Estimates of Covering Type and the Number of Vertices of Minimal Triangulations

Abstract

The covering type of a space X is a numerical homotopy invariant which in some sense measures the homotopical size of X. It was first introduced by Karoubi and Weibel (in Enseign Math 62(3–4):457–474, 2016) as the minimal cardinality of a good cover of a space Y taken among all spaces that are homotopy equivalent to X. We give several estimates of the covering type in terms of other homotopy invariants of X, most notably the ranks of the homology groups of X, the multiplicative structure of the cohomology ring of X and the Lusternik–Schnirelmann category of X. In addition, we relate the covering type of a triangulable space to the number of vertices in its minimal triangulations. In this way we derive within a unified framework several estimates of vertex-minimal triangulations which are either new or extensions of results that have been previously obtained by ad hoc combinatorial arguments.