### Abstract

Let G be a finite group. It is well known that a Mackey functor [H → M(H)} is a module over the Burnside ring functor {H → Ω(H)}. where H ranges over the set of all subgroups of G. For a fixed homomorphism w : G → {-1,1}, the Wall group functor {H → L_{n}^{h}(ℤ[H],w|_{H}}} is not a Mackey functor if w is nontrivial. In this paper, we show that the Wall group functor is a module over the Burnside ring functor as well as over the Grothendieck-Witt ring functor {H → GW_{0}(ℤ, H)}. In fact, we prove a more general result, that the functor assigning the equivariant surgery obstruction group on manifolds with middle-dimensional singular sets to each subgroup of G is a module over the Burnside ring functor as well as over the special Grothendieck-Witt ring functor. As an application, we obtain a computable property of the functor described with an element in the Burnside ring.

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

Pages (from-to) | 2341-2384 |

Number of pages | 44 |

Journal | Transactions of the American Mathematical Society |

Volume | 355 |

Issue number | 6 |

DOIs | |

Publication status | Published - Jun 2003 |

### Fingerprint

### Keywords

- Burnside ring
- Equivariant surgery
- Grothendieck group
- Induction
- Restriction
- Witt group

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Induction theorems of surgery obstruction groups.** / Morimoto, Masaharu.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 355, no. 6, pp. 2341-2384. https://doi.org/10.1090/S0002-9947-03-03266-5

}

TY - JOUR

T1 - Induction theorems of surgery obstruction groups

AU - Morimoto, Masaharu

PY - 2003/6

Y1 - 2003/6

N2 - Let G be a finite group. It is well known that a Mackey functor [H → M(H)} is a module over the Burnside ring functor {H → Ω(H)}. where H ranges over the set of all subgroups of G. For a fixed homomorphism w : G → {-1,1}, the Wall group functor {H → Lnh(ℤ[H],w|H}} is not a Mackey functor if w is nontrivial. In this paper, we show that the Wall group functor is a module over the Burnside ring functor as well as over the Grothendieck-Witt ring functor {H → GW0(ℤ, H)}. In fact, we prove a more general result, that the functor assigning the equivariant surgery obstruction group on manifolds with middle-dimensional singular sets to each subgroup of G is a module over the Burnside ring functor as well as over the special Grothendieck-Witt ring functor. As an application, we obtain a computable property of the functor described with an element in the Burnside ring.

AB - Let G be a finite group. It is well known that a Mackey functor [H → M(H)} is a module over the Burnside ring functor {H → Ω(H)}. where H ranges over the set of all subgroups of G. For a fixed homomorphism w : G → {-1,1}, the Wall group functor {H → Lnh(ℤ[H],w|H}} is not a Mackey functor if w is nontrivial. In this paper, we show that the Wall group functor is a module over the Burnside ring functor as well as over the Grothendieck-Witt ring functor {H → GW0(ℤ, H)}. In fact, we prove a more general result, that the functor assigning the equivariant surgery obstruction group on manifolds with middle-dimensional singular sets to each subgroup of G is a module over the Burnside ring functor as well as over the special Grothendieck-Witt ring functor. As an application, we obtain a computable property of the functor described with an element in the Burnside ring.

KW - Burnside ring

KW - Equivariant surgery

KW - Grothendieck group

KW - Induction

KW - Restriction

KW - Witt group

UR - http://www.scopus.com/inward/record.url?scp=0038030886&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038030886&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-03-03266-5

DO - 10.1090/S0002-9947-03-03266-5

M3 - Article

AN - SCOPUS:0038030886

VL - 355

SP - 2341

EP - 2384

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 6

ER -