# Topologically realizable homology endomorphism

*This term is nonstandard and is being used locally within the wiki. For its use outside the wiki, please define the term when using it.*

## 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 , all homology endomorphisms which act as identity on the zeroth homology, are topologically realizable

The special treatment at can be dispensed with if we instead work with reduced homology.