Cellular chain complex: Difference between revisions

Revision as of 00:31, 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 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:

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

We compose this with the natural map from $H_{n - 1} (X^{n - 1})$ 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.

@@ Line 4: / Line 4: @@
 * The <math>n^{th}</math> member is the 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 [[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>
@@ Line 10: / Line 10: @@
 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 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.
+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==