On the torus cobordant cohomology spheres
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.
scientific article; zbMATH DE number 5601778
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