Hamiltonian torus actions and the unimodality of odd Betti numbers

TL;DR

This paper investigates the unimodality of odd Betti numbers in closed symplectic manifolds with Hamiltonian torus actions, providing positive evidence in specific dimensions under certain symmetry conditions.

Contribution

It extends previous work by demonstrating unimodality of odd Betti numbers in dimensions 6, 8, and 10 with stronger symmetry assumptions.

Findings

Unimodality holds in dimension 6.

Unimodality holds in dimension 8.

Unimodality holds in dimension 10.

Abstract

This paper is dedicated to the question: Is the sequence of odd and even Betti numbers of a closed symplectic manifold with a non-trivial Hamiltonian torus action unimodal? Recently, there was some progress on the question for the sequence of even Betti numbers by Cho-Kim and the author. The results of this paper give positive evidence in the case of odd Betti numbers, in dimensions $6,8$ and $10$ under progressively stronger symmetry assumptions.