Faster access than browser!
53 relations: Abelian category, Abelian group, Algebra over a field, Algebraic geometry, Algebraic topology, Betti number, Bockstein homomorphism, Category (mathematics), Category of abelian groups, Category of topological spaces, Category theory, Cellular homology, Chain complex, Closed manifold, Cobordism, Cohomology, Cohomology operation, Contractible space, Cup product, Derived category, Differential graded algebra, Eilenberg–Steenrod axioms, Exact sequence, Excision theorem, Formal sum, Free abelian group, Free module, Functor, Homology (mathematics), Homomorphism, Homotopy, Homotopy category, Homotopy category of chain complexes, Hurewicz theorem, Injective function, Invariant theory, K-theory, Mathematics, Module (mathematics), Morphism, Quotient category, Quotient group, Relative homology, Ring (mathematics), Simplex, Simplicial complex, Simplicial homology, Snake lemma, Steenrod algebra, Subcategory, ..., Topological property, Topological space, Uncountable set. Expand index (3 more) »
Abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.
New!!: Singular homology and Abelian category · See more »
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
New!!: Singular homology and Abelian group · See more »
Algebra over a field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.
New!!: Singular homology and Algebra over a field · See more »
Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
New!!: Singular homology and Algebraic geometry · See more »
Algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.
New!!: Singular homology and Algebraic topology · See more »
Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.
New!!: Singular homology and Betti number · See more »
Bockstein homomorphism
In homological algebra, the Bockstein homomorphism, introduced by, is a connecting homomorphism associated with a short exact sequence of abelian groups, when they are introduced as coefficients into a chain complex C, and which appears in the homology groups as a homomorphism reducing degree by one, To be more precise, C should be a complex of free, or at least torsion-free, abelian groups, and the homology is of the complexes formed by tensor product with C (some flat module condition should enter).
New!!: Singular homology and Bockstein homomorphism · See more »
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is an algebraic structure similar to a group but without requiring inverse or closure properties.
New!!: Singular homology and Category (mathematics) · See more »
Category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms.
New!!: Singular homology and Category of abelian groups · See more »
Category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps or some other variant; for example, objects are often assumed to be compactly generated.
New!!: Singular homology and Category of topological spaces · See more »
Category theory
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).
New!!: Singular homology and Category theory · See more »
Cellular homology
In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes.
New!!: Singular homology and Cellular homology · See more »
Chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next.
New!!: Singular homology and Chain complex · See more »
Closed manifold
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary.
New!!: Singular homology and Closed manifold · See more »
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French bord, giving cobordism) of a manifold.
New!!: Singular homology and Cobordism · See more »
Cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex.
New!!: Singular homology and Cohomology · See more »
Cohomology operation
In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from F to itself.
New!!: Singular homology and Cohomology operation · See more »
Contractible space
In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map.
New!!: Singular homology and Contractible space · See more »
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 (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space X into a graded ring, H∗(X), called the cohomology ring.
New!!: Singular homology and Cup product · See more »
Derived category
In mathematics, the derived category D(A) of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects of D(A) should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes.
New!!: Singular homology and Derived category · See more »
Differential graded algebra
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded algebra with an added chain complex structure that respects the algebra structure.
New!!: Singular homology and Differential graded algebra · See more »
Eilenberg–Steenrod axioms
In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common.
New!!: Singular homology and Eilenberg–Steenrod axioms · See more »
Exact sequence
An exact sequence is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry.
New!!: Singular homology and Exact sequence · See more »
Excision theorem
In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology—given topological spaces X and subspaces A and U such that U is also a subspace of A, the theorem says that under certain circumstances, we can cut out (excise) U from both spaces such that the relative homologies of the pairs (X,A) and (X \ U,A \ U) are isomorphic.
New!!: Singular homology and Excision theorem · See more »
Formal sum
In mathematics, a formal sum, formal series, or formal linear combination may be.
New!!: Singular homology and Formal sum · See more »
Free abelian group
In abstract algebra, a free abelian group or free Z-module is an abelian group with a basis.
New!!: Singular homology and Free abelian group · See more »
Free module
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements.
New!!: Singular homology and Free module · See more »
Functor
In mathematics, a functor is a map between categories.
New!!: Singular homology and Functor · See more »
Homology (mathematics)
In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.
New!!: Singular homology and Homology (mathematics) · See more »
Homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).
New!!: Singular homology and Homomorphism · See more »
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.
New!!: Singular homology and Homotopy · See more »
Homotopy category
In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape.
New!!: Singular homology and Homotopy category · See more »
Homotopy category of chain complexes
In homological algebra in mathematics, the homotopy category K(A) of chain complexes in an additive category A is a framework for working with chain homotopies and homotopy equivalences.
New!!: Singular homology and Homotopy category of chain complexes · See more »
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.
New!!: Singular homology and Hurewicz theorem · See more »
Injective function
In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.
New!!: Singular homology and Injective function · See more »
Invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions.
New!!: Singular homology and Invariant theory · See more »
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme.
New!!: Singular homology and K-theory · See more »
Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
New!!: Singular homology and Mathematics · See more »
Module (mathematics)
In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.
New!!: Singular homology and Module (mathematics) · See more »
Morphism
In mathematics, a morphism is a structure-preserving map from one mathematical structure to another one of the same type.
New!!: Singular homology and Morphism · See more »
Quotient category
In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms.
New!!: Singular homology and Quotient category · See more »
Quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.
New!!: Singular homology and Quotient group · See more »
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.
New!!: Singular homology and Relative homology · See more »
Ring (mathematics)
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.
New!!: Singular homology and Ring (mathematics) · See more »
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.
New!!: Singular homology and Simplex · See more »
Simplicial complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration).
New!!: Singular homology and Simplicial complex · See more »
Simplicial homology
In algebraic topology, simplicial homology formalizes the idea of the number of holes of a given dimension in a simplicial complex.
New!!: Singular homology and Simplicial homology · See more »
Snake lemma
The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences.
New!!: Singular homology and Snake lemma · See more »
Steenrod algebra
In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology.
New!!: Singular homology and Steenrod algebra · See more »
Subcategory
In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms.
New!!: Singular homology and Subcategory · See more »
Topological property
In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms.
New!!: Singular homology and Topological property · See more »
Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
New!!: Singular homology and Topological space · See more »
Uncountable set
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable.
New!!: Singular homology and Uncountable set · See more »
Redirects here:
Homology functor, Singular chain, Singular chain complex, Singular complex, Singular homology group, Singular theory.
References
[1] https://en.wikipedia.org/wiki/Singular_homology
