Homology theory - Revision history

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2008-05-11T19:46:45Z

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Vipul: /* Axioms */

2007-05-22T02:09:39Z

Axioms

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	* '''Exactness axiom''': For every pair <math>(X,A)</math> with inclusions <math>(A,\emptyset) \to (X,\emptyset) \to (X,A)</math>, there is a long exact sequence:		* '''Exactness axiom''': For every pair <math>(X,A)</math> with inclusions <math>(A,\emptyset) \to (X,\emptyset) \to (X,A)</math>, there is a long exact sequence:

	<math>\ldots \to H_n(A,\emptyset) \to H_n(X,emptyset) \to H_n(X,A) \to H_{n-1}(A,\emptyset) \to \ldots </math>		<math>\ldots \to H_n(A,\emptyset) \to H_n(X,\emptyset) \to H_n(X,A) \to H_{n-1}(A,\emptyset) \to \ldots </math>
	* '''Excision axiom''': For every open subset <math>U</math> whose closure lies in the interior of <math>A</math>, the map of homotopy groups induced by the inclusion <math>(X - U, A - U) \to (X,A)</math> is an isomorphism		* '''Excision axiom''': For every open subset <math>U</math> whose closure lies in the interior of <math>A</math>, the map of homotopy groups induced by the inclusion <math>(X - U, A - U) \to (X,A)</math> is an isomorphism
			* '''Dimension axiom''': If <math>X</math> is a one-point space, then <math>H_n(X,\emptyset)</math> is trivial for all <math>n > 0</math>. One calls <math>H_0(X,\emptyset)</math> the '''coefficient group''' of the homology theory.

			For a homology theory, the homology of a topological space <math>X</math> is defined as the homology of the pair <math>(X,\emptyset)</math>.

Vipul at 02:06, 22 May 2007

2007-05-22T02:06:18Z

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==Definition==

Let <math>C</math> be the category of [[compact polyhedral pair]]s. A '''homology theory''' on <math>C</math> is defined as follows.

===Data===

* For every nonnegative integer <math>n</math>, a functor <math>H_n: C \to Ab</math> where <math>Ab</math> denotes the category of [[Abelian group]]s
* For every positive integer, a natural transformation <math>\partial_n: H_n \to H_{n-1} \circ R</math> where <math>R</math> is the functor that sends <math>(X,A)</math> to <math>(A,\emptyset)</math>.

===Axioms===

* '''Homotopy axiom''': If <math>f_0,f_1:(X,A) \to (Y,B)</math> are homotopic, then <math>H_n(f_0) = H_n(f_1)</math>
* '''Exactness axiom''': For every pair <math>(X,A)</math> with inclusions <math>(A,\emptyset) \to (X,\emptyset) \to (X,A)</math>, there is a long exact sequence:

<math>\ldots \to H_n(A,\emptyset) \to H_n(X,emptyset) \to H_n(X,A) \to H_{n-1}(A,\emptyset) \to \ldots </math>
* '''Excision axiom''': For every open subset <math>U</math> whose closure lies in the interior of <math>A</math>, the map of homotopy groups induced by the inclusion <math>(X - U, A - U) \to (X,A)</math> is an isomorphism