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

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 [3]. 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. MATH CLASSIFICATION: 20F34, 20L13, 20L15, 57S30 1 Groups acting on spaces In this section we show some of the theory of a group G acting on a topological space X , and describe the orbit topological space, which is written X/G. There arises the problem of relating topological invariants of the orbit space X/G to those of X and the group action. In particular, it is a complicated and interesting question to find, if at all possible, relations between the fundamental groups and groupoids of X and X/G. This we shall do for a particular family of actions which arise commonly, namely the discontinuous actions. The resulting theory generalises that of regular covering spaces, and has a number of important applications. A useful special case of a discontinuous ∗email: [email protected] †email: [email protected]