“Relative homology”的意思、由来-开放百科全书

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.

Definition

where

denotes the singular chains on the space X. The boundary map on

leaves

invariant^[1] and therefore descends to a boundary map

on the quotient. If we denote this quotient by

, we then have a complex

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).^[2]

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

The connecting map δ takes a relative cycle, representing a homology class in H_n(X, A), to its boundary (which is a cycle in A).^[3]

It follows that H_n(X, x₀), where x₀ is a point in X, is the n-th reduced homology group of X. In other words, H_i(X, x₀) = H_i(X) for all i > 0. When i = 0, H₀(X, x₀) is the free module of one rank less than H₀(X). The connected component containing x₀ becomes trivial in relative homology.

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 "local" homology of X close to x₀.

The exactness of the sequence implies that the Euler characteristic is additive, i.e. if Z ⊂ Y ⊂ X, one has

Functoriality

Just as in absolute homology, continuous maps between spaces induce homomorphisms between relative homology groups. In fact, this map is exactly the induced map on homology groups, but it descends to the quotient.

Let

and

be pairs of spaces such that

and

, and let

be a continuous map. Then there is an induced map

on the (absolute) chain groups. If

, then

. Let

be the natural projections which take elements to their equivalence classes in the quotient groups. Then the map

is a group homomorphism. Since

, this map descends to the quotient, inducing a well-defined map

such that the following diagram commutes:

Chain maps induce homomorphisms between homology groups, so

induces a map

on the relative homology groups.^[3]

Examples

One important use of relative homology is the computation of the homology groups of quotient spaces

. In the case that

is a subspace of

fulfilling the mild regularity condition that there exists a neighborhood of

that has

as a deformation retract, then the group

is isomorphic to

. We can immediately use this fact to compute the homology of a sphere. We can realize

as the quotient of an n-disk by its boundary, i.e.

. Applying the exact sequence of relative homology gives the following:

Because the disk is contractible, we know its reduced homology groups vanish in all dimensions, so the above sequence collapses to the short exact sequence: