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Lifting homotopies through fixed points

Published online by Cambridge University Press:  14 November 2011

M. A. Armstrong
M. A. Armstrong
Affiliation:
Department of Mathematics, Science Laboratories, University of Durham, Durham
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Synopsis

If G is a discontinuous group of homeomorphisms of a connected, locally path connected space X, which acts freely on X, then the projection π: X → X/G is a covering map and has the homotopy lifting property. Here we allow the elements of G to have fixed points and use work of Rhodes to investigate how two loops in X are related if their projections are homotopic in X/G. This enables us to establish a formula for the fundamental group of the orbit space of a discontinuous group under very general conditions. Finally we show by means of an example that some restriction on the action near fixed points is needed for the formula to be valid.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 93 , Issue 1-2 , 1982 , pp. 123 - 128
Copyright
Copyright © Royal Society of Edinburgh 1982

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