Estimates of covering type and the number of vertices of minimal triangulations

2017/10/09 by Govc, Dejan, Marzantowicz, Wacław, Pavešić, Petar · 1 citation
#57Q15 #57R05 #Algebraic Topology (math.AT) #FOS: Mathematics #Primary 55M #Secondary 55M30

paper · doi:10.48550/arxiv.1710.03333

Abstract

The covering type of a space X is defined as the minimal cardinality of a good cover of a space that is homotopy equivalent to X. We derive estimates for the covering type of X in terms of other invariants of X, namely the ranks of the homology groups, the multiplicative structure of the cohomology ring and the Lusternik-Schnirelmann category of X. By relating the covering type to the number of vertices of minimal triangulations of complexes and combinatorial manifolds, we obtain, within a unified framework, several estimates which are either new or extensions of results that have been previously obtained by ad hoc combinatorial arguments. Moreover, our methods give results that are valid for entire homotopy classes of spaces.

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• Fundamental groups of small simplicial complexes

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