In

^{th} relative homology group of the pair of spaces $(X,A)$ is
:$H\_n(X,A)\; :=\; \backslash ker\backslash partial\text{'}\_n/\backslash operatorname\backslash partial\text{'}\_.$
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).

$\backslash cdots\backslash to\; \backslash tilde\; H\_n(D^n)\backslash rightarrow\; H\_n(D^n,S^)\backslash rightarrow\; \backslash tilde\; H\_(S^)\backslash rightarrow\; \backslash tilde\; H\_(D^n)\backslash to\; \backslash cdots.$ 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: $0\backslash rightarrow\; H\_n(D^n,S^)\; \backslash rightarrow\; \backslash tilde\; H\_(S^)\; \backslash rightarrow\; 0.$ Therefore, we get isomorphisms $H\_n(D^n,S^)\backslash cong\; \backslash tilde\; H\_(S^)$. We can now proceed by induction to show that $H\_n(D^n,S^)\backslash cong\; \backslash Z$. Now because $S^$ is the deformation retract of a suitable neighborhood of itself in $D^n$, we get that $H\_n(D^n,S^)\backslash cong\; \backslash tilde\; H\_n(S^n)\backslash cong\; \backslash Z$. Another insightful geometric example is given by the relative homology of $(X=\backslash Complex^*,\; D\; =\; \backslash )$ where $\backslash alpha\; \backslash neq\; 0,\; 1$. Then we can use the long exact sequence :$\backslash begin\; 0\; \&\backslash to\; H\_1(D)\backslash to\; H\_1(X)\; \backslash to\; H\_1(X,D)\; \backslash \backslash \; \&\; \backslash to\; H\_0(D)\backslash to\; H\_0(X)\; \backslash to\; H\_0(X,D)\; \backslash end\; =\; \backslash begin\; 0\; \&\; \backslash to\; 0\; \backslash to\; \backslash Z\; \backslash to\; H\_1(X,D)\; \backslash \backslash \; \&\; \backslash to\; \backslash Z^\; \backslash to\; \backslash Z\; \backslash to\; 0\; \backslash end$ Using exactness of the sequence we can see that $H\_1(X,D)$ contains a loop $\backslash sigma$ counterclockwise around the origin. Since the cokernel of $\backslash phi\backslash colon\; \backslash Z\; \backslash to\; H\_1(X,D)$ fits into the exact sequence :$0\; \backslash to\; \backslash operatorname(\backslash phi)\; \backslash to\; \backslash Z^\; \backslash to\; \backslash Z\; \backslash to\; 0$ it must be isomorphic to $\backslash Z$. One generator for the cokernel is the $1$-chain $[1,\backslash alpha]$ since its boundary map is :$\backslash partial([1,\backslash alpha])\; =\; [\backslash alpha]\; -\; [1]$

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 $A\backslash subseteq\; X$, one may form the short exact sequence :$0\backslash to\; C\_\backslash bullet(A)\; \backslash to\; C\_\backslash bullet(X)\backslash to\; C\_\backslash bullet(X)\; /C\_\backslash bullet(A)\; \backslash to\; 0\; ,$ where $C\_\backslash bullet(X)$ denotes the singular chains on the space ''X''. The boundary map on $C\_\backslash bullet(X)$ leaves $C\_\backslash bullet(A)$ invariant and therefore descends to a boundary map $\backslash partial\text{'}\_\backslash bullet$ on the quotient. If we denote this quotient by $C\_n(X,A):=C\_n(X)/C\_n(A)$, we then have a complex :$\backslash cdots\backslash longrightarrow\; C\_n(X,A)\; \backslash xrightarrow\; C\_(X,A)\; \backslash longrightarrow\; \backslash cdots\; .$ By definition, theProperties

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 :$\backslash cdots\; \backslash to\; H\_n(A)\; \backslash stackrel\; H\_n(X)\; \backslash stackrel\; H\_n\; (X,A)\; \backslash stackrel\; H\_(A)\; \backslash to\; \backslash cdots\; .$ The connecting map ''$\backslash partial$'' 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 $Z\; \backslash subset\; A$ 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 $Z\; \backslash subset\; Y\; \backslash subset\; X$. One can define the Euler characteristic for a pair $Y\; \backslash subset\; X$ by :$\backslash chi\; (X,\; Y)\; =\; \backslash sum\; \_\; ^n\; (-1)^j\; \backslash operatorname\; H\_j\; (X,\; Y)\; .$ The exactness of the sequence implies that the Euler characteristic is ''additive'', i.e., if $Z\; \backslash subset\; Y\; \backslash subset\; X$, one has :$\backslash chi\; (X,\; Z)\; =\; \backslash chi\; (X,\; Y)\; +\; \backslash chi\; (Y,\; Z)\; .$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 $H\_n(X,X\backslash setminus\; \backslash )$. 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 :$CX\; =\; (X\backslash times\; I)/(X\backslash times\backslash )\; ,$ where $X\; \backslash times\; \backslash $ has the subspace topology. Then, the origin $x\_0\; =\; 0$ is the equivalence class of points $[X\backslash times\; 0]$. 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\; \backslash setminus\; \backslash $ has a homotopy retract to $X$. Computing the local cohomology can then be done using the long exact sequence in homology :$\backslash begin\; \backslash to\; \&H\_n(CX\backslash setminus\; \backslash )\backslash to\; H\_n(CX)\; \backslash to\; H\_(CX)\backslash \backslash \; \backslash to\; \&\; H\_(CX\backslash setminus\; \backslash )\backslash to\; H\_(CX)\; \backslash to\; H\_(CX).\; \backslash end$ Because the cone of a space is Contractible space, contractible, the middle homology groups are all zero, giving the isomorphism :$\backslash begin\; H\_(CX)\; \&\; \backslash cong\; H\_(CX\; \backslash setminus\; \backslash )\; \backslash \backslash \; \&\; \backslash cong\; H\_(X),\; \backslash end$ since $CX\; \backslash setminus\; \backslash $ 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 $\backslash mathbb^n\; =\; \backslash $ and let $U\; =\; M\; \backslash setminus\; K$. Using the excision theorem there is an isomorphism of relative homology groups :$\backslash begin\; H\_n(M,M\backslash setminus\backslash )\; \&\backslash cong\; H\_n(M\backslash setminus\; U,\; M\backslash setminus\; (U\backslash cup\; \backslash ))\; \backslash \backslash \; \&=\; H\_n(K,\; K\backslash setminus\backslash ),\; \backslash end$ hence the local homology of a point reduces to the local homology of a point in a closed ball $\backslash mathbb^n$. Because of the homotopy equivalence :$\backslash mathbb^n\; \backslash setminus\; \backslash \; \backslash simeq\; S^$ and the fact :$H\_k(\backslash mathbb^n)\; \backslash cong\; \backslash begin\; \backslash Z\; \&\; k\; =\; 0\; \backslash \backslash \; 0\; \&\; k\; \backslash neq\; 0\; ,\; \backslash end$ the only non-trivial part of the long exact sequence of the pair $(\backslash mathbb,\backslash mathbb\backslash setminus\backslash )$ is :$0\; \backslash to\; H\_(\backslash mathbb^n)\; \backslash to\; H\_(S^)\; \backslash to\; 0\; ,$ hence the only non-zero local homology group is $H\_(\backslash mathbb^n)$.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 $A\backslash subseteq\; X$ and $B\backslash subseteq\; Y$, and let $f\backslash colon\; X\backslash to\; Y$ be a continuous map. Then there is an induced map $f\_\backslash \#\backslash colon\; C\_n(X)\backslash to\; C\_n(Y)$ on the (absolute) chain groups. If $f(A)\backslash subseteq\; B$, then $f\_\backslash \#(C\_n(A))\backslash subseteq\; C\_n(B)$. Let $\backslash begin\; \backslash pi\_X\&:C\_n(X)\backslash longrightarrow\; C\_n(X)/C\_n(A)\; \backslash \backslash \; \backslash pi\_Y\&:C\_n(Y)\backslash longrightarrow\; C\_n(Y)/C\_n(B)\; \backslash \backslash \; \backslash end$ be the Quotient group#Properties, natural projections which take elements to their equivalence classes in the quotient groups. Then the map $\backslash pi\_Y\backslash circ\; f\_\backslash \#\backslash colon\; C\_n(X)\backslash to\; C\_n(Y)/C\_n(B)$ is a group homomorphism. Since $f\_\backslash \#(C\_n(A))\backslash subseteq\; C\_n(B)=\backslash ker\backslash pi\_Y$, this map descends to the quotient, inducing a well-defined map $\backslash varphi\backslash colon\; C\_n(X)/C\_n(A)\backslash to\; C\_n(Y)/C\_n(B)$ such that the following diagram commutes: Chain maps induce homomorphisms between homology groups, so $f$ induces a map $f\_*\backslash colon\; H\_n(X,A)\backslash to\; H\_n(Y,B)$ 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 $\backslash tilde\; H\_n(X/A)$ 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:$\backslash cdots\backslash to\; \backslash tilde\; H\_n(D^n)\backslash rightarrow\; H\_n(D^n,S^)\backslash rightarrow\; \backslash tilde\; H\_(S^)\backslash rightarrow\; \backslash tilde\; H\_(D^n)\backslash to\; \backslash cdots.$ 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: $0\backslash rightarrow\; H\_n(D^n,S^)\; \backslash rightarrow\; \backslash tilde\; H\_(S^)\; \backslash rightarrow\; 0.$ Therefore, we get isomorphisms $H\_n(D^n,S^)\backslash cong\; \backslash tilde\; H\_(S^)$. We can now proceed by induction to show that $H\_n(D^n,S^)\backslash cong\; \backslash Z$. Now because $S^$ is the deformation retract of a suitable neighborhood of itself in $D^n$, we get that $H\_n(D^n,S^)\backslash cong\; \backslash tilde\; H\_n(S^n)\backslash cong\; \backslash Z$. Another insightful geometric example is given by the relative homology of $(X=\backslash Complex^*,\; D\; =\; \backslash )$ where $\backslash alpha\; \backslash neq\; 0,\; 1$. Then we can use the long exact sequence :$\backslash begin\; 0\; \&\backslash to\; H\_1(D)\backslash to\; H\_1(X)\; \backslash to\; H\_1(X,D)\; \backslash \backslash \; \&\; \backslash to\; H\_0(D)\backslash to\; H\_0(X)\; \backslash to\; H\_0(X,D)\; \backslash end\; =\; \backslash begin\; 0\; \&\; \backslash to\; 0\; \backslash to\; \backslash Z\; \backslash to\; H\_1(X,D)\; \backslash \backslash \; \&\; \backslash to\; \backslash Z^\; \backslash to\; \backslash Z\; \backslash to\; 0\; \backslash end$ Using exactness of the sequence we can see that $H\_1(X,D)$ contains a loop $\backslash sigma$ counterclockwise around the origin. Since the cokernel of $\backslash phi\backslash colon\; \backslash Z\; \backslash to\; H\_1(X,D)$ fits into the exact sequence :$0\; \backslash to\; \backslash operatorname(\backslash phi)\; \backslash to\; \backslash Z^\; \backslash to\; \backslash Z\; \backslash to\; 0$ it must be isomorphic to $\backslash Z$. One generator for the cokernel is the $1$-chain $[1,\backslash alpha]$ since its boundary map is :$\backslash partial([1,\backslash alpha])\; =\; [\backslash alpha]\; -\; [1]$

See also

*Excision theorem, Excision Theorem *Mayerâ€“Vietoris sequenceNotes

wiktionary:i.e., i.e., the boundary $\backslash partial\backslash colon\; C\_n(X)\backslash to\; C\_(X)$ 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