the cone. Recall that the cone is defined as the quotient space where has the subspace topology. Then, the origin is the equivalence class of points . Using the intuition that the local homology group of at captures the homology of "near" the origin, we should expect this is the homology of since has a homotopy retract to . Computing the local homology can then be done using the long exact sequence in homology Because the cone of a space is contractible, the middle homology groups are all zero, giving the isomorphism since is contractible to . In algebraic geometry Note the previous construction can be proven in algebraic geometry using the affine cone of a projective variety using Local cohomology. Local homology of a point on a smooth manifold Another computation for local homology can be computed on a point of a manifold . Then, let be a compact neighborhood of isomorphic to a closed disk and let . Using the excision theorem there is an isomorphism of relative homology groups hence the local homology of a point reduces to the local homology of a point in a closed ball . Because of the homotopy equivalence and the fact the only non-trivial part of the long exact sequence of the pair is hence the only non-zero local homology group is . 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. 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