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...

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

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

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

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...

However, a technical result implies that the singular homology groups coincide with smooth singular homology groups. This shows that the de Rham theorem...

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...

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

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...

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

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...

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

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

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

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

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

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...

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

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...

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

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

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

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

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...

standard idea in singular homology of "probing" a target topological space with standard topological n-simplices. Furthermore, the singular functor S is right...

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

topological invariance of simplicial homology groups. In 1918, Alexander introduced the concept of singular homology. Henceforth, most of the invariants...

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...