The purpose of this thesis is to present a fairly complete account of equivariant Ktheory on compact spaces. Equivariant Ktheory is a generalisation of Ktheory, a rather wellknown cohomology theory arising from consideration of the vectorbundles on a space. Equivariant Ktheory, or K_{G}theory, is defined not on a space but on Gspaces, i.e. pairs (X,α), where X is a space and α is an action of a fixed group G on X, and it arises from consideration of Gvectorbundles on X, i.e. vectorbundles on whose total space G acts in a suitable way (of 3.1). In this thesis G will always be a compact group. But K_{G}theory does not appear in the first three chapters, which are introductory. Chapter 1 consists of preliminary discussions of little relevance to the sequel, but which permit me to make a few propositions in the later chapters shorter or more elegant. It was intended to be amusing, and the reader may prefer to omit it. Chapter 2 is devoted to the representationtheory of compact groups. When X is a point a Gvectorbundle on X is just a representationmodule for G, so the representationring, or characterring, R(G) plays a fundamental role in K_{G}theory. In chapter 2 I investigate its algebraic structure, and in particular when G is a compact Lie group I determine completely its prime ideals. To do this I have to discuss first the space of conjugacyclasses of a compact Lie group, and outline an inducedrepresentation construction for obtaining finitedimensional modules for G from modules for suitable subgroups not of finite index. Chapter 3 is a rather full collection of technical results concerning Gvectorbundles: they are all essentially wellknown, but have not been stated in the equivariant case. Chapter 4 presents basic equivariant Ktheory. I show that it can be defined in three ways: by Gvectorbundles, by complexes of Gvectorbundles, and by Fredholm complexes of infinitedimensional Gvectorbundles. This chapter also treats the continuity of K_{G} with respect to inverse limits of Gspaces, the Thorn homomorphism for a Gvectorbundle and the periodicityisomorphism, and the question of extending K_{G} to noncompact spaces. In chapter 5 I obtain for K_{G}(X) a filtration and spectral sequence generalising those of [6], but without dissecting the space X. My method is based on a Cech approach: for each open covering of X I construct an auxiliary space homotopyequivalent to X which has the natural filtration that X lacks. Also in chapter 5 I prove the localisationtheorem (5.3), which, together with the theory of chapter 6, is one of the most important tools in applied K_{G}theory. K_{G}(X) is a module over the characterring R(G), so one can localise it at the prime ideals of R(G), which I have determined in 2.5. The simplest and most important case of the localisationtheorem states that, if β is the prime ideal of characters of G vanishing at a conjugacyclass γ, and if X^{γ} is the part of X where elements in γ have fixedpoints, then the natural restrictionmap K_{G}(X) → K^{G}(X^{γ}) induces an isomorphism when localised at β. In chapter 6 I show how to associate to certain maps f : X → Y of (Gspaces a homomorphism f_{!} : K_{G}(X) → K_{G}(Y). It is the analogue of the Gysin homomorphism in ordinary cohomologytheory; but it can also be regarded as a generalisation of the inducedrepresentation construction of 2.4. In the important special case when f is a fibration whose fibre is a rational algebraic variety I prove that f_{!} is leftinverse to the natural map f^{!} : K_{G}(Y) → K_{G}(X); and I apply that to obtain the general Thom isomorphism theorem. Finally in chapter 7 I prove the theorem towards which my thesis was originally directed. Just as a Gmodule defines a vectorbundle on the classifyingspace B_{G} for G (of [1]), so a G~vectorbundle on X defines a vectorbundle on the space X_{G} fibred over B_{G} with fibre X. Thus one gets a homomorphism α : K_{G}(X) → K(X_{G}). I prove that if K_{G}(X) and K(X_{G}) are given suitable topologies then in certain circumstances K(X_{G})is complete and α induces an isomorphism of the completion of K_{G}(X) with K(X_{G}). This generalises the theorem of AtiyahHirsebruch that R(G)^ ≅ K(B_{G}).
