Vipul: /* Definition */

2011-01-11T21:57:41Z

Definition

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	The '''homology''' of <math>C</math>, denoted <math>H_*(C)</math>, is a collection of groups <math>H_n(C), n \in \mathbb{Z}</math>, defined as follows:		The '''homology''' of <math>C</math>, denoted <math>H_*(C)</math>, is a collection of groups <math>H_n(C), n \in \mathbb{Z}</math>, defined as follows:

	<math>H_n(C) = \operatorname{Ker}(\partial_n)/\operatorname{Im}(\partial_{n-1}})</math>.		<math>H_n(C) = \operatorname{Ker}(\partial_n)/\operatorname{Im}(\partial_{n-1})</math>.

	Note that if <math>C</math> is being viewed simply as a chain complex of abelian groups, this is a quotient in the abelian group sense. If <math>C</math> is being viewed as a chain complex of modules over a commutative unital ring, the quotient is a quotient module and the homology groups also get module structures over that ring.		Note that if <math>C</math> is being viewed simply as a chain complex of abelian groups, this is a quotient in the abelian group sense. If <math>C</math> is being viewed as a chain complex of modules over a commutative unital ring, the quotient is a quotient module and the homology groups also get module structures over that ring.

Vipul: Created page with '==Definition== Suppose $C$ is a chain complex, i.e., a collection of groups $C_n, n \in \mathbb{Z}$ , along with boundary maps $\partial_n: C_n \to...'$

2011-01-11T21:57:13Z

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==Definition==

Suppose <math>C</math> is a [[chain complex]], i.e., a collection of groups <math>C_n, n \in \mathbb{Z}</math>, along with boundary maps <math>\partial_n: C_n \to C_{n-1}</math> for all <math>n \in \mathbb{Z}</math> such that <math>\partial_{n-1} \circ \partial_n = 0</math> for all <math>n</math>. In other words, we have:

<math>\dots \stackrel{\partial_{n+1}}{\to} C_n \stackrel{\partial_n}{\to} C_{n-1} \stackrel{\partial_{n-1}}{\to} C_{n-2} \stackrel{\partial_{n-2}}{\to} \dots</math>

The '''homology''' of <math>C</math>, denoted <math>H_*(C)</math>, is a collection of groups <math>H_n(C), n \in \mathbb{Z}</math>, defined as follows:

<math>H_n(C) = \operatorname{Ker}(\partial_n)/\operatorname{Im}(\partial_{n-1}})</math>.

Note that if <math>C</math> is being viewed simply as a chain complex of abelian groups, this is a quotient in the abelian group sense. If <math>C</math> is being viewed as a chain complex of modules over a commutative unital ring, the quotient is a quotient module and the homology groups also get module structures over that ring.

==Functoriality and invariance==

===Basic statement===

Homology is functorial, in the sense that each <math>H_n</math> is a covariant functor:

* For abelian groups: For each fixed <math>n</math>, the association <math>C \mapsto H_n(C)</math> is a functor from the [[category of chain complexes with chain maps]] (over abelian groups) to the category of abelian groups with group homomorphisms. In particular, a [[chain map]] between chain complexes induces a homomorphism between the corresponding homology groups for each <math>n</math>.
* For modules over a commutative unital ring <math>R</math>: For each fixed <math>n</math>, the association <math>C \mapsto H_n(C)</math> is a functor from the [[category of chain complexes with chain maps]] (of <math>R</math>-modules) to the category of <math>R</math>-modules with module maps. In particular, a [[chain map]] between chain complexes induces a homomorphism between the corresponding homology modules for each <math>n</math>.

===Homotopy invariance===

Two [[chain-homotopic chain map]]s <math>f,g:A \to B</math> between a pair of chain complexes induce identical maps on the corresponding homology groups.

===Composition with other functors===

A chain complex may itself arise by the application of a functor to a topological, algebraic, or differential construct. For instance, we can associate to any [[topological space]] a [[singular chain complex]]. This in turn has a homology as defined here. Composing these two functors, we get a bunch of functor from the [[category of topological spaces]] to the category of abelian groups with group homomorphisms. Somewhat confusingly, these groups are often called the homology groups of the topological space.

Homology of a chain complex - Revision history

Vipul: /* Definition */

Vipul: Created page with '==Definition== Suppose C is a chain complex, i.e., a collection of groups C_n, n \in \mathbb{Z}, along with boundary maps \partial_n: C_n \to...'

Vipul: Created page with '==Definition== Suppose $C$ is a chain complex, i.e., a collection of groups $C_n, n \in \mathbb{Z}$ , along with boundary maps $\partial_n: C_n \to...'$