Induction theorems of surgery obstruction groups

Author:

Morimoto Masaharu

Abstract

Let G G be a finite group. It is well known that a Mackey functor { H M ( H ) } \{ H \mapsto M(H) \} is a module over the Burnside ring functor { H Ω ( H ) } \{ H \mapsto \Omega (H) \} , where H H ranges over the set of all subgroups of G G . For a fixed homomorphism w : G { 1 , 1 } w : G \to \{ -1, 1 \} , the Wall group functor { H L n h ( Z [ H ] , w | H ) } \{ H \mapsto L_n^h ({\mathbb Z}[H], w|_H) \} is not a Mackey functor if w 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 G W 0 ( Z , H ) } \{ H \mapsto {\mathrm {GW}}_0 ({\mathbb 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 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.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

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