Cellular chain complex - Revision history

Vipul: /* Facts */

2011-01-09T20:31:16Z

Facts

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	==Facts==		==Facts==

	The <math>n^{th}</math> homology group of the cellular chain complex, is isomorphic to the <math>n^{th}</math> homology of the pair <math>(X,X^{-1})</math> (<math>X^{-1}</math> can be viewed as the ''base space'').		The <math>n^{th}</math> homology group of the cellular chain complex, is isomorphic to the <math>n^{th}</math> [[singular homology]] of the pair <math>(X,X^{-1})</math> (<math>X^{-1}</math> can be viewed as the ''base space''). In particular, if <math>X^{-1}</math> is empty, i.e., the filtration begins with the empty set, then the cellular homology of the filtration equals the singular homology of <math>X</math>. {{further\|[[cellular homology equals singular homology]]}}

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

Vipul at 00:33, 26 December 2010

2010-12-26T00:33:52Z

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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 '''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>		<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]]}}		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==		==Facts==

Vipul at 00:31, 26 December 2010

2010-12-26T00:31:33Z

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	* 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 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>		<math>H_n(X^n, X^{n-1}) \to H_{n-1}(X^{n-1})</math>
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	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>.		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==		==Facts==

Vipul: 2 revisions

2008-05-11T19:36:59Z

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Vipul at 23:56, 2 November 2007

2007-11-02T23:56:19Z

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	The <math>n^{th}</math> homology group of the cellular chain complex, is isomorphic to the <math>n^{th}</math> homology of the pair <math>(X,X^{-1})</math> (<math>X^{-1}</math> can be viewed as the ''base space'').		The <math>n^{th}</math> homology group of the cellular chain complex, is isomorphic to the <math>n^{th}</math> homology of the pair <math>(X,X^{-1})</math> (<math>X^{-1}</math> can be viewed as the ''base space'').

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

			==Functoriality==

			{{further\|[[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.

Vipul at 23:00, 24 October 2007

2007-10-24T23:00:39Z

New page

==Definition==

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 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>

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.

==Facts==

The <math>n^{th}</math> homology group of the cellular chain complex, is isomorphic to the <math>n^{th}</math> homology of the pair <math>(X,X^{-1})</math> (<math>X^{-1}</math> can be viewed as the ''base space'').

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