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 (M,ω) is a 2n-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points, then the sequence of Betti numbers {b0(M),b2(M),...,b2n(M)} is unimodal, i.e. bi(M) ≤ bi+2(M) for every i < n. 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 H : M → R is index-increasing, which means that ind(p) < ind(q) implies H(p) < H(q) for every pair of critical points p and q of H, where ind(p) is the Morse index of p with respect to H. © World Scientific Publishing Company