ON THE ASYMPTOTICS OF MORSE NUMBERS OF FINITE COVERS OF MANIFOLDS

Abstract

Let M be a closed connected manifold. We denote by M(M) the Morse number of M, i.e. the minimal possible number of critical points of a Morse function f on M. M.Gromov posed the following question: Let N k, k∈ N be a sequence of manifolds, such that each N k is an a k -fold cover of M where a k →∞ as k→∞. What are the asymptotic properties of the sequence M(N k) as k→∞? In this paper we study the case π 1(M)≈ Z m, dim M⩾6 . Let ξ∈H 1(M, Z), ξ≠0 . Let M( ξ) be the infinite cyclic cover corresponding to ξ, with generating covering translation t: M( ξ)→ M( ξ). Let M(ξ, k) be the quotient M( ξ)/ t k . We prove that lim k→∞ M(M(ξ, k))/k exists. For ξ outside a subset M⊂H 1(M) which is the union of a finite family of hyperplanes, we obtain the asymptotics of M(M(ξ, k)) as k→∞ in terms of homotopy invariants of M related to the Novikov homology of M. It turns out that the limit above does not depend on ξ (if ξ∉ M ). Similar results hold for the stable Morse numbers. Generalizations for the case of non-cyclic coverings are obtained.