(G, X, k) = Θ(G, X′, k), W_{γ} is an oriented cobordism between X_{γ} and X′_{γ}.

Let Cob Cob Cob(G;fff,fff′) denote the set of all admissible, G-framed cobordisms between fff and fff′.

8 MASAHARU MORIMOTO

Lemma 2.5. Let fff, fff′ and F F F ∈ Cob Cob Cob(G;fff,fff′) be as above. Then F F F is G-framed cobordant rel. ∂W ∪ Sing(G, W) (by G-surgery of dimension ≦ k − 1 on Free(G, W) \ ∂W) to a k-connected cobordism F F F′ ∈ Cob Cob Cob(G;fff,fff′).

Proof. This is clear form the fact that Sing(G, W) + (k − 1) < dim W. □

Suppose that fff, fff′ and F F F are k-connected. Then the next sequence for the coefficient ring ℤ is exact:

0 → K_{k+1}(W) → K_{k+1}(W, ∂W) ⟶^{∂} K_{k}(∂W) → K_{k}(W) → K_{k}(W, ∂W) → 0.

By the universal coefficient theorem, the sequence for the coefficient ring R is also exact. Furthermore suppose K_{k}(W, ∂W; R) = 0 = K_{k+1}(W; R) and regard K_{k+1}(W, ∂W; R) as a submodule of K_{k}(∂W; R) via the connecting homomorphism ∂. The restriction ∂F F F of F F F to the boundary determines a special quadratic (A A A, Θ(G, X, k))-module

\overline{M M M}_{∂F F F} = (K_{k}(∂W; R), B_{∂F} , q_{∂F F F} , \overline{α}_{∂F})

as follows. The map

B_{∂F} : K_{k}(∂W; R) × K_{k}(∂W; R) → R[G]

is the equivariant intersection form and the map

q_{∂F F F} : K_{k}(∂W; R) → R[G \ S(G, X)]

is the equivariant selfintersection form. Let

InterSec_{∂W} : K_{k}(∂W; R) × K_{k}(∂W; R) → R

denote the ordinary intersection form. Then, by definition

B_{∂F} (x, y) = ∑_{g∈G} InterSec_{∂W} (x, g^{−1}y) g (x, y ∈ K_{k}(∂W; R)).

Immediately B_{∂F} = (−B_{f}) ⊕ B_{f′} holds. Similarly, q_{∂F F F} = (−q_{f f f}) ⊕ q_{f f f}′ also holds. The G-map \overline{α}_{∂F F F} : Θ(G, X, k) → K_{k}(∂W; R) is defined by

\overline{α}_{∂F} (γ) = π_{∂F} (j_{∂W_{γ}} _{∗}(∂[W_{γ}])) (γ ∈ Θ(G, X, k)),

where [W_{γ}] ∈ H_{k+1}(W_{γ}, ∂W_{γ}; R) is the orientation class, j_{∂W_{γ}} is the inclusion map ∂W_{γ} → ∂W, and π_{∂F} : H_{k}(∂W; R) → K_{k}(∂W; R) is the canonical projection. Thus we obtain the commutative diagram:

Θ(G, X, ℓ)  ⟶^{\overline{α}_{∂F}}  K_{k}(∂W; R)
diagonal↓  ↓^{canonical iso.}
Θ(G, X, ℓ) × Θ(G, X′, ℓ) ⟶^{−α_{f} ×α_{f′}} K_{k}(X; R) ⊕ K_{k}(X′; R) .

Remark 2.6. In [2, Lemma 7.1] we invoked Y ∈ Mnf Mnf Mnf_{cl,sg}^{n}(G)_{1}. But it is superfluous and the condition Y ∈ Mnf Mnf Mnf_{cl}^{n}(G)_{1} suffices to obtain the conclusion in [2, Lemma 7.1], namely to guarantee M M M_{f f f} ∈ SQ SQ SQ(A A A, Θ(G, X, k))_{C} ( for the definition of M M M_{f f f}, see [2, pp. 286–288]).

Lemma 2.7. Let fff and fff′ be as above. Suppose that f : X → Y and f′ : X′ → Y are k-connected. Let F F F ∈ Cob Cob Cob(G;fff,fff′) be as above. Suppose that F : W → I × Y is k-connected, and

K_{k}(W, ∂W; R) = 0 = K_{k+1}(W; R).

Then the following hold.

(2.7.1) \overline{M M M}_{∂F F F} = −M M M_{f f f} ⊕ M M M_{f f f}′.

(2.7.2) The submodule K_{k+1}(W, ∂W; R) of K_{k}(∂W; R) includes Im(\overline{α}_{∂F}).

(2.7.3) The submodule K_{k+1}(W, ∂W; R) is totally isotropic in \overline{M M M}_{∂F F F}, namely B_{∂F} and q_{∂F F F} are trivial on K_{k+1}(W, ∂W; R).

Proof. (2.7.1): The conclusion immediately follows from ∂W = (−X) ⨿ X and the definition of B_{∂F}, q_{∂F F F} and \overline{α}_{∂W}.

(2.7.2): The conclusion is obtained from observation of the following commutative diagram.

H_{k+1}(W_{γ}, ∂W_{γ}; R)  ⟶^{j_{Wγ} _{∗}} H_{k+1}(W, ∂W; R) ⟶^{π_{F}} K_{k+1}(W, ∂W; R)
↓^{∂}  ↓^{∂}  ↓^{∂}
H_{k}(∂W_{γ}; R)  ⟶^{j_{∂Wγ} _{∗}} H_{k}(∂W; R) ⟶^{π_{∂F}} K_{k}(∂W; R)
∥
H_{k}(−X_{γ}; R) ⊕ H_{k}(X′_{γ}; R)  .

(2.7.3): Now we prove the last conclusion. Here the coefficient ring of homology groups is R. It is well known that the ordinary intersection form satisfies

InterSec_{∂W} (K_{k+1}(W, ∂W), K_{k+1}(W, ∂W)) = 0.

Since K_{k+1}(W, ∂W) is G-invariant, it follows that

B_{∂F} (K_{k+1}(W, ∂W), K_{k+1}(W, ∂W)) = 0.

Let us decompose G to a disjoint union

G = {1} ⨿ G(2) ⨿ G(≧ 3)_{h} ⨿ G(≧ 3)_{h}^{−1}

such that G(≧ 3)_{h} is a subset of G consisting of elements of order ≧ 3, and G(≧ 3)_{h}^{−1} = {g^{−1} | g ∈ G(≧ 3)_{h}}. Let

\begin{matrix}G^{λ}(2) = \{g ∈ G(2) | w_{X}(g) = −λ\} and \\ G^{−λ}(2) = \{g ∈ G(2) | w_{X}(g) = λ\}. \end{matrix}

Then we can regard the map q_{∂F F F} : K_{k}(∂W) → R[G \ S]/Λ as a map

K_{k}(∂W) ⟶ R/(1 − λ)[{1}] ⊕ R/2R[G^{λ}(2) \ Q] ⊕ R[G^{−λ}(2) \ S] ⊕ R[G(≧ 3)_{h}].

(cf. [2, p.274]). In the case where g ∈ G(≧ 3)_{h}, ε_{g}(q_{∂W} (x)) = 0 for any x ∈ K_{k+1}(W, ∂W) because

ε_{g}(q_{∂W} (x)) = InterSec_{∂W} (x, g^{−1}x).

Let g ∈ {1} ∪ (G^{λ}(2) \ Q) ∪ (G^{−λ}(2) \ S). Then one has

ε_{g}(q_{∂F F F} (x)) = ε_{g}(q_{∂(res_{⟨g⟩}^{G}F F F)}(x))

for any x ∈ K_{k}(∂W) = K_{k}(res_{⟨g⟩}^{G}∂W). However, we have

dim(Sing_{⟨g⟩}(res_{⟨g⟩}^{G}∂W)) + (k + 1) ≦ {(k − 2) + 1} + (k + 1)
= 2k
< dim(res_{⟨g⟩}^{G}∂W).

(In other words, res_{⟨g⟩}^{G}∂W satisfies the strong gap condition.) In this case, a similar argument to Wall [4, p.53, lines 7–11] proves that q_{res_{⟨g⟩}^{G}∂F F F} (x) = 0 for all x ∈ K_{k+1}(res_{⟨g⟩}^{G}W, ∂(res_{⟨g⟩}^{G}W)). Consequently we get q_{∂F F F} (x) = 0 for all x ∈ K_{k+1}(W, ∂W). □

3. Main results

Throughout this section let C denote the category ((proj)), ((s-free)) or ((free)). We begin the section with a lemma which is a key to Theorems 3.2 and 3.3.

Lemma 3.1 Let X, X′ ∈ Mnf Mnf Mnf_{cl,sg}^{n}(G)_{0} and Y ∈ Mnf Mnf Mnf_{cl}^{n}(G)_{1} such that n = 2k ≧ 6 is even and let fff = (f : X → Y, b : T (X) ⊕ f^{∗}η → f^{∗}(ξ ⊕ η)) and fff′ = (f′ : X′ → Y, b′ : T (X′) ⊕ f′^{∗}η → f′^{∗}(ξ ⊕ η)) be degree 1, k-connected, G-framed maps such that K_{k}(X; R) and K_{k}(X′; R) are modules in C. Further let F F F = (F : (W ∂W) → (I × Y, ∂I × Y), B : T (W) ⊕ (p_{Y} ∘ F)^{∗}η → (p_{Y} ∘ F)^{∗}(ε_{Y} (ℝ) ⊕ ξ ⊕ η)) be an admissible, G-framed cobordism between fff and fff′. Then F F F is G-framed cobordant rel. X ∪ Sing(G, W) (by G-surgery of dimension ≦ k − 1 and k-dimensional handle subtraction in Free(G, W)) to a k-connected, admissible, G-framed cobordism

\begin{matrix}F F F′ = (F′ : & (W′, ∂W′) → (I × Y, ∂I × Y), \\ B′ : T (W) ⊕ (p_{Y} ∘ F′)^{∗}η → (p_{Y} ∘ F′)^{∗}(ε_{Y} (ℝ) ⊕ ξ ⊕ η)) \end{matrix}

such that for the degree 1, G-framed map fff′′ = (f′′ : X′′ → Y, b′′ : T (X′′) ⊕ f′′^{∗}η → f′′^{∗}(ξ ⊕ η)) obtained by

∂F F F′ = (−fff) ∐fff′′,

the following hold:

(3.1.1) f′′ : X′′ → Y is k-connected,

(3.1.2) K_{k}(W′, ∂W′; ℤ) = 0 = K_{k+1}(W′; ℤ) and

(3.1.3) M M M_{f′′} ≅ M M M_{f′} ⊕ 𝕄(x_{1}, y_{1}, β_{1}) ⊕ ⋯ ⊕ 𝕄(x_{ℓ}, y_{ℓ}, β_{ℓ}) in SQ SQ SQ^{λ}(A A A, Θ(G, X, k))_{C} for some nonnegative integer ℓ and β_{i} = (c_{i}, r_{i}), i = 1, …, ℓ, consisting of c_{i} : Θ(G, X, k) → ℤ and r_{i} : S(G, X) → ℤ such that c_{i}(γ) ≡ r_{i}(g) mod 2ℤ for γ ∈ Θ(G, X, k) and g ∈ S(G, X) with g ∈ ρ(γ).

In the lemma, Θ(G, X, k) is identified with S(G, X) as in Section 2, and 𝕄(x_{i}, y_{i}, β_{i}) are the metabolic planes define in [2, Definition 5.1] (cf. [2, Lemma 5.5]). The lemma will be proven in Section 4.

Theorem 3.2. Let R = ℤ_{(m)} (m = 0, 1 or a prime), let X, X′ ∈ Mnf Mnf Mnf_{cl,sg}^{n}(G)_{0} and Y ∈ Mnf Mnf Mnf_{cl}^{n}(G)_{1} such that n = 2k ≧ 6 is even, and let fff = (f : X → Y, b : T (X) ⊕ f^{∗}η → f^{∗}(ξ ⊕ η)) and fff′ = (f′ : X′ → Y, b′ : T (X′) ⊕ f′^{∗}η → f′^{∗}(ξ ⊕ η)) be degree 1, k-connected, G-framed maps such that K_{k}(X; R) and K_{k}(X′; R) are modules in C. If there exists admissible, G-framed cobordism F F F = (F, B) between fff and fff′ such that F : W → I × Y satisfies F_{m}-Condition (or P_{0}-Condition and K̃_{0}(ℤ[G]) = 0) then σ(fff) = σ(fff′) in W_{n}(A A A, Θ(G, X, k))_{C}.

For the definition of W_{n}(A A A, Θ(G, X, k))_{C}, see [2, p.280, (4.4)].

Proof. By Lemma 3.1, there exist a degree 1, G-framed map fff′′ and a cobordism F F F′ ∈ Cob Cob Cob(G;fff,fff′′) satisfying Conditions (3.1.1)–(3.1.3). The equality σ(fff′) = σ(fff′′) in W_{n}(A A A, Θ(G, X, k))_{C} is clear from (3.1.3). Since K_{k}(W′; R) is stably free over R[G] (cf. Remark 2.3), the universal coefficient theorem and the Poincaré-Lefschetz duality theorem guarantee that K_{k+1}(W′, ∂W′; R) is also stably free over R[G]. In addition, it follows that

K_{k}(∂W′; R) ≅ K_{k}(W′; R) ⊕ K_{k+1}(W′, ∂W′; R) and
rank_{R[G]}K_{k+1}(W′, ∂W′; R) = rank_{R[G]}K_{k}(W′; R).

Thus, we can deduce form (2.7.2) and (2.7.3) that K_{k+1}(W′, ∂W′; R) is a stably free Lagrangian of \overline{M M M}_{∂F F F′}, hence

[\overline{M M M}_{∂F F F′}] = 0 ∈ W_{n}(A A A, Θ(G, X, k))_{((s−free))} = W_{n}(A A A, Θ(G, X, k))_{((free))}.

By (2.7.1), we get σ(fff) = σ(fff′′) in W_{n}(A A A, Θ(G, X, k))_{C}. □

Let R = ℤ_{(m)}, let X, X′ ∈ Mnf Mnf Mnf_{cl,sg}^{n}(G)_{0} and Y ∈ Mnf Mnf Mnf_{cl}^{n}(G)_{1}, where n = 2k ≧ 6, and let fff = (f, b) be a degree 1, G-framed map from X to Y. Suppose that f : X → Y satisfies F_{m}-Condition (or P_{0}-Condition and K̃_{0}(ℤ[G]) = 0). Since Sing(G, X) ≦ k, fff is G-framed cobordant rel. Sing(G, X) to fff′ = (f′, b′) such that f′ : X′ → Y is k-connected. (Such a

EQUIVARIANT SURGERY 13

G-framed cobordism can be obtained by G-surgery of dimension ≦ k − 1 on Free(G, X).) Then by F_{m}-Condition (or P_{0}-Condition and K̃_{0}(ℤ[G]) = 0), K_{k}(X′; R) is stably free over R[G]. By [2, Lemma 7.1], we obtain the element

σ(fff′) = [M M M_{f f f}′] ∈ W_{n}(A A A, Θ(G, X, k))_{((free))}.

Now define

σ(fff) := σ(fff′) ∈ W_{n}(A A A, Θ(G, X, k))_{((free))}.

Theorem 3.2 guarantees that σ(fff) is defined independently of the choice of fff′.

Theorem 3.3. Let R = ℤ_{(m)}, where m = 0, 1 or a prime and let X, X′ ∈ Mnf Mnf Mnf_{cl,sg}^{n}(G)_{0} and Y ∈ Mnf Mnf Mnf_{cl}^{n}(G)_{1} such that n = 2k ≧ 6 is even, and let fff = (f : X → Y, b : T (X) ⊕ f^{∗}η → f^{∗}(ξ ⊕ η)) be a degree 1, G-framed map such that f : X → Y satisfies F_{m}-Condition (or P_{0}-Condition and K̃_{0}(ℤ[G]) = 0) (cf. Remark 2.3). Then σ(fff) = 0 in W_{n}(A A A, Θ(G, X, k))_{((free))} if and only if fff can be converted by G-surgery of isotropy type {1} and of dimension ≦ k to a degree 1, k-connected, G-framed map fff′ = (f′ : X′ → Y, b′ : T (X′) ⊕ f′^{∗}η → f′^{∗}(ξ ⊕ η)) (thus fff ∼ fff′ rel. Sing(G, X)) such that f′ : X′ → Y is an R-homology equivalence.

Proof. The ‘if part’ of the claim is obvious from the definition of σ(fff). We have to prove the ‘only if part’, namely σ(fff) = 0 ⟹ fff ∼ fff′ rel. Sing(G, X) as in the claim. Since Sing(G, X) ≦ k, there exists a k-connected, degree 1, G-framed map fff′′ = (f′′ : X′′ → Y, b′′ : T (X′′) ⊕ f′′^{∗}η → f′′^{∗}(ξ ⊕ η)) such that fff ∼ fff′′ rel. Sing(G, X) by G-surgery of dimension ≦ k − 1. Then apply Theorem 7.3 of [2] to fff′′ and obtain a desired fff′. □

4. Proof of Lemma 3.1

In this section we prove Lemma 3.1 using equivariant handle subtraction.

Firstly, using Lemma 2.5, we adjust F F F so that F : W → I × Y is k-connected. Fix a small, closed, n-dimensional disk D_{0} in general position of Free(G, X′). Thus we may assume that D_{0} ∩ gD_{0} = ∅ if g ∈ G \ {1}. For each γ ∈ Θ(G, X, k) = Θ(G, X′, k), let us take a point x_{γ} ∈ X′_{γ} such that G_{x_{γ}} = ρ_{X′}^{G}(X′_{γ}) (= ρ(γ)) and small closed, n-dimensional, G_{x_{γ}} -invariant disks D_{γ} with center x_{γ} in X′ \ GD_{0} so that gD_{γ} ∩ g′D_{γ′} = ∅ unless γ = γ′ and gG_{x_{γ}} = g′G_{x_{γ′}} (i.e. gx_{γ} = g′x_{γ′}), where γ and γ′ range over Θ(G, X, k) and g and g′

14 MASAHARU MORIMOTO

range over ∈ G. Here GD_{γ} becomes a G-equivariant tubular neighborhood of Gx_{γ}. Take connecting tubes T_{γ} from ∂D_{0} to ∂D_{γ} in general position of

Free(G, X′) \ G(Int(D_{0})) \ ⋃_{γ∈Θ(G,X,k)}G(Int(D_{γ})).

Then,

D := D_{0} ∪ ⋃_{γ∈Θ(G,X,k)} (D_{γ} ∪ T_{γ})

is homeomorphic to the n-dimensional disk D^{n}. Since K_{k}(W; ℤ) → K_{k}(W, ∂W; ℤ) is surjective, we can take finitely many embeddings

h_{i} : S^{k} → Int(W),

i = 1, ⋯, ℓ say, which together generate K_{k}(W, ∂W; ℤ). Without loss of generality, we can assume that gh_{i} and g′h_{j} are disjoint unless i = j. For each i, take a path

p_{i} : [0, 1] ↪ Free(G, W)

in general position, such that for t ∈ [0, 1], p_{i}(t) ∈ X′ ⇔ t = 0 and p_{i}(t) ∈ Im(h_{i}) ⇔ t = 1, and p_{i}(0) ∈ IntD_{0}. Choose a thin band u_{i} : [0, 1]×D^{k} ↪ Free(G, W) along the path p_{i} (i.e., u_{i}(t, 0) = p_{i}(t) for all t ∈ [0, 1]) such that for t ∈ [0, 1] and z ∈ D^{k}, u_{i}(t, z) ∈ X′ ⇔ t = 0 and u_{i}(t, z) ∈ Im(h_{i}) ⇔ t = 1, and u_{i}(0, D^{k}) ⊂ Int(D_{0}). Then,

u_{i}([0, 1] × S^{k−1}) ∪ Im(h_{i}) \ u_{i}(1, Int(D^{k}))

gives an embedding

h′_{i} : (D^{k}, S^{k−1}) → (W, Free(G, X′)).

Figure 4.1.

EQUIVARIANT SURGERY 15

Here the restriction ∂h′_{i} : S^{k−1} → Free(G, X′) of h′_{i} is a trivial embedding close to p_{i}(0). We may suppose that gh_{i} and g′h_{j} are disjoint unless i = j. For γ ∈ Θ(G, X, k), let c_{i}(γ) denote the ordinary intersection number of W_{γ} and Im(h_{i}). Thus we obtain a map c_{i} : Θ(G, X, k) → ℤ. Without loss of generality, we can assume that for γ ∈ Θ(G, X, k), h_{i} meets with W_{γ} at distinct m-points A(i, γ, 1), ⋯, A(i, γ, m) of isotropy subgroup ρ(γ), where m = |c_{i}(γ)|. Take distinct points B(i, γ, 1), ⋯, B(i, γ, m) in h′_{i}(S^{k−1}) and C(i, γ, 1), ⋯, C(i, γ, m) in Int(D_{γ}) ∩ X′_{γ}^{=ρ(γ)}. Furthermore take disjoint embedded triangles △(i, γ, j) = △A(i, γ, j)B(i, γ, j)C(i, γ, j) (which is homeomorphic to D^{2}) in W so that

\overline{A(i, γ, j)B(i, γ, j)} ⊂ Im(h′_{i}),
\overline{A(i, γ, j)C(i, γ, j)} ⊂ W_{γ}^{=ρ(γ)}, and
\overline{B(i, γ, j)C(i, γ, j)} ⊂ D.

Figure 4.2.

We may assume that g(Int(△(i, γ, j))) ∩ g′(Int(△(i′, γ′, j′))) = ∅ unless γ = γ′, j = j′, i = i′ and g = g′. Moreover we may assume that △(i, γ, j) is perpendicular to W_{γ}, X′ and Im(h′_{i}). For each i, delete the intersection points A(i, γ, j) of h′_{i} with W_{γ} along the triangle △(i, γ, j) by [3, Lemma 1.2], where γ ∈ Θ(G, X, k) and j = 1, ⋯, m, and obtain an embedding

h′′_{i} : (D^{k}, S^{k−1}) → (Free(G, W), Free(G, X′)).

Figure 4.3.

Let c : Θ(G, X, k) → ℤ be a map. Applying the construction of [2, p.298, ℓ.−10 – p.299, ℓ.18] to the data X replaced by X′, S_{c+} = {s ∈ S | c(s) > 0}, S_{c−} = {s ∈ S | c(s) < 0}, S′_{c} = S_{c+} ∪ S_{c−} and Γ_{c} = {(s, c) | s ∈ S′, 1 ≦ i ≦ |c(s)|}, we obtain an embedding

\begin{matrix}h_{D_{c}} : (D^{k} ≅) D_{c} → X′ and \\ h_{c} = h_{D_{c}}|_{∂D_{c}} : (S^{k−1} ≅) ∂D_{c} → Free(G, X′). \end{matrix}

With this notation, the restriction ∂h′′_{i}| : S^{k−1} → Free(G, X′) of h′′_{i} is the connected sum of ∂h′_{i} and h_{c_{i}}.

Thickening h′′_{i}, we obtain embeddings

H′′_{i} : (D^{k} × D^{k+1}, S^{k−1} × D^{k+1}) → (Free(G, W), Free(G, X′)), i = 1, ⋯, ℓ.

Now we may suppose that gH′′_{i} and g′H′′_{j} are disjoint unless g = g′ and i = j. Further we may suppose that F (Im(H′′_{i})) is a point in {1} × Y. Set

V = ⋃_{i=1}^{ℓ} G(Im(H′′_{i})),

∂_{+}V = ⋃_{i=1}^{ℓ} G(H′′_{i}(S^{k−1} × D^{k+1})),

W′ = Closure(W \ V),

X′′ = (X′ ∪ V) \ ⋃_{i=1}^{ℓ} G(H′′_{i}(D^{k} × Int(D^{k+1}))),

F′ = F |_{W′} : W′ → I × Y,

B′ = B|_{W} : T (W′) ⊕ (p_{Y} ∘ F′)^{∗}η → (p_{Y} ∘ F′)^{∗}(ε_{Y} (ℝ) ⊕ ξ ⊕ η), f′′ = F |_{X′′} : X′′ → {1} × Y = Y.

By [2, Proposition 10.1], we may assume that B′ has the form on X′′:

id_{X′′} ⊕ b′′ = B′|_{X′′} : ε_{X′′}(ℝ) ⊕ T (X′′) ⊕ f′′^{∗}η → ε_{X′′}(ℝ) ⊕ f′′^{∗}(ξ ⊕ η).

It remains to check that the above F F F′ = (F′, B′) and fff′′ = (f′′, b′′) are G-framed maps desired in Lemma 3.1. By excision, K_{k}(W′, ∂W′; R) = K_{k}(W, ∂W ∪ V; R). There is an exact sequence:

K_{k}(∂W ∪ V, ∂W; R) ⟶ K_{k}(W, ∂W; R) ⟶ K_{k}(W, ∂W ∪ V; R) ⟶ 0.

Since the first arrow is surjective, K_{k}(W, ∂W ∪ V; R) = 0. Thus we get

K_{k}(W′, ∂W′; R) = 0.

By the universal coefficient theorem, K^{k}(W′, ∂W′; R) = 0. The Poincaré-Lefschetz duality implies K_{k+1}(W′; R) = 0. Note that fff′′ can be obtained from fff′ by G-surgery along the embeddings ∂h′′_{1}, ⋯, ∂h′′_{ℓ} : S^{k−1} → X′. These embeddings are isotopic in general position of X′ to h_{c_{1}}, ⋯, h_{c_{ℓ}}, respectively. By the same argument as [2, §9, p.299, ℓ.−12 – p.300, ℓ.−1], we obtain

M M M_{f f f}′′ ≅ M M M_{f f f}′ ⊕ 𝕄(x_{1}, y_{1}, β_{1}) ⊕ ⋯ ⊕ 𝕄(x_{ℓ}, y_{ℓ}, β_{ℓ}). □

18 MASAHARU MORIMOTO

References

[1] Bak, A. and Morimoto, M., Equivariant surgery on compact manifolds with half dimensional singular sets, Preprint 1992.

[2] Bak, A. and Morimoto, M., Equivariant surgery with middle dimensional singular sets. I, Forum Math. 8 (1996), 267–302.

[3] Morimoto, M., Bak groups and equivariant surgery II, K-Theory 3 (1990), 505–521.

[4] Wall, C. T. C., Surgery on Compact Manifolds, Academic Press, London, 1970.