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

*(English)*Zbl 1431.55003The 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.

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.

Reviewer: José Calcines (La Laguna)

##### MSC:

55M30 | Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) |

57Q15 | Triangulating manifolds |

57R05 | Triangulating |

##### References:

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