The first step in studing self maps of cyclic Moore spaces, Pm(k) = Pm(ℤ/kℤ), is to look at what is induced in mod k homology. If the map induces 0 in mod k homology then it will factor through the homotopy theoretic fibre Fm{k} of the pinch map q: Pm(k) → Sm. We begin this chapter by studying the homology of the fibre of the pinch map and then use it to determine the homotopy classes of self maps of cyclic Moore spaces.

We use the results on self maps to determine the precise exponents of the homotopy groups with cyclic coefficients. Furthermore, if the self map of a Moore space k: Pm(k) → Pm(k), that is, k times the identity, is null homotopic, it implies that the smash products Pm(k) ∧ Pn(k) have the homotopy type of a bouquet, Pm+n(k) ∨ Pm+n−1(k).

These decompositions of smash products into bouquets are uniquely determined up to compositions with Whitehead products by the decomposition of the mod k homology of the smash into a direct sum. In order to see this, we use the fact that, if p is an odd prime, then the odd-dimensional Moore space P2n+1(pr) is equivalent modulo Whitehead products through a range of dimensions with the fibre S2n+1{pr} of the degree pr map pr: S2n+1 → S2n+1. Surprisingly, we also have the stronger result that the spaces ΩP2n+2(pr) and S2n+1{pr} are equivalent through all dimensions up to compositions with multiplicative extensions of Samelson products.