On the Torus Cobordant Cohomology Spheres

Proceedings Mathematical Sciences

DoÄŸan Dönmez¹  Ali Özkurt² 

[1] $$;Department of Mathematics, Çukurova University, 00-Adana, Turkey$$

关键词: Equivariant cohomology;    integral weight;    Serre spectral sequence;    cobordism.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Let 𝐺 be a compact Lie group. In 1960, P A Smith asked the following question: ``Is it true that for any smooth action of 𝐺 on a homotopy sphere with exactly two fixed points, the tangent 𝐺-modules at these two points are isomorphic?" A result due to Atiyah and Bott proves that the answer is `yes’ for $mathbb{Z}_p$ and it is also known to be the same for connected Lie groups. In this work, we prove that two linear torus actions on $S^n$ which are 𝑐-cobordant (cobordism in which inclusion of each boundary component induces isomorphisms in $mathbb{Z}$-cohomology) must be linearly equivalent. As a corollary, for connected case, we prove a variant of Smith’s question.

【 授权许可】

Unknown