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A coincidence theorem for commuting involutions

Author: Pedro L. Q. Pergher
Journal: Proc. Amer. Math. Soc. 140 (2012), 2537-2541
MSC (2010): Primary 55M20; Secondary 57R75, 57R85
Published electronically: October 28, 2011
MathSciNet review: 2898715
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Abstract: Let $ M^{m}$ be an $ m$-dimensional, closed and smooth manifold, and let $ S, T:M^{m} \to M^{m}$ be two smooth and commuting diffeomorphisms of period $ 2$. Suppose that $ S \not = T$ on each component of $ M^{m}$. Denote by $ F_{S}$ and $ F_{T}$ the respective sets of fixed points. In this paper we prove the following coincidence theorem: if $ F_{T}$ is empty and the number of points of $ F_{S}$ is of the form $ 2p$, with $ p$ odd, then $ Coinc(S,T)=\{x \in M^{m} \ \vert \ S(x)=T(x) \}$ has at least some component of dimension $ m-1$. This generalizes the classic example given by $ M^{m}=S^{m}$, the $ m$-dimensional sphere, $ S(x_{0},x_{1},...,x_{m}) = (-x_{0},-x_{1},...,-x_{m-1},x_{m})$ and $ T$ the antipodal map.

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Additional Information

Pedro L. Q. Pergher
Affiliation: Departamento de Matemática, Universidade Federal de São Carlos, Caixa Postal 676, CEP 13.565-905, São Carlos, SP, Brazil

Keywords: Coincidence point, involution, characteristic class, projective space bundle, characteristic number, singular manifold, unoriented cobordism class, equivariant diffeomorphism
Received by editor(s): December 6, 2010
Received by editor(s) in revised form: February 16, 2011
Published electronically: October 28, 2011
Additional Notes: The author was partially supported by CNPq and FAPESP
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2011 American Mathematical Society