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

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

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.

• Language: English
• Keywords: covering type, minimal triangulation, Lusternik-Schnirelmann category, cup-length
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2020
• Number of pages: Str. 31-48
• Numbering: Vol. 63, iss. 1
• PID: 20.500.12556/RUL-115223
• UDC: 515.14
• ISSN on article: 0179-5376
• DOI: 10.1007/s00454-019-00092-z
• COBISS.SI-ID: 18627417
• Publication date in RUL: 18.04.2020

Record is a part of a journal

• Title: Discrete & computational geometry
• Shortened title: Discrete comput. geom.
• Publisher: Springer
• ISSN: 0179-5376
• COBISS.SI-ID: 25342208

Secondary language

• Language: Slovenian
• Title: Ocene tipa pokritja in števila oglišč v minimalnih triangulacijah

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 s 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