Singular homology: Difference between revisions

Revision as of 02:22, 22 May 2007

Definition

Singular homology over a ring $R$ is a homology theory that can be defined for any pair $(X, A)$ where $X$ is a topological space and $A$ is a subspace. It is defined as the homology of the singular complex associated with the pair $(X, A)$ with coefficients in $R$ .

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 ==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==