Topologically realizable homology endomorphism

Definition
Given a nonempty topological space, a homology endomorphism (viz, an endomorphism for each of its homology groups) is termed topologically realizable if it occurs by applying the homology functor to some continuous map from the topological space]] to itself.

The set of homology endomorphisms is closed under composition (and is hence a submonoid of the monoid of homology endomorphisms under composition). However, it is not in general additively closed.

(We may also sometimes be interested in studying realizable endomorphisms for only one homology group).

Related notions

 * Topologically realizable homology automorphism
 * Topologically realizable fundamental group automorphism
 * Topologically realizable fundamental group endomorphism

Facts

 * The map which is identity on zeroth homology and zero elsewhere is always topologically realizable: we can consider a continuous map that sends everything to a single point
 * The identity map is always topologically realizable: it occurs as the image of a continuous map
 * For all spheres $$S^n$$, all homology endomorphisms which act as identity on the zeroth homology, are topologically realizable

The special treatment at $$n=0$$ can be dispensed with if we instead work with reduced homology.