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

Dejan Govc (Author), Wacław Marzantowicz (Author), Petar Pavešić (Author)

Abstract

Tip pokritja prostora ▫$X$▫ je homotopska invarianta, ki v določenem smislu meri homotopsko velikost ▫$X$▫. Vpeljala sta jo Karoubi in Weibel kot minimalno moč dobrega pokritja prostora ▫$Y$▫ med vsemi prostori ▫$Y$▫, ki so homotopsko ekvivalentni ▫$X$▫. V članku podamo vrsto ocen za tip pokritja pomočjo drugih homotopskih invariant, med katerimi izstopajo homološke grupe, kohomološki kolobar in Lusternik-Schnirelmannova kategorija. Poleg tega v članku povežemo tip pokritja poliedra s številom oglišč v minimalni triangulaciji. Tako izpeljemo na enovit način vrsto ocen, ki so bodisi nove, bodisi posplošitve ocen, ki so v preteklosti slonele na ad hoc kombinatornih ocenah.

Keywords

tip pokritja;minimalna triangulacija;Lusternik-Schnirelmannova kategorija;dolžina kohomološkega produkta;covering type;minimal triangulation;Lusternik-Schnirelmann category;cup-length;

Data

Language:	English
Year of publishing:	2020
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	515.14
COBISS:	18627417
ISSN:	0179-5376
Views:	496
Downloads:	239
Average score:	0 (0 votes)
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Other data

Secondary language:	Slovenian
Secondary title:	Ocene tipa pokritja in števila oglišč v minimalnih triangulacijah
Secondary abstract:	The covering type of a space ▫$X$▫ is a numerical homotopy invariant which in some sense measures the homotopical size of ▫$X$▫. It was first introduced by Karoubi and Weibel (in Enseign Math 62(3-4):457-474, 2016) as the minimal cardinality of a good cover of a space ▫$Y$▫ taken among all spaces that are homotopy equivalent to ▫$X$▫. 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.
Secondary keywords:	tip pokritja;minimalna triangulacija;Lusternik-Schnirelmannova kategorija;dolžina kohomološkega produkta;
Pages:	str. 31-48
Volume:	ǂVol. ǂ63
Issue:	ǂiss. ǂ1
Chronology:	Jan. 2020
DOI:	10.1007/s00454-019-00092-z
ID:	11551956