TY - JOUR

T1 - Equivariant surgery with middle dimensional singular sets. II

T2 - Equivariant framed cobordism invariance

AU - Morimoto, Masaharu

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2001

Y1 - 2001

N2 - Let G be a finite group and let f : X → Y be a degree 1, G-framed map such that X and Y are simply connected, closed, oriented, smooth manifolds of dimension n = 2k ≧ 6 and such that the dimension of the singular set of the G-space X is at most k. In the previous article, assuming f is k-connected, we defined the G-equivariant surgery obstruction σ(f) in a certain abelian group. There it was shown that if σ(f) = 0 then f is G-framed cobordant to a homotopy equivalence f′ : X′ → Y. In the present article, we prove that the obstruction σ(f) is a G-framed cobordism invariant. Consequently, the G-surgery obstruction σ(f) is uniquely associated to f : X σ Y above even if it is not k-connected.

AB - Let G be a finite group and let f : X → Y be a degree 1, G-framed map such that X and Y are simply connected, closed, oriented, smooth manifolds of dimension n = 2k ≧ 6 and such that the dimension of the singular set of the G-space X is at most k. In the previous article, assuming f is k-connected, we defined the G-equivariant surgery obstruction σ(f) in a certain abelian group. There it was shown that if σ(f) = 0 then f is G-framed cobordant to a homotopy equivalence f′ : X′ → Y. In the present article, we prove that the obstruction σ(f) is a G-framed cobordism invariant. Consequently, the G-surgery obstruction σ(f) is uniquely associated to f : X σ Y above even if it is not k-connected.

KW - Cobordism invariant

KW - Equivariant surgery

KW - Quadratic module

KW - Surgery obstruction

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U2 - 10.1090/s0002-9947-01-02728-3

DO - 10.1090/s0002-9947-01-02728-3

M3 - Article

AN - SCOPUS:23044527020

SN - 0002-9947

VL - 353

SP - 2427

EP - 2440

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 6

ER -