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

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

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.

• Jezik: Angleški jezik
• Ključne besede: covering type, minimal triangulation, Lusternik-Schnirelmann category, cup-length
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Leto izida: 2020
• Št. strani: Str. 31-48
• Številčenje: Vol. 63, iss. 1
• PID: 20.500.12556/RUL-115223
• UDK: 515.14
• ISSN pri članku: 0179-5376
• DOI: 10.1007/s00454-019-00092-z
• COBISS.SI-ID: 18627417
• Datum objave v RUL: 18.04.2020

GOVC, Dejan, MARZANTOWICZ, Wacław in PAVEŠIĆ, Petar, 2020, Estimates of covering type and the number of vertices of minimal triangulations. Discrete & computational geometry [na spletu]. 2020. Vol. 63, no. 1, p. 31–48. [Dostopano 18 april 2025]. DOI 10.1007/s00454-019-00092-z. Pridobljeno s: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=slv&id=115223

Gradivo je del revije

• Naslov: Discrete & computational geometry
• Skrajšan naslov: Discrete comput. geom.
• Založnik: Springer
• ISSN: 0179-5376
• COBISS.SI-ID: 25342208

Sekundarni jezik

• Jezik: Slovenski jezik
• Naslov: Ocene tipa pokritja in števila oglišč v minimalnih triangulacijah

Izvleček

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.

• Ključne besede: tip pokritja, minimalna triangulacija, Lusternik-Schnirelmannova kategorija, dolžina kohomološkega produkta