The fundamental groupoid of the quotient of a Hausdorff space by a discontinuous action of a discrete group is the orbit groupoid of the induced action

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Brown, Ronald ; Higgins, Philip J. (2002)
  • Subject: 20F34, 20L13, 20L15 | Mathematics - Algebraic Topology | Mathematics - Category Theory

The main result is that the fundamental groupoid of the orbit space of a discontinuous action of a discrete group on a Hausdorff space which admits a universal cover is the orbit groupoid of the fundamental groupoid of the space. We also describe work of Higgins and of Taylor which makes this result usable for calculations. As an example, we compute the fundamental group of the symmetric square of a space. The main result, which is related to work of Armstrong, is due to Brown and Higgins in 1985 and was published in sections 9 and 10 of Chapter 9 of the first author's book on Topology (Ellis Horwood, 1988). This is a somewhat edited, and in one point (on normal closures) corrected, version of those sections. Since the book is out of print, and the result seems not well known, we now advertise it here. It is hoped that this account will also allow wider views of these results, for example in topos theory and descent theory. Because of its provenance, this should be read as a graduate text rather than an article. The Exercises should be regarded as further propositions for which we leave the proofs to the reader. It is expected that this material will be part of a new edition of the book.
  • References (12)
    12 references, page 1 of 2

    [1] M. A. Armstrong, On the Fundamental Group of an Orbit Space, Proc. Camb. Phil. Soc., 61, (1965), 639-646.

    [2] A. F. Beardon, 1983, The Geometry of Discrete Groups, volume 91 of Graduate Texts in Mathematics, Springer-Verlag, Berlin-Heidelberg-New York.

    [3] R. Brown, 1988, Topology: a geometric account of general topology, homotopy types, and the fundamental groupoid, Ellis-Horwood, Chichester.

    [4] R. Brown and G. Danesh-Naruie, The Fundamental Groupoid as a Topological Groupoid, Proc. Edinburgh Math. Soc., 19, (1975), 237-244.

    [5] P. J. Higgins, Algebras with a Scheme of Operators, Math. Nach., 27, (1963), 115-132.

    [6] P. J. Higgins, 1971, Categories and Groupoids, van Nostrand, New York.

    [7] P. J. Higgins and J. Taylor, 1982, The Fundamental Groupoid and Homotopy Crossed Complex of an Orbit Space, in K. H. Kamps et al., ed., Category Theory: Proceedings Gummersbach 1981, volume 962 of Lecture Notes in Math., 115-122, Springer-Verlag.

    [8] S. Iyanaga and Y. Kawada, eds., 1987, Encyclopaedic Dictionary of Mathematics, MIT Press, Cambridge, Mass. and London, England, third edition, produced by Mathematical Society of Japan; reviewed by K. O. May.

    [9] F. Rhodes, On the Fundamental Group of a Transformation Group, Proc. London Math. Soc., 3, (1966), 635-650.

    [10] F. Rhodes, On Lifting Transformation Groups, Proc. Amer. Math. Soc., 19, (1968), 905-908.

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