Topologically realizable homology endomorphism - Revision history

Vipul: 3 revisions

2008-05-11T19:59:35Z

3 revisions

← Older revision	Revision as of 19:59, 11 May 2008
(No difference)

Vipul at 02:02, 27 October 2007

2007-10-27T02:02:41Z

← Older revision		Revision as of 02:02, 27 October 2007
Line 16:		Line 16:
	==Facts==		==Facts==

	* The zero ~~map~~ is always topologically realizable: we can consider a continuous map that sends everything to a single point		* 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		* 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		* For all spheres <math>S^n</math>, all homology endomorphisms which act as identity on the zeroth homology, are topologically realizable

			The special treatment at <math>n=0</math> can be dispensed with if we instead work with reduced homology.

Vipul: /* Facts */

2007-10-27T02:00:41Z

Facts

← Older revision		Revision as of 02:00, 27 October 2007
Line 18:		Line 18:
	* 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

Vipul at 01:58, 27 October 2007

2007-10-27T01:58:39Z

New page

{{wikilocal}}

==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