Covering numbers of manifolds and critical points of a morse function

For a closed connected triangulatedn-manifoldM, we study some numerical invariants (namedcategory andcovering numbers) ofM which are strictly related to the topological structure ofM. We complete the classical results of 3-manifold topology and then we prove some characterization theorems in higher dimensions. Finally some applications are given about the minimal number of critical points (resp. values) of Morse functions defined on a closed connected smoothn-manifold. Work performed under the auspices of the G.N.S.A.G.A. of the C.N.R. and financially supported by the M.P.I. of Italy within the project “Geometria delle Varietà Differenziabili”.