Unimodality of Betti numbers for Hamiltonian circle actions with index-increasing moment Maps

Abstract

The unimodality conjecture posed by Tolman in [L. Jeffrey, T. Holm, Y. Karshon, E. Lerman and E. Meinrenken, Moment maps in various geometries, http://www.birs.ca/workshops/2005/05w5072/report05w5072.pdf ] states that if [Formula: see text] is a [Formula: see text]-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points, then the sequence of Betti numbers [Formula: see text] is unimodal, i.e. [Formula: see text] for every [Formula: see text]. Recently, the author and Kim [Y. Cho and M. Kim, Unimodality of the Betti numbers for Hamiltonian circle action with isolated fixed points, Math. Res. Lett. 21(4) (2014) 691–696] proved that the unimodality holds in eight-dimensional case by using equivariant cohomology theory. In this paper, we generalize the idea in [Y. Cho and M. Kim, Unimodality of the Betti numbers for Hamiltonian circle action with isolated fixed points, Math. Res. Lett. 21(4) (2014) 691–696] to an arbitrary dimensional case. We prove the conjecture in arbitrary dimension under the assumption that the moment map [Formula: see text] is index-increasing, which means that [Formula: see text] implies [Formula: see text] for every pair of critical points [Formula: see text] and [Formula: see text] of [Formula: see text], where [Formula: see text] is the Morse index of [Formula: see text] with respect to [Formula: see text].