• On the Torus Cobordant Cohomology Spheres

• Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/119/01/0101-0108

• Keywords

Equivariant cohomology; integral weight; Serre spectral sequence; cobordism.

• Abstract

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.

• Author Affiliations

1. Department of Mathematics, Çukurova University, 01330-Adana, Turkey

• Proceedings – Mathematical Sciences

Volume 132, 2022
All articles
Continuous Article Publishing mode

• Editorial Note on Continuous Article Publication

Posted on July 25, 2019