On almost free torus actions and Horrocks conjecture

TL;DR

This paper develops a cohomology model for spaces with torus actions, proving the toral rank conjecture for certain cases and deriving new bounds on cohomological dimensions using homological algebra.

Contribution

It introduces a new cohomology model for spaces with torus actions and proves the toral rank conjecture for spaces with formal quotients when the torus dimension is at most five.

Findings

Constructed a model for cohomology with formal homotopy orbit spaces.

Derived new bounds on the homological rank of spaces with almost free torus actions.

Proved the toral rank conjecture for spaces with formal quotients for torus dimension ≤ 5.

Abstract

We construct a model for cohomology of a space $X$ equipped with a torus $T$ action, whose homotopy orbit space $X_{T}$ is formal. This model represents Koszul complex of its equivariant cohomology. Studying homological properties of modules over polynomial ring we derive new bounds on homological rank (dimension of cohomology ring) of $X$ equipped with almost free torus action. We give a proof of toral rank conjecture for spaces with formal quotient in the case of torus dimension $\le5$.