Homotopy Groups of Transformation Groups

Rhodes, F.

Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1123-1136

Résumé

In a previous paper (2) I defined the fundamental group σ(X, x 0, 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, x 0, G) can be represented as a group extension of π 1(X, x 0) 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, x 0) 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, x 0). 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.

Détail

DOI : 10.4153/CJM-1969-123-x

Bibliographie

[1] 1. Fox, R. H., Homotopy groups and torus homotopy groups, Ann. of Math. (2) 49 (1948), 471–510. Google Scholar

[2] 2. Rhodes, F., On the fundamental group ofa transformation group, Proc. London Math. Soc. (3) 16 (1966), 635–650. Google Scholar