Topologically realizable homology endomorphism: Difference between revisions
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* The zero map is always topologically realizable: we can consider a continuous map that sends everything to a single point | * The zero map 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 | * The identity map is always topologically realizable: it occurs as the image of a continuous map | ||
* For all spheres <math>S^n</math> with <math>n > 0</math>, all homology endomorphisms which act as identity on the zeroth homology, are topologically realizable | |||
Revision as of 02:00, 27 October 2007
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 zero map 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 with , all homology endomorphisms which act as identity on the zeroth homology, are topologically realizable