Induction theorems of surgery obstruction groups

Abstract:

Let $G$ be a finite group. It is well known that a Mackey functor $\{ H \mapsto M(H) \}$ is a module over the Burnside ring functor $\{ H \mapsto \Omega (H) \}$, where $H$ ranges over the set of all subgroups of $G$. For a fixed homomorphism $w : G \to \{ -1, 1 \}$, the Wall group functor $\{ H \mapsto L_n^h ({\mathbb Z}[H], w|_H) \}$ 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 $\{ H \mapsto {\mathrm {GW}}_0 ({\mathbb Z}, H) \}$. 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 $G$ 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.