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algebraic topology 250px, A torus, one of the most frequently studied objects in algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathem ...
, a branch of
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for topological pair, 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

Given a subspace A\subseteq 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 \partial'_\bullet on the quotient. If we denote this quotient by C_n(X,A):=C_n(X)/C_n(A), we then have a complex :\cdots\longrightarrow C_n(X,A) \xrightarrow C_(X,A) \longrightarrow \cdots . By definition, the th relative homology group of the pair of spaces (X,A) is :H_n(X,A) := \ker\partial'_n/\operatorname\partial'_. 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).


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) \stackrel H_n(X) \stackrel H_n (X,A) \stackrel H_(A) \to \cdots . The connecting map ''\partial'' takes a relative cycle, representing a homology class in H_n(X,A), to its boundary (which is a cycle in ''A''). It follows that H_n(X,x_0), where x_0 is a point in ''X'', is the ''n''-th reduced homology group of ''X''. In other words, H_i(X,x_0) = H_i(X) for all i > 0. When i = 0, H_0(X,x_0) is the free module of one rank less than H_0(X). The connected component containing x_0 becomes trivial in relative homology. The excision theorem says that removing a sufficiently nice subset Z \subset 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. Relative homology readily extends to the triple (X,Y,Z) for Z \subset Y \subset X. One can define the Euler characteristic for a pair Y \subset X by :\chi (X, Y) = \sum _ ^n (-1)^j \operatorname H_j (X, Y) . The exactness of the sequence implies that the Euler characteristic is ''additive'', i.e., if Z \subset Y \subset X, one has : \chi (X, Z) = \chi (X, Y) + \chi (Y, Z) .


Local homology

The n-th local homology group of a space X at a point x_0, denoted :H_(X) is defined to be the relative homology group H_n(X,X\setminus \). Informally, this is the "local" homology of X close to x_0.


Local homology of the cone CX at the origin

One easy example of local homology is calculating the local homology of the cone (topology) of a space at the origin of the cone. Recall that the cone is defined as the quotient space :CX = (X\times I)/(X\times\) , where X \times \ has the subspace topology. Then, the origin x_0 = 0 is the equivalence class of points [X\times 0]. Using the intuition that the local homology group H_(CX) of CX at x_0 captures the homology of CX "near" the origin, we should expect this is the homology of H_*(X) since CX \setminus \ has a homotopy retract to X. Computing the local cohomology can then be done using the long exact sequence in homology :\begin \to &H_n(CX\setminus \)\to H_n(CX) \to H_(CX)\\ \to & H_(CX\setminus \)\to H_(CX) \to H_(CX). \end Because the cone of a space is Contractible space, contractible, the middle homology groups are all zero, giving the isomorphism :\begin H_(CX) & \cong H_(CX \setminus \) \\ & \cong H_(X), \end since CX \setminus \ is contractible to X.


In algebraic geometry

Note the previous construction can be proven in Algebraic geometry using the Cone (algebraic geometry), affine cone of a projective variety X using Local cohomology.


Local homology of a point on a smooth manifold

Another computation for local homology can be computed on a point p of a manifold M. Then, let K be a compact neighborhood of p isomorphic to a closed disk \mathbb^n = \ and let U = M \setminus K. Using the excision theorem there is an isomorphism of relative homology groups :\begin H_n(M,M\setminus\) &\cong H_n(M\setminus U, M\setminus (U\cup \)) \\ &= H_n(K, K\setminus\), \end hence the local homology of a point reduces to the local homology of a point in a closed ball \mathbb^n. Because of the homotopy equivalence :\mathbb^n \setminus \ \simeq S^ and the fact :H_k(\mathbb^n) \cong \begin \Z & k = 0 \\ 0 & k \neq 0 , \end the only non-trivial part of the long exact sequence of the pair (\mathbb,\mathbb\setminus\) is :0 \to H_(\mathbb^n) \to H_(S^) \to 0 , hence the only non-zero local homology group is H_(\mathbb^n).


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 (X,A) and (Y,B) be pairs of spaces such that A\subseteq X and B\subseteq Y, and let f\colon X\to Y be a continuous map. Then there is an induced map f_\#\colon C_n(X)\to C_n(Y) on the (absolute) chain groups. If f(A)\subseteq B, then f_\#(C_n(A))\subseteq C_n(B). Let \begin \pi_X&:C_n(X)\longrightarrow C_n(X)/C_n(A) \\ \pi_Y&:C_n(Y)\longrightarrow C_n(Y)/C_n(B) \\ \end be the Quotient group#Properties, natural projections which take elements to their equivalence classes in the quotient groups. Then the map \pi_Y\circ f_\#\colon C_n(X)\to C_n(Y)/C_n(B) is a group homomorphism. Since f_\#(C_n(A))\subseteq C_n(B)=\ker\pi_Y, this map descends to the quotient, inducing a well-defined map \varphi\colon C_n(X)/C_n(A)\to C_n(Y)/C_n(B) such that the following diagram commutes: Chain maps induce homomorphisms between homology groups, so f induces a map f_*\colon H_n(X,A)\to H_n(Y,B) on the relative homology groups.


Examples

One important use of relative homology is the computation of the homology groups of quotient spaces X/A. In the case that A is a subspace of X fulfilling the mild regularity condition that there exists a neighborhood of A that has A as a deformation retract, then the group \tilde H_n(X/A) is isomorphic to H_n(X,A). We can immediately use this fact to compute the homology of a sphere. We can realize S^n as the quotient of an n-disk by its boundary, i.e. S^n = D^n/S^. Applying the exact sequence of relative homology gives the following:
\cdots\to \tilde H_n(D^n)\rightarrow H_n(D^n,S^)\rightarrow \tilde H_(S^)\rightarrow \tilde H_(D^n)\to \cdots. 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: 0\rightarrow H_n(D^n,S^) \rightarrow \tilde H_(S^) \rightarrow 0. Therefore, we get isomorphisms H_n(D^n,S^)\cong \tilde H_(S^). We can now proceed by induction to show that H_n(D^n,S^)\cong \Z. Now because S^ is the deformation retract of a suitable neighborhood of itself in D^n, we get that H_n(D^n,S^)\cong \tilde H_n(S^n)\cong \Z. Another insightful geometric example is given by the relative homology of (X=\Complex^*, D = \) where \alpha \neq 0, 1. Then we can use the long exact sequence : \begin 0 &\to H_1(D)\to H_1(X) \to H_1(X,D) \\ & \to H_0(D)\to H_0(X) \to H_0(X,D) \end = \begin 0 & \to 0 \to \Z \to H_1(X,D) \\ & \to \Z^ \to \Z \to 0 \end Using exactness of the sequence we can see that H_1(X,D) contains a loop \sigma counterclockwise around the origin. Since the cokernel of \phi\colon \Z \to H_1(X,D) fits into the exact sequence : 0 \to \operatorname(\phi) \to \Z^ \to \Z \to 0 it must be isomorphic to \Z. One generator for the cokernel is the 1-chain [1,\alpha] since its boundary map is :\partial([1,\alpha]) = [\alpha] - [1]


See also

*Excision theorem, Excision Theorem *Mayer–Vietoris sequence


Notes

wiktionary:i.e., i.e., the boundary \partial\colon C_n(X)\to C_(X) maps C_n(A) to C_(A)


References

* * Joseph J. Rotman, ''An Introduction to Algebraic Topology'', Springer Science+Business Media, Springer-Verlag, ;Specific {{Reflist Homology theory