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 that in some sense measures the homotopical size of X. It was first introduced by Karoubi and Weibel [9] as the minimal cardinality of a good cover of a space Y taken among all spaces Y that are homotopy equivalent to X. In this paper 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.2010 Mathematics Subject Classification. Primary 55M; Secondary 55M30, 57Q15, 57R05 .

Triangulations with few vertices of manifolds with non-free fundamental group

“…Let us begin with a slight improvement of the theorem first proved by Brehm and Kühnel [3]. Our approach is based on the notion of covering type [8] and is much simpler than the original one. Recall that Poincaré duality together with the positive answer to the Poincaré conjecture imply that every simply-connected closed d-manifold, whose reduced homology is trivial in dimensions less or equal to d/2 is homeomorphic to the d-sphere.…”

“…Govc, Marzantowicz and Pavešić [7] applied techniques from Lusternik-Schnirelmann category to obtain further estimates of the covering type of a space and proved the following results: vertices.…”

Estimates of covering type and minimal triangulations based on category weight

“…The covering type is obviously a homotopy invariant of the space and can thus be related to other homotopy invariants -cf. [19] and [14].…”

“…In [14] we introduced several new estimates for the minimal number of vertices that are needed to triangulate a compact triangulable space X based on the Lusternik-Schnirelmann category of X and on the structure of the cohomology ring of X. In the case of manifolds these estimates were improved by using information obtained from the fundamental group in [26] and on the Lower Bound Theorem in [15].…”

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In a recent publication [14] we have introduced a new method, based on the Lusternik-Schnirelmann category and the cohomology ring of a space X, that yields lower bounds for the size of a triangulation of X. In this paper we present an important extension that takes into account the fundamental group of X.

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