Induction theorems of surgery obstruction groups

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6 Citations (Scopus)

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 → 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.

Original languageEnglish
Pages (from-to)2341-2384
Number of pages44
JournalTransactions of the American Mathematical Society
Volume355
Issue number6
DOIs
Publication statusPublished - Jun 1 2003

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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