# Category: Homology theory

Cohomology ring
In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multipl
Aspherical space
In topology, a branch of mathematics, an aspherical space is a topological space with all homotopy groups equal to 0 when . If one works with CW complexes, one can reformulate this condition: an asphe
Excision theorem
In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms. Given a topological space and subspaces and such tha
Hurewicz theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after W
Polar homology
In complex geometry, a polar homology is a group which captures holomorphic invariants of a complex manifold in a similar way to usual homology of a manifold in differential topology. Polar homology w
Poincaré complex
In mathematics, and especially topology, a Poincaré complex (named after the mathematician Henri Poincaré) is an abstraction of the singular chain complex of a closed, orientable manifold. The singula
Persistent homology
Persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of spatial scales and are deemed
Acyclic space
In mathematics, an acyclic space is a nonempty topological space X in which cycles are always boundaries, in the sense of homology theory. This implies that integral homology groups in all dimensions
Relative contact homology
In mathematics, in the area of symplectic topology, relative contact homology is an invariant of spaces together with a chosen subspace. Namely, it is associated to a contact manifold and one of its L
Cellular homology
In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homolog
Eilenberg–Moore spectral sequence
In mathematics, in the field of algebraic topology, the Eilenberg–Moore spectral sequence addresses the calculation of the homology groups of a pullback over a fibration. The spectral sequence formula
Homological connectivity
In algebraic topology, homological connectivity is a property describing a topological space based on its homology groups.
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
Morse homology
In mathematics, specifically in the field of differential topology, Morse homology is a homology theory defined for any smooth manifold. It is constructed using the smooth structure and an auxiliary m
Cyclic category
In mathematics, the cyclic category or cycle category or category of cycles is a category of finite cyclically ordered sets and degree-1 maps between them. It was introduced by .
Khovanov homology
In mathematics, Khovanov homology is an oriented link invariant that arises as the homology of a chain complex. It may be regarded as a categorification of the Jones polynomial. It was developed in th
Reduced homology
In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to
Bivariant theory
In mathematics, a bivariant theory was introduced by Fulton and MacPherson, in order to put a ring structure on the Chow group of a singular variety, the resulting ring called an operational Chow ring
Compactly-supported homology
In mathematics, a homology theory in algebraic topology is compactly supported if, in every degree n, the relative homology group Hn(X, A) of every pair of spaces (X, A) is naturally isomorphic to the
Cup product
In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative (an
Poincaré duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if M is an n-dimensional
Topological data analysis
In applied mathematics, topological based data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimension
Chern–Simons form
In mathematics, the Chern–Simons forms are certain secondary characteristic classes. The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteris
Homology (mathematics)
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups
Singular homology
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
Steenrod problem
In mathematics, and particularly homology theory, Steenrod's Problem (named after mathematician Norman Steenrod) is a problem concerning the realisation of homology classes by singular manifolds.
Mayer–Vietoris sequence
In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homolog
Good cover (algebraic topology)
In mathematics, an open cover of a topological space is a family of open subsets such that is the union of all of the open sets. A good cover is an open cover in which all sets and all non-empty inter
Floer homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-di
Continuation map
In differential topology, given a family of Morse-Smale functions on a smooth manifold X parameterized by a closed interval I, one can construct a vector field on X × I whose critical points occur onl
Bump and hole
The bump-and-hole method is a tool in chemical genetics for studying a specific isoform in a protein family without perturbing the other members of the family. The unattainability of isoform-selective
Products in algebraic topology
In algebraic topology, several types of products are defined on homological and cohomological theories.
Eilenberg–Steenrod axioms
In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology t
K-homology
In mathematics, K-homology is a homology theory on the category of locally compact Hausdorff spaces. It classifies the elliptic pseudo-differential operators acting on the vector bundles over a space.
Pontryagin product
In mathematics, the Pontryagin product, introduced by Lev Pontryagin, is a product on the homology of a topological space induced by a product on the topological space. Special cases include the Pontr
Cap product
In algebraic topology the cap product is a method of adjoining a chain of degree p with a cochain of degree q, such that q ≤ p, to form a composite chain of degree p − q. It was introduced by Eduard Č
Stratifold
In differential topology, a branch of mathematics, a stratifold is a generalization of a differentiable manifold where certain kinds of singularities are allowed. More specifically a stratifold is str
Homology sphere
In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer . That is, and for all other i. Therefore X is a connected space, with one non-z
Pushforward (homology)
In algebraic topology, the pushforward of a continuous function : between two topological spaces is a homomorphism between the homology groups for . Homology is a functor which converts a topological
Simplicial volume
In the mathematical field of geometric topology, the simplicial volume (also called Gromov norm) is a certain measure of the topological complexity of a manifold. More generally, the simplicial norm m
Borel–Moore homology
In topology, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Armand Borel and John Moore in 1960. For reasonable compact spaces, Bor
Kan-Thurston theorem
In mathematics, particularly algebraic topology, the Kan-Thurston theorem associates a discrete group to every path-connected topological space in such a way that the group cohomology of is the same a
Steenrod homology
In algebraic topology, Steenrod homology is a homology theory for compact metric spaces introduced by Norman Steenrod , based on regular cycles.It is similar to the homology theory introduced rather s
Meshulam's game
In graph theory, Meshulam's game is a game used to explain a theorem of Roy Meshulam related to the homological connectivity of the independence complex of a graph, which is the smallest index k such
Kirby–Siebenmann class
In mathematics, more specifically in geometric topology, the Kirby–Siebenmann class is an obstruction for topological manifolds to allow a PL-structure.
Graph homology
In algebraic topology and graph theory, graph homology describes the homology groups of a graph, where the graph is considered as a topological space. It formalizes the idea of the number of "holes" i
Hodge conjecture
In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subva