Title:

Algebraic topology : endomorphisms of complete spaces

Let X be a based, connected CW complex of finite type, where by finite type we mean H_{i}(X,Z/p) is finite dimensional for each i, where p is a prime. Under various assumptions, we study homology and cohomology representations of [X,X], the set of homology classes of based self maps of X. The main results of the thesis are as follows. When X is pcomplete, [X,X] has a natural profinite topology, with a zero given by the constant map. LetN = (fε[X,X]/fg is topologically nilpotent for all gε[X,X]). Hubbuck has shown that if X is an Hspace or a coHspace, then [X,X]/N is well defined and has a natural ring structure as a product of matrix algebras over finite fields of characteristic p. We construct an Hspace such that [X,X]/N is any given finite field. The methods are related to work of Adams and Kuhn. Also working in the Pcomplete category, we then turn to consider the group of based self equivalences, ε(X), of X. With the subspace topology induced from [X,X], it is a profinite group. For any profinite group G, we write O_p(G) = (g ε G/h^{p ∞ =} 1 → (gh)^p ∞ = 1). Then we establish the following,Theorem (7.1) We have the isomorphisms of the following topological groups:goodbreakmidinsertvskip 2.0cm endinsert2) If X is an Hspace or a coHspace, thengoodbreakmidinsertvskip 2.0cm endinsertwhere, as usual, GL(n_i,F_i) denotes the n_i x n_i general linear group over a finite field F_i.
