Relative homology

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In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace.

1 Definition
2 Properties
3 Functoriality
4 References
5 See also

[edit] Definition

Given a subspace $A\subset X$ , one may form the short exact sequence

$0\to C_\bullet(A) \to C_\bullet(X)\to C_\bullet(X) /C_\bullet(A) \to 0$

where $C_\bullet(X)$ denotes the singular chains on the space X. The boundary map on $C_\bullet(X)$ leaves $C_\bullet(A)$ invariant and therefore descends to a boundary map on the quotient. The correponding homology is called relative homology:

$H_n(X,A) = H_n (C_\bullet(X) /C_\bullet(A)).$

One says that relative homology is given by the relative cycles, chains whose boundaries are chains on A, modulo the relative boundaries (chains that are homologous to a chain on A, i.e. chains that would be boundaries, modulo A again).

[edit] Properties

The above short exact sequences specifying the relative chain groups gives rise to a chain complex of short exact sequences. An application of the snake lemma then yields a long exact sequence

$\cdots \to H_n(A) \to H_n(X) \to H_n (X,A) \stackrel{\delta}{\to} H_{n-1}(A) \to \cdots .$

The connecting map δ takes a relative cycle, representing a homology class in H_n(X, A), to its boundary (which is a chain in A).

It follows that H_n(X, x₀), where x₀ is a point in X, is the n-th reduced homology group of X.

The Excision theorem says that removing a sufficiently nice subset Z ⊂ A leaves the relative homology groups H_n(X, A) unchanged. Using the long exact sequence of pairs and the excision theorem, one can show that H_n(X, A) is the same as the n-th reduced homology groups of the quotient space X/A.

The n-th local homology group of a space X at a point x₀ is defined to be H_n(X, X - x₀). Informally, this is the homology of a small sphere containing x₀.