On the asymptotics of Morse numbers of finite covers of manifolds

Abstract

Let M be a closed connected manifold. Let m(M) be the Morse number of M, that is, the minimal number of critical points of a Morse function on M. Let N be a finite cover of M of degree d. M.Gromov posed the following question: what are the asymptotic properties of m(N) as d goes to infinity? In this paper we study the case of high dimensional manifolds M with free abelian fundamental group. Let x be a non-zero element of H^1(M), let M(x) be the infinite cyclic cover corresponding to x, and t be a generator of the structure group of this cover. Set M(x,k)=M(x)/t^k. We prove that the sequence m(M(x,k))/k converges as k goes to infinity. For x outside of a finite union of hyperplanes in H^1(M) we obtain the asymptotics of m(M(x,k)) as k goes to infinity, in terms of homotopy invariants of M related to Novikov homology of M.