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 .
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In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups Intuitively, singular homology counts, for each dimension n, the n-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions (see also the related theory simplicial homology).
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper. Sheaves, sheaf cohomology, and spectral sequences were introduced by Jean Leray at the prisoner-of-war camp Oflag XVII-A in Austria.
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of .
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