Unimodality of the Betti numbers for Hamiltonian circle action with isolated fixed points

TL;DR

This paper proves that for an 8-dimensional symplectic manifold with a Hamiltonian circle action and isolated fixed points, the Betti numbers follow a unimodal sequence, specifically increasing up to the middle dimension.

Contribution

It establishes the unimodality of Betti numbers for a specific class of symplectic manifolds with Hamiltonian circle actions, extending understanding of their topological structure.

Findings

Betti numbers are unimodal in the specified setting

Betti number sequence satisfies b0 ≤ b2 ≤ b4

Results contribute to symplectic topology and Hamiltonian group actions

Abstract

Let $(M,\omega)$ be an eight-dimensional closed symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points. In this article, we will show that the Betti numbers of $M$ are unimodal, i.e. $b_0(M) \leq b_2(M) \leq b_4(M)$.