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Induction theorems of surgery obstruction groups


Author: Masaharu Morimoto
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
Published electronically: February 4, 2003
MathSciNet review: 1973993
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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 $\{ H \mapsto L_n^h ({\mathbb Z}[H], w\vert _H) \}$ 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.


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Additional 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

DOI: https://doi.org/10.1090/S0002-9947-03-03266-5
Keywords: Induction, restriction, Burnside ring, Grothendieck group, Witt group, equivariant surgery
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)
Dedicated: Dedicated to Professor Anthony Bak for his sixtieth birthday
Article copyright: © Copyright 2003 American Mathematical Society

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