### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 2427-2440 |

Number of pages | 14 |

Journal | Transactions of the American Mathematical Society |

Volume | 353 |

Issue number | 6 |

Publication status | Published - 2001 |

### Fingerprint

### Keywords

- Cobordism invariant
- Equivariant surgery
- Quadratic module
- Surgery obstruction

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Equivariant surgery with middle dimensional singular sets. II : Equivariant framed cobordism invariance.** / Morimoto, Masaharu.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 353, no. 6, pp. 2427-2440.

}

TY - JOUR

T1 - Equivariant surgery with middle dimensional singular sets. II

T2 - Equivariant framed cobordism invariance

AU - Morimoto, Masaharu

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

UR - http://www.scopus.com/inward/record.url?scp=23044527020&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23044527020&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:23044527020

VL - 353

SP - 2427

EP - 2440

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 6

ER -