Induction Theorems of Surgery Obstruction Groups

Let G be a finite group. It is well known that a Mackey functor {H 7→ M(H)} is a module over the Burnside ring functor {H 7→ 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 7→ Ln(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 7→ 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.