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In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages.The most direct usage of the term is to take the homology of a chain complex, resulting in a sequence of abelian groups called homology groups.
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological ...
In mathematics, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra, and possibly the abelian aspects as special cases. . The homotopical nomenclature stems from the fact that a common approach to such generalizations is via abstract homotopy theory, as in nonabelian algebraic topology, and in particular the theory of closed model categor
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 ...
In homotopy theory and algebraic topology, the word "space" denotes a topological space.In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated weak Hausdorff or a CW complex.
Given two directed paths γ and δ, a directed homotopy from γ to δ is a morphism of directed spaces h whose underlying map U(h) is a homotopy –in the usual sense– between the underlying paths U(γ) and U(δ). In algebraic topology, there is a homotopy from α to β if and only if there is a homotopy from β to α. Due to non ...
A homotopy between two embeddings of the torus into : as "the surface of a doughnut" and as "the surface of a coffee mug".This is also an example of an isotopy.. Formally, a homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function: [,] from the product of the space X with the unit interval [0, 1] to Y such that ...
In general one uses singular homology; but if X and Y happen to be CW complexes, then this can be replaced by cellular homology, because that is isomorphic to singular homology. The simplest case is when the coefficient ring for homology is a field F. In this situation, the Künneth theorem (for singular homology) states that for any integer k,