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...

Borel–Moore homology Cellular homology Cyclic homology Hochschild homology Floer homology Intersection homology K-homology Khovanov homology Morse homology Persistent...

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...

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...

{\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...

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

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...

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

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

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...

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

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

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

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

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...

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

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

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...

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

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

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

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

space, we can define the nth Betti number bn as the rank of the n-th singular homology group. The Euler characteristic can then be defined as the alternating...

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...

then a consequence of the fact that over the reals, singular cohomology is the dual of singular homology. Separately, a 1927 paper of Solomon Lefschetz used...

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

reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of...

resulting homology is an invariant of the manifold (that is, independent of the function and metric) and isomorphic to the singular homology of the manifold;...