New bounds on the toral rank with application to cohomologically symplectic spaces

TL;DR

This paper introduces new lower bounds on the cohomology dimension related to the toral rank, with significant implications for c-symplectic spaces and the toral rank conjecture.

Contribution

It applies Boij-S"oderberg theory to establish novel bounds on cohomology dimensions, proving the toral rank conjecture for certain c-symplectic spaces.

Findings

Established two new lower bounds for cohomology dimension

Proved the toral rank conjecture for c-symplectic spaces of dimension ≤ 8

Demonstrated effectiveness of bounds using Boij-S"oderberg theory

Abstract

We use Boij-S\"oderberg theory to give two lower bounds for the dimension of the cohomology of a finite CW-complex in terms of the toral rank and certain Betti numbers of the space. One of our bounds turns out to be particularly effective for c-symplectic spaces, proving the toral rank conjecture for c-symplectic spaces of formal dimension $\leq 8$.