Hostname: page-component-84b7d79bbc-5lx2p Total loading time: 0 Render date: 2024-08-02T10:18:29.887Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Polytopes and polyhedra PL-topology

Published online by Cambridge University Press:  15 January 2019

Petar Pavešić
Petar Pavešić*
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenija (petar.pavesic@fmf.uni-lj.si)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study lower bounds for the number of vertices in a PL-triangulation of a given manifold M. While most of the previous estimates are based on the dimension and the connectivity of M, we show that further information can be extracted by studying the structure of the fundamental group of M and applying techniques from the Lusternik-Schnirelmann category theory. In particular, we prove that every PL-triangulation of a d-dimensional manifold (d ⩾ 3) whose fundamental group is not free has at least 3d + 1 vertices. As a corollary, every d-dimensional homology sphere that admits a combinatorial triangulation with less than 3d vertices is PL-homeomorphic to Sd. Another important consequence is that every triangulation with small links of M is combinatorial.

Keywords

minimal triangulationPL-manifoldhomology spheregood coverLusternik-Schnirelmann category

MSC classification

Primary: 57Q15: Triangulating manifolds
Secondary: 52B70: Polyhedral manifolds
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 6 , December 2019 , pp. 1453 - 1463
Copyright
Copyright © Royal Society of Edinburgh 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bagchi, B. and Datta, B.. Combinatorial triangulations of homology spheres. Discrete Math. 305 (2005), 1–17.Google Scholar
2Borghini, E. and Minian, E. G.. The covering type of closed surfaces and minimal triangulations, arXiv:1712.02833 (2017), 6.Google Scholar
3Brehm, U. and Kühnel, W.. Combinatorial manifolds with few vertices. Topol. 26 (1987), 465–473.Google Scholar
4Bryant, J. L.. Piecewise linear topology. In Handbook of geometric topology (ed.Daverman, R. J.. et al. ), pp. 219–259 (Amsterdam: Elsevier, 2002).Google Scholar
5Cornea, O., Lupton, G., Oprea, J. and Tanre, D.. Lusternik-Schnirelmann category. Mathematical surveys and monographs, vol. 103 (Providence, RI: American Mathematical Society, 2003).Google Scholar
6Dranishnikov, A., Katz, M. and Rudyak, Y.. Small values of the Lusternik–Schnirelmann category for manifolds. Geom. Topol. 12 (2008), 1711–1727.Google Scholar
7Eels, J. and Kuiper, N.. Manifolds which are like projective planes. Publications mathématiques de l'I.H.É.S. 14 (1962), 5–46.Google Scholar
8Govc, D., Marzantowicz, W. and Pavešić, P.. Estimates of covering type and the number of vertices of minimal triangulations, arXiv:1710.03333 (2017), 19.Google Scholar
9Hatcher, A.. Algebraic topology (Cambridge: Cambridge University Press, 2002).Google Scholar
10 Joswig, M., Lutz, F. H. and Tsuruga, M.. Heuristics for sphere recognition, In Mathematical software – ICMS 2014, (eds. Hong, H. and Yap, C.) Lecture Notes in Computer Science, vol 8592 (Berlin, Heidelberg: Springer, 2014).Google Scholar
11Karoubi, M. and Weibel, C.. On the covering type of a space. Enseign Math. 62 (2016), 457–474.Google Scholar
12 Klee, S. and Novik, I.. Face enumeration on simplicial complexes, In Recent trends in combinatorics, (eds. Beveridge, A. et al. ) The IMA volumes in Mathematics and its applications, vol. 159,pp. 653–685 (Cham: Springer, 2016).Google Scholar
13Kühnel, W.. Higherdimensional analogues of Császár's torus. Result. Math. 9 (1986), 95–106.Google Scholar
14Lutz, F. H.. Triangulated manifolds with few vertices: combinatorial manifolds, arXiv:math.CO/0506372v1 (2005), 37.Google Scholar
15Spanier, E. H.. Algebraic topology (New York: Springer-Verlag, 1966).Google Scholar
16Wall, C. T. C.. Classification of (n − 1)-connected 2n-manifolds. Ann. of Math. 75 (1962), 163–189.Google Scholar