# Bounds on the toral rank of cohomologically symplectic spaces

## Leopold Zoller

created by daniele on 14 Mar 2019
modified on 17 May 2019

23 may 2019 -- 14:30

Aula Tricerri, DiMaI, Firenze

Abstract.

The toral rank conjecture states that in order for a torus $T$ to act freely on a compact manifold $M$, it is necessary that $\dim_{\mathbb{Q}} H^*(M;\mathbb{Q}) \geq \dim_{\mathbb{Q}} H^*(T;\mathbb{Q})$. While the conjecture has been open for more than 30 years it was solved in a number of cases, including that of compact K\"ahler manifolds. In the talk, we investigate the algebraic topology of free torus actions on compact symplectic manifolds and try to attack the conjecture for this type of space. We obtain new cohomological obstructions to the existence of free torus actions and prove the conjecture for symplectic manifolds of dimension $\leq 8$.

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