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Homotopy Groups of Transformation Groups

Published online by Cambridge University Press:  20 November 2018

F. Rhodes
F. Rhodes*
Affiliation:
University of Southampton, Southampton, England Wesley an University, Middletown, Connecticut
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In a previous paper (2) I defined the fundamental group σ(X, x0, G) of a group Gof homeomorphisms of a space X, and showed that if the transformation group admits a family of preferred paths, then σ(X, x0, G) can be represented as a group extension of π1(X, x0) by G. In this paper the homotopy groups of a transformation group are defined. The nth absolute homotopy group of a transformation group which admits a family of preferred paths is shown to be representable as a split extension of the nth absolute torus homotopy group τn(X, x0) by G.

In § 6 it is shown that the action of G on X induces a homomorphism of Ginto a quotient group of a subgroup of the group of automorphisms of τn(X, x0). This homomorphism is used to obtain a necessary condition for the embedding of one transformation group in another, in particular, for the embedding of a discrete flow in a continuous flow with the same phase space.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1123 - 1136
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Fox, R. H., Homotopy groups and torus homotopy groups, Ann. of Math. (2) 49 (1948), 471–510.Google Scholar
2. Rhodes, F., On the fundamental group ofa transformation group, Proc. London Math. Soc. (3) 16 (1966), 635–650.Google Scholar