Equivariant algebraic topology and the equivariant Brown representability theorem

Abstract

The main purpose of this thesis is to give a complete proof of the Equivariant Brown Represcentability Theorem, in the process developing the equivariant algebraic topology needed in the final proof. The proof of the theorem in the category of path-connected G-spaces is given in Chapter 4 and follows the proof of the non-equivariant case given in Spanier ([Sp], pp. 406-411). There is another account of the proof given in Switzer ([Sw], pp. 152-157), which is closer to the original account given by Brown [Br1]. Equivariant versions of the theorem are announced in [LMS] and [V], for example, but no details of the proofs are given. -- In Chapter 1, the basic theory of G-spaces and G-maps is presented. G-final and G-initial structures on a set are defined and sufficient conditions are given which allow such G-space structures to be constructed. -- The equivariant homotopy groups are defined in Chapter 2 and the isomorphisms πHn(X) ≅ πn(XH) and πHn(X,A) ≅ πn(XH,AH) are established. These two results are then used to prove the results about equivariant homotopy groups needed in Chapter 4. -- In Chapter 3, G-CW-complexes are defined and all the necessary homotopic properties of G-CW-complexes are developed, culminating in the proof of the Equivariant Whitehead Theorem. -- Finally, in Chapter 5, we prove an equivariant version of the statement that if a functor satisfies the Wedge Axiom and the Mayer-Victoris Axiom given in Brown's original version of his theorem, then it also satisfies the Equalizer Axiom given in Spanier's version of the theorem. This immediately gives a Switzer style version of the main result.