Singular homology - Revision history

Vipul: 4 revisions

2008-05-11T19:58:29Z

4 revisions

← Older revision	Revision as of 19:58, 11 May 2008
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Vipul: /* Definition */

2007-05-22T02:22:59Z

Definition

← Older revision		Revision as of 02:22, 22 May 2007
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	==Definition==		==Definition==

	'''Singular homology''' over a ring <math>R</math> is a homology theory that can be defined for any pair <math>(X,A)</math> ~~of a topological space and a subspace, the following [[chain complex of Abelian groups]]:~~		'''Singular homology''' over a ring <math>R</math> is a homology theory that can be defined for any pair <math>(X,A)</math> where <math>X</math> is a topological space and <math>A</math> is a subspace. It is defined as the homology of the [[singular complex]] associated with the pair <math>(X,A)</math> with coefficients in <math>R</math>.

	* The <math>n^{th}</math> chain group, denoted <math>~~C_n(~~X)</math>, is ~~defined as the free <math>R</math>-module genereated by all continuous maps from <math>n</math>-simplices to <math>X</math> where~~ <math>A</math> is the ~~image~~ of ~~a subsimplex.~~
	* The boundary map <math>~~\partial:C_n(X) \to C_{n-1}~~(X)</math> ~~sends each continuous map from an <math>n</math>-simplex to <math>X</math>, to a suitable alternating sum of continuous maps from its codimension one faces to <math>X</math>.~~

	~~Note that when no ring is specified, we can assume that we are working over <math>\mathbb{Z}</math>,~~ in ~~which case the chain group is simply the free Abelian group generated by all continuous maps from~~ <math>~~n</math>-simplices to <math>X~~</math>.

	==Relation with other homology theories==		==Relation with other homology theories==

Vipul at 02:20, 22 May 2007

2007-05-22T02:20:43Z

← Older revision		Revision as of 02:20, 22 May 2007
Line 3:		Line 3:
	==Definition==		==Definition==

	'''Singular homology''' over a ring <math>R</math> is a homology theory that can be defined for any [[topological space~~]]. It is defined as the homology of~~ the following [[chain complex of Abelian groups]] ~~associated with a topoligicalspace <math>X</math>~~:		'''Singular homology''' over a ring <math>R</math> is a homology theory that can be defined for any pair <math>(X,A)</math> of a topological space and a subspace, the following [[chain complex of Abelian groups]]:

	* The <math>n^{th}</math> chain group, denoted <math>C_n(X)</math>, is defined as the free <math>R</math>-module genereated by all continuous maps from <math>n</math>-simplices to <math>X</math>		* The <math>n^{th}</math> chain group, denoted <math>C_n(X)</math>, is defined as the free <math>R</math>-module genereated by all continuous maps from <math>n</math>-simplices to <math>X</math> where <math>A</math> is the image of a subsimplex.
	* The boundary map <math>\partial:C_n(X) \to C_{n-1}(X)</math> sends each continuous map from an <math>n</math>-simplex to <math>X</math>, to a suitable alternating sum of continuous maps from its codimension one faces to <math>X</math>.		* The boundary map <math>\partial:C_n(X) \to C_{n-1}(X)</math> sends each continuous map from an <math>n</math>-simplex to <math>X</math>, to a suitable alternating sum of continuous maps from its codimension one faces to <math>X</math>.

Vipul: Simplicial homology moved to Singular homology

2007-05-22T02:17:20Z

Simplicial homology moved to Singular homology

← Older revision	Revision as of 02:17, 22 May 2007
(No difference)

Vipul at 02:17, 22 May 2007

2007-05-22T02:17:04Z

New page

{{homology theory}}

==Definition==

'''Singular homology''' over a ring <math>R</math> is a homology theory that can be defined for any [[topological space]]. It is defined as the homology of the following [[chain complex of Abelian groups]] associated with a topoligicalspace <math>X</math>:

* The <math>n^{th}</math> chain group, denoted <math>C_n(X)</math>, is defined as the free <math>R</math>-module genereated by all continuous maps from <math>n</math>-simplices to <math>X</math>
* The boundary map <math>\partial:C_n(X) \to C_{n-1}(X)</math> sends each continuous map from an <math>n</math>-simplex to <math>X</math>, to a suitable alternating sum of continuous maps from its codimension one faces to <math>X</math>.

Note that when no ring is specified, we can assume that we are working over <math>\mathbb{Z}</math>, in which case the chain group is simply the free Abelian group generated by all continuous maps from <math>n</math>-simplices to <math>X</math>.

==Relation with other homology theories==