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2024 | OriginalPaper | Chapter

Smith-Gysin Sequence

Authors : J. I. Royo Prieto, M. Saralegi-Aranguren, R. Wolak

Published in: Differential Geometric Structures and Applications

Publisher: Springer Nature Switzerland

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Abstract

Given a smooth semifree action of \(S^3\) on a manifold M, we have the Smith-Gysin sequence:
$$\begin{aligned} \cdots \rightarrow {H}^{^{*}}{\big ( M \big )} {\rightarrow } {H}^{^{*-3}}{\big ( M/S^3, M^{S^3} \big )} \oplus {H}^{^{*}}{\big ( M^{S^3} \big )} {\rightarrow } {H}^{^{*+1}}{\big ( M/S^3, M^{S^3} \big )} {\rightarrow } {H}^{^{*+1}}{\big ( M \big )}\rightarrow \cdots \end{aligned}$$
In this paper, we construct a Smith-Gysin sequence that does not require the semifree condition. This sequence includes a new term, referred to as the exotic term, which depends on the subset \(M^{S^1}\):
$$\begin{aligned} \cdots \rightarrow {H}^{^{*}}{\big ( M \big )} \rightarrow {H}^{^{*-3}}{\big ( M/S^3, \Sigma /S^3 \big )} \oplus {H}^{^{*}}{\big ( M^{S^3} \big )} \oplus \big ({H}^{^{*-2}}{\big ( M^{S^1} \big )}\big )^{-{\mathbb {Z}}_{_2}} \\ \rightarrow {H}^{^{*+1}}{\big ( M/S^3,M^{S^3} \big )} \rightarrow {H}^{^{*+1}}{\big ( M \big )} \rightarrow \cdots \end{aligned}$$
Here, \(\Sigma \subset M\) is the subset of points in M whose isotropy group is infinite. The group \(\mathbb {Z}_2\) acts on \(M^{S^1}\) by \(j \in S^3\).

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Footnotes
1
Recall that \({H}^{^{*}}{\big ( M,M^{S^3} \big )} = {H}^{^{*}}{\big ( M \backslash M^{S^3},T\backslash M^{S^3} \big )}\); here, T is a tubular neighborhood of the fixed point set \(M^{S^3}\).
 
2
We refer to [5, Sect. I] for the notions appearing in this presentation.
 
3
The result obtained by Bredon is more general than the one presented in this Remark, which only applies to smooth manifolds.
 
Literature
1.
Bredon, G. E.: Cohomology fibre spaces, the Smith-Gysin sequence, and orientation in generalized manifolds. Michigan Mathematical Journal, 10(4): 321–333, 1963MathSciNetCrossRef
2.
Bredon, G. E.: Introduction to compact transformation groups. Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46
3.
Eilenberg, S. and Moore, J. C.: Foundations of relative homological algebra. Mem. Amer. Math. Soc. No. 55:39, 1965MathSciNet
4.
Greub, W., Halperin, S. and Vanstone R.: Connections, curvature, and cohomology. Vol. I: De Rham cohomology of manifolds and vector bundles Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Pure and Applied Mathematics, Vol. 47-I
5.
Royo Prieto, J.I. and Saralegi-Aranguren, M.: The Gysin sequence for \(S^3\)-actions on manifolds. Publ. Math. Debrecen, 83(3): 275–289, 2013
6.
Saralegi, M.: A Gysin sequence for semifree actions of \(S^3\). Proc. Amer. Math. Soc., 118(4): 1335–1345, 1993
7.
Saralegi-Aranguren, M.: de Rham intersection cohomology for general perversities. Illinois J. Math., 49(3): 737–758 (electronic), 2005
8.
Verona, A.: Le théorème de de Rham pour les préstratifications abstraites. C. R. Acad. Sci. Paris Sér. A-B, 273: A886–A889, 1971
Metadata
Title
Smith-Gysin Sequence
Authors
J. I. Royo Prieto
M. Saralegi-Aranguren
R. Wolak
Copyright Year
2024
Book
Differential Geometric Structures and Applications

Print ISBN: 978-3-031-50585-0

Electronic ISBN: 978-3-031-50586-7

Copyright Year: 2024

DOI
https://doi.org/10.1007/978-3-031-50586-7_12

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