### 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 1 2003 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics