[For the entire collection see Zbl 0682.00015.]
Let X be a p-complete based CW complex of finite type. The space X is called atomic if each \(f\in [X,X]\) is an equivalence or is topologically nilpotent (i.e. \(\{f^ n\}\) converges to 0 in the profinite compact space [X,X]). Modifying work of Adams, the author shows that, for the space X, f is topologically nilpotent if, and only if, \(f^*\in end(Q\bar H^ k(X:{\mathbb{Z}}_ p))\) is nilpotent for each k. Define \(N=\{f\in [X,X]:\) fg is topologically nilpotent for all \(g\in [X,X]\}\). Then [X,X]/N is shown to be a ring which, with the quotient topology, is isomorphic to the image of [X,X]\(\to \prod end(Q\bar H^ k(X:{\mathbb{Z}}_ p))\) divided by its Jacobson radical and therefore is isomorphic to a countable product of matrix algebras \(\prod_{i\geq 1}M(n_ i,F_ i)\) with the product topology where each \(F_ i\) is a finite field. In the case where X is an atomic H-space, [X,X]/N is a finite field. The image of the representation [X,X]\(\to \prod end(\bar H^ k(X:{\mathbb{Z}}_ 2))\) is calculated for a number of Lie groups and loop spaces X.