## Induction theorems of surgery obstruction groups

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- by Masaharu Morimoto
- Trans. Amer. Math. Soc.
**355**(2003), 2341-2384 - DOI: https://doi.org/10.1090/S0002-9947-03-03266-5
- Published electronically: February 4, 2003
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## 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.## References

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## Bibliographic Information

**Masaharu Morimoto**- Affiliation: Department of Environmental and Mathematical Sciences, Faculty of Environmental Science and Technology, Okayama University, Okayama, 700-8530 Japan
- Email: morimoto@ems.okayama-u.ac.jp
- Received by editor(s): January 1, 2002
- Published electronically: February 4, 2003
- Additional Notes: Partially supported by a Grant-in-Aid for Scientific Research (Kakenhi)
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**355**(2003), 2341-2384 - MSC (2000): Primary 19G12, 19G24, 19J25; Secondary 57R67
- DOI: https://doi.org/10.1090/S0002-9947-03-03266-5
- MathSciNet review: 1973993

Dedicated: Dedicated to Professor Anthony Bak for his sixtieth birthday