ON THE ASYMPTOTICS OF MORSE NUMBERS OF FINITE COVERS OF MANIFOLDS

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 Nk, k∈N be a sequence of manifolds, such that each Nk is an ak-fold cover of M where ak→∞ as k→∞. What are the asymptotic properties of the sequence M(Nk) as k→∞? s paper we study the case π1(M)≈Zm, dim M⩾6. Let ξ∈H1(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)/tk. We prove that limk→∞ M(M(ξ, k))/k exists. For ξ outside a subset M⊂H1(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.