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Lifting homotopies through fixed points II

Published online by Cambridge University Press:  14 November 2011

M. A. Armstrong
M. A. Armstrong
Affiliation:
Department of Mathematical Sciences, University of Durham, Durham
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Synopsis

This note complements an earlier paper of the same title. Let G be a discontinuous group of homeomorphisms of a connected, locally path connected, Hausdorff space X, and let ∏:X → X/G denote the associated projection. We work relative to a G-invariant subgroup H of the fundamental group of X and investigate the quotient group ∏1(X/G)/∏*(H). By choosing H appropriately, we can calculate ∏1(X/G) and show that ∏1(X/G)/∏*(∏1(X)) is isomorphic to G/F, where F is the normal subgroup of G generated by those elements which have fixed points. In a final section, we give analogous results for actions of a compact Lie group.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 96 , Issue 3-4 , 1984 , pp. 201 - 205
Copyright
Copyright © Royal Society of Edinburgh 1984

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References

1Armstrong, M. A.. Lifting homotopies through fixed points. Proc. Roy. Soc. Edinburgh Sect. A 93 (1982), 123–128.CrossRefGoogle Scholar
2Armstrong, M. A.. Calculating the fundamental group of an orbit space. Proc. Amer. Math. Soc. 84 (1982), 267–271.CrossRefGoogle Scholar
3Bredon, G. E.. Introduction to compact transformation groups (New York: Academic Press, 1972).Google Scholar
4Spanier, E. H.. Algebraic Topology (New York: McGraw-Hill, 1966).Google Scholar