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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Equivariant surgery with middle dimensional singular sets. II: Equivariant framed cobordism invariance


Author: Masaharu Morimoto
Journal: Trans. Amer. Math. Soc. 353 (2001), 2427-2440
MSC (2000): Primary 57R67, 57R91, 19G24
Published electronically: January 16, 2001
MathSciNet review: 1814076
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Abstract:

Let $G$ be a finite group and let $f : X \to Y$ be a degree 1, $G$-framed map such that $X$ and $Y$ are simply connected, closed, oriented, smooth manifolds of dimension $n = 2k \geqq 6$ and such that the dimension of the singular set of the $G$-space $X$ is at most $k$. In the previous article, assuming $f$ is $k$-connected, we defined the $G$-equivariant surgery obstruction $\sigma (f)$ in a certain abelian group. There it was shown that if $\sigma (f) = 0$ then $f$ is $G$-framed cobordant to a homotopy equivalence $f' : X' \to Y$. In the present article, we prove that the obstruction $\sigma (f)$ is a $G$-framed cobordism invariant. Consequently, the $G$-surgery obstruction $\sigma (f)$ is uniquely associated to $f : X \to Y$ above even if it is not $k$-connected.


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Additional Information

Masaharu Morimoto
Affiliation: Department of Environmental and Mathematical Sciences, Faculty of Environmental Science and Technology, Okayama University, Tsushimanaka 3-1-1, Okayama, 700-8530 Japan
Email: morimoto@math.ems.okayama-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02728-3
PII: S 0002-9947(01)02728-3
Keywords: Equivariant surgery, surgery obstruction, cobordism invariant, quadratic module
Received by editor(s): October 12, 1999
Published electronically: January 16, 2001
Additional Notes: Research partially supported by Max-Plank-Institut für Mathematik in Bonn and also by Grant-in-Aid for Scientific Research
Dedicated: Dedicated to Professor Mamoru Mimura on his sixtieth birthday
Article copyright: © Copyright 2001 American Mathematical Society