Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Calculating the fundamental group of an orbit space

Author: M. A. Armstrong
Journal: Proc. Amer. Math. Soc. 84 (1982), 267-271
MSC: Primary 55Q05; Secondary 57S99
MathSciNet review: 637181
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose $ G$ acts effectively as a group of homeomorphisms of the connected, locally path connected, simply connected, locally compact metric space $ X$. Let $ \overline G $ denote the closure of $ G$ in $ {\text{Homeo}}(X)$, and $ N$ the smallest normal subgroup of $ \overline G $ which contains the path component of the identity of $ \overline G $ and all those elements of $ \overline G $ which have fixed points. We show that $ {\pi _1}(X/G)$ is isomorphic to $ \overline G /N$ subject to a weak path lifting assumption for the projection $ X \to X/\overline G $.

References [Enhancements On Off] (What's this?)

  • [1] M. A. Armstrong, On the fundamental group of an orbit space, Proc. Cambridge Philos. Soc. 61 (1965), 639-646. MR 0187244 (32:4697)
  • [2] -, The fundamental group of the orbit space of a discontinuous group, Proc. Cambridge Philos. Soc. 64 (1968), 299-301. MR 0221488 (36:4540)
  • [3] G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. MR 0413144 (54:1265)
  • [4] R. S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295-323. MR 0126506 (23:A3802)
  • [5] F. Rhodes, On the fundamental group of a transformation group, Proc. London Math. Soc. (3) 16 (1966), 635-650. MR 0203715 (34:3564)
  • [6] S. Smale, A note on open maps, Proc. Amer. Math. Soc. 8 (1957), 391-393. MR 0086295 (19:158f)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55Q05, 57S99

Retrieve articles in all journals with MSC: 55Q05, 57S99

Additional Information

Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society