Induction theorems of surgery obstruction groups

Let be a finite group. It is well known that a Mackey functor is a module over the Burnside ring functor , where ranges over the set of all subgroups of . For a fixed homomorphism , the Wall group functor is not a Mackey functor if 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 . 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 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.