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Equivariant covering type and the number of vertices in equivariant triangulations

Part of: Differential topology PL-topology Low-dimensional topology Classical Topics in Algebraic Topology

Published online by Cambridge University Press:  22 November 2024

Dejan Govc ,
Wacław Marzantowicz  and
Petar Pavešić
Dejan Govc
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, and the Institute of Mathematics, Physics, and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenija (dejan.govc@gmail.com)
Wacław Marzantowicz
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University of Poznań, ul. Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland (marzan@amu.edu.pl)
Petar Pavešić
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, and the Institute of Mathematics, Physics, and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenija (petar.pavesic@fmf.uni-lj.si) (corresponding author)
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Abstract

We introduce the notion of the equivariant covering type of a space X on which a finite group G acts and study its properties. The equivariant covering type measures the size of G-equivariant good covers of X and is thus an extension of the covering type of a space, introduced by Karoubi and Weibel. We show that the equivariant covering type is a G-homotopy invariant and describe its relation with other G-invariants, like the equivariant LS-category, G-genus, and the multiplicative structures of equivariant cohomology theories. We also compute the G-covering type of regular G-graphs, give estimates for orientation-preserving actions on surfaces and for the projectivizations of complex representations of G and cohomology spheres. As an application, we derive estimates of sizes of minimal G-triangulations for various G-spaces.

Keywords

G-actioncovering typecup-lengthequivariantLusternik–Schnirelmann G-category

MSC classification

Primary: 57M60: Group actions in low dimensions
Secondary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirel'man) category of a space 57Q15: Triangulating manifolds 57R05: Triangulating
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 32
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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