G-category versus orbifold category
DOI:
https://doi.org/10.12775/TMNA.2022.055Keywords
Orbifolds, $G$-spaces, Lusternik-Schnirelman category, Hilsum-Skandalis maps, path groupoidAbstract
We present a comparative study of certain invariants defined for group actions and their analogues defined for orbifolds. In particular, we prove that Fadell's equivariant category for $G$-spaces coincides with the Lusternik-Schnirelmann category for orbifolds when the group is finite.References
A. Angel and H. Colman, Free and based path groupoids, Algebr. Geom. Topol. (2022). (to appear)
A. Angel, H. Colman, M. Grant and J. Oprea, Morita invariance of equivariant Lusternik–Schnirelmann category and invariant topological complexity, Theory Appl. Categ. 35 (2020), no. 7, 179–195.
G.E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Academic Press, New York, London, 1972.
M. Clapp and D. Puppe, Invariants of the Lusternik–Schnirelmann type and the topology of critical sets, Trans. Amer. Math. Soc. 298 (1986), no. 2, 603–620.
H. Colman, Equivariant LS-category for finite group actions, In Lusternik–Schnirelmann category and related topics (South Hadley, MA, 2001), Contemp. Math., vol. 316, Amer. Math. Soc., Providence, RI, 2002, pp. 35–40.
O. Cornea, G. Lupton, J. Oprea and D. Tanré, Lusternik–Schnirelmann Category, Mathematical Surveys and Monographs, vol. 103, American Mathematical Society, Providence, RI, 2003.
E. Fadell, The equivariant Lusternik–Schnirelmann method for invariant functionals and relative cohomological index theories, Topological Methods in Nonlinear Analysis, Sém. Math. Sup., vol. 95, Presses Univ. Montréal, Montreal, QC, 1985, pp. 41–70.
W. Marzantowicz, A G-Lusternik–Schnirelman category of space with an action of a compact lie group, Topology 28 (1089), no. 4, 403–412.
J.P. May, Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics, vol. 91, The Conference Board of the Mathematical Sciences, Washington, DC; American Mathematical Society, Providence, RI, 1996.
I. Moerdijk and J. Mrčun, Lie groupoids, sheaves and cohomology, Poisson Geometry, Deformation Quantisation and Group Representations, Lecture Note Ser., vol. 323, London Math. Soc., Cambridge Univ. Press, Cambridge, 2005, pp. 145–272.
I. Moerdijk and D. Pronk., Orbifolds, sheaves and groupoids, K-Theory 12 (1997), 3–21.
J. Mrčun, Stability and invariants of Hilsum–Skandalis maps, PhD thesis, 2005.
J. Pardon, Enough vector bundles on orbispaces, Compos. Math. 158 (2022), no. 11, 2046–2081.
D. Pronk, Etendues and stacks as bicategories of fractions, Compos. Math. 102 (1996), no. 3, 243–303.
D. Pronk and L. Scull, Translation groupoids and orbifold cohomology, Canad. J. Math. 62 (2010), no. 3, 614–645.
I. Satake, On a generalization of the notion of manifold, Proc. Natl. Acad. Sci. USA 42 (1956), 359–363.
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Copyright (c) 2023 Andrés Ángel, Hellen Colman
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