On the length of cohomology spheres

TL;DR

This paper introduces a new cohomological index called length for G-spaces that are cohomology spheres, leading to generalized Borsuk-Ulam and Bourgin-Yang theorems, with bounds and sharper results for manifolds.

Contribution

It develops the length index for cohomology spheres with p-torus or torus group actions, extending classical theorems and providing bounds and sharper results.

Findings

Defined the length index for G-spaces as cohomology spheres.

Derived Borsuk-Ulam and Bourgin-Yang type theorems in this setting.

Established bounds for the length index in various cases.

Abstract

We present the length, a numerical cohomological index theory, of $ G $-spaces which are cohomology spheres and $ G $ is a $p$-torus or a torus group, where $p$ is a prime. As a consequence, we obtain Borsuk-Ulam and Bourgin-Yang type theorems in this context. A sharper version of the Bourgin-Yang theorem for topological manifolds is also proved. Also, we give some general results regarding the upper and lower bound for the length.