The covering type of closed surfaces and minimal triangulations

Borghini, Eugenio; Minian, Elías Gabriel

doi:10.48550/arxiv.1712.02833

Cited by 2 publications

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“…It appears to be much harder to distinguish between ct(M ) and ∆ P L (M ) when M is a closed manifold. Borghini and Minian [2] have recently proved that with one exception ct(M ) = ∆ P L (M ) for all closed surfaces (both orientable and non-orientable). The only exception is the genus two orientable surface T #T , as they prove that ct(T #T ) = 9 while ∆ P L (T #T ) = 10, which means that the minimal triangulation of T #T has 10 vertices but there exists a 2-dimensional complex on 9 vertices whose geometric realization is homotopy equivalent to T #T .…”

Section: Introductionmentioning

confidence: 99%

Estimates of Covering Type and the Number of Vertices of Minimal Triangulations

Govc

Marzantowicz

Pavešić

2019

Discrete Comput Geom

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The covering type of a space X is a numerical homotopy invariant that in some sense measures the homotopical size of X. It was first introduced by Karoubi and Weibel [9] as the minimal cardinality of a good cover of a space Y taken among all spaces Y that are homotopy equivalent to X. In this paper we give several estimates of the covering type in terms of other homotopy invariants of X, most notably the ranks of the homology groups of X, the multiplicative structure of the cohomology ring of X and the Lusternik-Schnirelmann category of X. In addition, we relate the covering type of a triangulable space to the number of vertices in its minimal triangulations. In this way we derive within a unified framework several estimates of vertex-minimal triangulations which are either new or extensions of results that have been previously obtained by ad hoc combinatorial arguments.2010 Mathematics Subject Classification. Primary 55M; Secondary 55M30, 57Q15, 57R05 .

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Section: Introductionmentioning

confidence: 99%

Estimates of Covering Type and the Number of Vertices of Minimal Triangulations

Govc

Marzantowicz

Pavešić

2019

Discrete Comput Geom

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“…However, if M is a closed triangulable manifold then there is some evidence that ∆(M ) are ct(M ) close and often equal. Notably, Borghini and Minian [2] showed that for closed surfaces ∆(M ) and ct(M ) coincide, with the sole exception of the orientable surface of genus 2, where the two quantities differ by one.…”

Section: Homotopy Triangulations and Covering Typementioning

confidence: 99%

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

Pavešić¹

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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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 (Zp-)homology sphere that admits a PL-triangulation with less than 3d vertices is homeomorphic to S d . Another important consequence is that every triangulation with small links of M is combinatorial.

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The covering type of closed surfaces and minimal triangulations

Cited by 2 publications

References 0 publications

Estimates of Covering Type and the Number of Vertices of Minimal Triangulations

Estimates of Covering Type and the Number of Vertices of Minimal Triangulations

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

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