On the asymptotics of Morse numbers of finite covers of manifolds

Details On the asymptotics of Morse numbers of finite covers of manifolds On the asymptotics of Morse numbers of finite covers of manifolds

1998-10-22

arxiv.org/abs/math/9810136v1

Topology 38 (1999) 529-541

Mathematics

Differential Geometry

Amslatex file, 13 pages. To be published in "Topology"

Scientific paper

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.

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