Relative homology

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. Given a subspace , one may form the short exact sequence where denotes the singular chains on the space X. The boundary map on descends to and therefore induces a boundary map on the quotient. If we denote this quotient by , we then have a complex By definition, the th relative homology group of the pair of spaces is 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). 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 takes a relative cycle, representing a homology class in , to its boundary (which is a cycle in A). It follows that , where is a point in X, is the n-th reduced homology group of X. In other words, for all . When , is the free module of one rank less than . The connected component containing becomes trivial in relative homology. The excision theorem says that removing a sufficiently nice subset leaves the relative homology groups unchanged. Using the long exact sequence of pairs and the excision theorem, one can show that is the same as the n-th reduced homology groups of the quotient space . Relative homology readily extends to the triple for . One can define the Euler characteristic for a pair by The exactness of the sequence implies that the Euler characteristic is additive, i.e., if , one has The -th local homology group of a space at a point , denoted is defined to be the relative homology group .