On the length of cohomology spheres

Abstract

In [2], T. Bartsch provided detailed and broad exposition of a numerical cohomological index theory for G-spaces, known as the length, where G is a compact Lie group. We present the length 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.