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Estimates of covering type and the number of vertices of minimal triangulations. (English) Zbl 1431.55003
The covering type of a space \(X\), denoted as \(\mbox{ct}(X),\) is a numerical homotopy invariant which was first introduced by M. Karoubi and C. A. Weibel in [Enseign. Math. (2) 62, No. 3–4, 457–474 (2016; Zbl 1378.55002)]. It is defined as the minimum of the strict covering types of spaces \(Y\) having the homotopy type of \(X\): \[ \mathrm{ct}(X):=\min \{\mathrm{sct}(Y)\mid Y\simeq X\} \] In turn, the strict covering type of \(Y\), \(\mbox{sct}(Y),\) means the minimum number of open subsets that determine a good cover of \(Y\) (i.e., all elements of the cover and all their finite nonempty intersections are contractible).
In the paper under review the authors do a deeper study of this relatively new homotopy invariant. In this sense they provide several interesting estimates of the covering type in terms of other well-known homotopy invariants. Among these we can mention the Lusternik-Schnirelmann category, the cohomology ring and the homology groups of the space under study. Moreover, when dealing with a triangulable space, the authors also give a relationship of the covering type of the space with the number of vertices in its minimal triangulation. Several examples displaying their new results are also given throughout the paper.

MSC:
55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
57Q15 Triangulating manifolds
57R05 Triangulating
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