topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H n...

the same homology. The resulting homology theory is often named according to the type of chain complex prescribed. For example, singular homology, Morse...

relating the homology of two objects to the homology of their product. The classical statement of the Künneth theorem relates the singular homology of two topological...

due to André Weil. There is also a version of the theorem involving singular homology instead of cohomology. It says the pairing ( ω , σ ) ↦ ∫ σ ω {\displaystyle...

{\partial _{i}}{\to }}\ C_{i-1}\to \cdots } By definition, the singular homology of X is the homology of this chain complex (the kernel of one homomorphism modulo...

restrict to the boundary of the simplex. The homology of this chain complex is called the singular homology of X, and is a commonly used invariant of a...

abelianisation of G , {\displaystyle G,} and therefore the first singular homology group H 1 ( H ) {\displaystyle H_{1}(\mathbb {H} )} is isomorphic...

(singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is...

sequence holds for a variety of cohomology and homology theories, including simplicial homology and singular cohomology. In general, the sequence holds for...

of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky...

In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of...

Carpenter Singular: Act II, a 2019 studio album by Sabrina Carpenter Singular homology SINGULAR, an open source Computer Algebra System (CAS) Singular matrix...

isomorphic to singular homology. Morse homology also serves as a model for the various infinite-dimensional generalizations known as Floer homology theories...

and Voevodsky (1996). It is sometimes called singular homology as it is analogous to the singular homology of topological spaces. By definition, given...

The symplectic Floer homology of a Hamiltonian symplectomorphism of a compact manifold is isomorphic to the singular homology of the underlying manifold...

its homology groups with coefficients in A, for any abelian group A: Hi(X; A) Here Hi might be the simplicial homology, or more generally the singular homology...

1960. For reasonable compact spaces, Borel−Moore homology coincides with the usual singular homology. For non-compact spaces, each theory has its own...

mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide...

A. In the case of topological spaces, we arrive at the notion of singular homology, which plays a fundamental role in investigating the properties of...

isomorphism: Subdivision does not change the homology of the complex. To compute the singular homology groups of a topological space X {\displaystyle...

singular homology is a special case of a simplicial homology; indeed, for each space X, there is the singular simplicial complex of X whose homology is...

Euler characteristic Genus Riemann–Hurwitz formula Singular homology Cellular homology Relative homology Mayer–Vietoris sequence Excision theorem Universal...

{\displaystyle (X,A)} are isomorphic. This assists in computation of singular homology groups, as sometimes after excising an appropriately chosen subspace...

Shape theory associates with the Čech homology theory while homotopy theory associates with the singular homology theory. Shape theory was invented and...

space and R a coefficient ring. The cap product is a bilinear map on singular homology and cohomology ⌢ : H p ( X ; R ) × H q ( X ; R ) → H p − q ( X ; R...

approach to basic algebraic topology, without needing a basis in singular homology, or the method of simplicial approximation. It contains a lot of material...

Eilenberg–MacLane spaces. On simplicial complexes, these theories coincide with singular homology and cohomology. Spectrum: H (Eilenberg–MacLane spectrum of the integers...

can choose an orientation on the tangent space at a point or we use singular homology to define orientation. Then for every open, oriented subset of M we...

and μ (id ∧ η) ~ id ~ μ(η ∧ id). Examples of ring spectra include singular homology with coefficients in a ring, complex cobordism, K-theory, and Morava...

that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homology, developed...