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

The fixed-point construction in equivariant bordism


Author: Russell J. Rowlett
Journal: Trans. Amer. Math. Soc. 246 (1978), 473-481
MSC: Primary 57R85
MathSciNet review: 515553
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Abstract: Consider the bordism $ {\Omega _ {\ast}}(G)$ of smooth G-actions. If K is a subgroup of G, with normalizer NK, there is a standard $ NK/K$-action on $ {\Omega _ {\ast}}(K)$(All, Proper). If M has a smooth G-action, a tubular neighborhood of the fixed set of K in M represents an element of $ {\Omega _ {\ast}}(K){({\text{All, Proper}})^{NK/K}}$. One thus obtains the ``fixed point homomorphism'' $ \phi $ carrying $ {\Omega _ {\ast}}(G)$ to the sum of the $ {\Omega _ {\ast}}(K){({\text{All, Proper}})^{NK/K}}$, summed over conjugacy classes of subgroups K. Let P be the collection of primes not dividing the order of G. We show that the P-localization of $ \phi $ is an isomorphism, and give several applications.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0515553-5
PII: S 0002-9947(1978)0515553-5
Keywords: Finite group actions, equivalent bordism, fixed point homomorphism
Article copyright: © Copyright 1978 American Mathematical Society