Algebraic topology : endomorphisms of complete spaces
Let X be a based, connected CW complex of finite type, where by finite type we mean Hi(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 p-complete, [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 H-space or a co-H-space, 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 H-space such that [X,X]/N is any given finite field. The methods are related to work of Adams and Kuhn.- Also working in the P-complete 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/hp ∞ = 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 H-space or a co-H-space, 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.