Cellular chain complex: Difference between revisions

Revision as of 00:33, 26 December 2010

Definition

The cellular chain complex of a cellular space $X$ (viz, a topological space $X$ equipped with a cellular filtration $X^{n}$ ) is described as follows:

The $n^{t h}$ member is the relative homology group $H_{n} (X^{n}, X^{n - 1})$
The boundary map is defined as follows. First note that the long exact sequence of homology of a pair $(X^{n}, X^{n - 1})$ gives a map:

$H_{n} (X^{n}, X^{n - 1}) \to H_{n - 1} (X^{n - 1})$

The long exact sequence of homology of a pair $(X^{n - 1}, X^{n - 2})$ gives a map:

$H_{n - 1} (X^{n - 1}) \to H_{n - 1} (X^{n - 1}, X^{n - 2})$ .

Composing these two maps, we get the boundary map for the chain complex:

$H_{n} (X^{n}, X^{n - 1}) \to H_{n - 1} (X^{n - 1}, X^{n - 2})$

The fact that the composite of two boundary maps is zero, follows from the trick of writing each chain map as a composite of the two maps as above, and then noting that in the composite, we get a composite of two consecutive terms of a long exact sequence of homology. Further information: composite of consecutive maps of cellular chain complex is zero

Facts

The $n^{t h}$ homology group of the cellular chain complex, is isomorphic to the $n^{t h}$ homology of the pair $(X, X^{- 1})$ ( $X^{- 1}$ can be viewed as the base space).

Cellular homology is typically used only for cellular filtrations arising from CW-complex structures.

Functoriality

Further information: Cellular chain complex functor

The cellular chain complex can be viewed as a functor from the category of cellular spaces with cellular maps, to the category of chain complexes with chain maps.

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 The '''cellular chain complex''' of a [[cellular space]] <math>X</math> (viz, a [[topological space]] <math>X</math> equipped with a cellular filtration <math>X^n</math>) is described as follows:
-* The <math>n^{th}</math> member is the group <math>H_n(X^n, X^{n-1})</math>
+* The <math>n^{th}</math> member is the [[relative homology group]] <math>H_n(X^n, X^{n-1})</math>
-* The boundary map is defined as follows. First note that the [[long exact sequence of homology of a pair]] <math>(X^n,X^{n-1})</math> gives a map:
+* The boundary map is defined as follows. First note that the [[defining ingredient::long exact sequence of homology of a pair]] <math>(X^n,X^{n-1})</math> gives a map:
 <math>H_n(X^n, X^{n-1}) \to H_{n-1}(X^{n-1})</math>
-We compose this with the natural map from <math>H_{n-1}(X^{n-1})</math> to <math>H_{n-1}(X^{n-1},X^{n-2})</math>.
+The [[long exact sequence of homology of a pair]] <math>(X^{n-1},X^{n-2})</math> gives a map:
+<math>H_{n-1}(X^{n-1}) \to H_{n-1}(X^{n-1},X^{n-2})</math>.
+Composing these two maps, we get the boundary map for the chain complex:
+<math>H_n(X^n,X^{n-1}) \to H_{n-1}(X^{n-1},X^{n-2})</math>
 The fact that the composite of two boundary maps is zero, follows from the trick of writing each chain map as a composite of the two maps as above, and then noting that in the composite, we get a composite of two consecutive terms of a long exact sequence of homology. {{further|[[composite of consecutive maps of cellular chain complex is zero]]}}
 ==Facts==