zbMATH — the first resource for mathematics

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.

55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
57Q15 Triangulating manifolds
57R05 Triangulating
Full Text: DOI arXiv
[1] Arnoux, P.; Marin, A., The Kühnel triangulation of the complex projective plane from the view-point of complex cristallography. II., Mem. Fac. Sci. Kyushu Univ. Ser. A, 45, 2, 167-244 (1991) · Zbl 0753.52002
[2] Bagchi, B.; Datta, B., Minimal triangulations of sphere bundles over the circle, J. Combin. Theory Ser. A, 115, 5, 737-752 (2008) · Zbl 1146.52007
[3] Borghini, E.; Minian, Eg, The covering type of closed surfaces and minimal triangulations, J. Combin. Theory Ser. A, 166, 1-10 (2019) · Zbl 1439.57033
[4] Brehm, U.; Kühnel, W., Combinatorial manifolds with few vertices, Topology, 26, 4, 465-473 (1987) · Zbl 0681.57009
[5] Cornea, O.; Lupton, G.; Oprea, J.; Tanré, D., Lusternik-Schnirelmann Category. Mathematical Surveys and Monographs (2003), Providence: American Mathematical Society, Providence · Zbl 1032.55001
[6] Datta, B., Minimal triangulations of manifolds, J. Indian Inst. Sci., 87, 4, 429-449 (2007) · Zbl 1226.52005
[7] Dranishnikov, Alexander N.; Katz, Mikhail G.; Rudyak, Yuli B., Small values of the Lusternik-Schnirelmann category for manifolds, Geometry & Topology, 12, 3, 1711-1727 (2008) · Zbl 1152.55002
[8] Hatcher, A., Algebraic Topology (2002), Cambridge: Cambridge University Press, Cambridge
[9] Karoubi, M.; Weibel, C., On the covering type of a space, Enseign. Math., 62, 3-4, 457-474 (2016) · Zbl 1378.55002
[10] Lutz, F.H.: Triangulated manifolds with few vertices: combinatorial manifolds (2005). arXiv:math/0506372
[11] Whitehead, Gw, Elements of Homotopy Theory. Graduate Texts in Mathematics (1978), Berlin: Springer, Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.