On the torus cobordant cohomology spheres
type: Item
instance of: scholarly article
title: On the torus cobordant cohomology spheres
zbMATH Open document ID: 1179.57047
author: Dogan Dönmez
author: Ali Arslan Özkurt
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 2009-09-04T00:00:00Z
review text: Smith has proposed the question ``Is it true that for any smooth action of \(G\) on a homotopy sphere with exactly two fixed points, the tangent bundle \(G\)-modules at these points are isomorphic?''. It turns out that in some cases the answer is yes and in other cases the answer is no. A positive answer was obtained by [\textit{M. F. Atiyah} and \textit{R. Bott}, Ann. Math. 88, 451--491 (1968; Zbl 0167.21703)] under the hypothesis that the group \(G\) is connected. The paper under review considers a question which is a variation of Smith's question. In the introduction the authors describe several works which illustrate when Smith's question has a positive answer and when it has negative answer. The authors show that the same positive result as the one shown by Atiyah and Bott holds if the homotopy sphere is replaced by a smooth integral cohomology sphere. The main result of the paper is: Theorem 1.1. Let \(G\) be a compact connected Lie group, \(\Sigma\) a smooth integral cohomology \(G\)-sphere with exactly two fixed points namely \(p\) and \(q\). Then the tangent representations of \(G\) at \(T_p\Sigma\) and \(T_q\Sigma\) are linearly equivalent. As main steps to obtain the result first the authors reduce to the case where the group is the torus, and then they compare the cohomology of the fixed points of a \(T\)-actions on a cohomology integral sphere \(X\), which is a \(T\)-\(c\)-cobordism, with the fixed points of each boundary component of \(X\). The equivariant Euler class and integral weight values of the action play an important technical rôle.
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 57Q20
Mathematics Subject Classification ID: 57Q60
zbMATH DE Number: 5601778
zbMATH Keywords: equivariant cohomology
zbMATH Keywords: integral cohomology sphere
zbMATH Keywords: Smith's question
zbMATH Keywords: Lie groups
zbMATH Keywords: linear actions
zbMATH Keywords: equivariant Euler class
reviewed by: Daciberg Lima Gonçalves
MaRDI profile type: Publication
full work available at URL: s12044-009-0010-7
OpenAlex ID: W2007276219
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DOI: 10.1007/S12044-009-0010-7
description: scientific article; zbMATH DE number 5601778