CHAPTER 1 Transformation Groups This is an introductory chapter. It is used to set notation and recount basic ideas of group actions on general spaces as well as related material on fiber bundles, classifying spaces, and the Borel construction. Locally compact Lie groups are significant players in Seifert fiberings, and the notion of proper G-spaces plays an important role. So a background on proper actions is included leading to Palais’ proof of the existence of a slice for proper G-spaces. The final section sets the stage for how the preliminary work will be applied in the Seifert Constructions. 1.1. Introduction 1.1.1. Our spaces, in general, will be path-connected, completely regular and Hausdorff. When using covering space theory, we shall also assume our spaces locally path-connected and semilocally simply connected (that is, every point x in X has a neighborhood U such that the homomorphism from the fundamental group of U to the fundamental group of X, induced by the inclusion map of U into X, is trivial). Notation 1.1.2. Let G be a Lie group, and let K be a subgroup of G. NG(K) = Normalizer of K in G CG(K) = Centralizer of K in G Aut(G) = The group of continuous automorphisms of G Inn(G) = The group of inner automorphisms of G Aut(G,K) = {α∈ Aut(G) : α|K Aut(K)} Inn(G,K) = Inn(G) Aut(G,K) Out(G,K) = Aut(G,K)/Inn(G,K) μ(a) = Conjugation by a so μ(a)(x) = axa−1 for x∈G 1.1.3. Let P be a property on a group. We say a group is virtually P if it contains a normal subgroup of finite index which is P. For example, a crystallographic group is virtually free Abelian see Theorem 8.1.2. 1.1.4. A left action of a topological group G on a topological space X is a continuous function ϕ : G×X −→X such that (i) ϕ(gh,x) = ϕ(g,ϕ(h,x)) for all g,h∈G and x∈X, and (ii) ϕ(1,x) = x, for all x∈X, where 1 is the identity element of G. The point ϕ(g,x)∈X is usually written simply as gx,g(x), or sometimes g·x. Clearly, each element g ∈G can be viewed as a homeomorphism of X onto itself. We may denote this action by (G,X,ϕ), or more simply suppress the ϕ, (G,X), and call X a G-space. If X and Y are G-spaces, then a G-map is a continuous function 1
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