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...
19 KB (3,239 words) - 23:11, 10 September 2024
the same homology. The resulting homology theory is often named according to the type of chain complex prescribed. For example, singular homology, Morse...
54 KB (8,222 words) - 01:06, 11 September 2024
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...
10 KB (1,708 words) - 22:11, 8 April 2024
De Rham theorem (section Singular-homology version)
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...
4 KB (643 words) - 15:16, 4 September 2024
Cohomology (redirect from Extraordinary homology theory)
{\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...
43 KB (6,691 words) - 21:02, 23 March 2024
Chain complex (section Singular homology)
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...
13 KB (2,029 words) - 20:38, 17 December 2023
Hawaiian earring (section First singular homology)
abelianisation of G , {\displaystyle G,} and therefore the first singular homology group H 1 ( H ) {\displaystyle H_{1}(\mathbb {H} )} is isomorphic...
11 KB (1,752 words) - 00:19, 21 March 2024
(singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is...
11 KB (2,477 words) - 12:00, 16 July 2024
Mayer–Vietoris sequence (category Homology theory)
sequence holds for a variety of cohomology and homology theories, including simplicial homology and singular cohomology. In general, the sequence holds for...
26 KB (3,768 words) - 02:49, 19 August 2024
of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky...
15 KB (2,760 words) - 19:19, 27 October 2022
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...
17 KB (2,171 words) - 08:50, 19 June 2024
Carpenter Singular: Act II, a 2019 studio album by Sabrina Carpenter Singular homology SINGULAR, an open source Computer Algebra System (CAS) Singular matrix...
1,019 bytes (169 words) - 05:05, 21 May 2023
isomorphic to singular homology. Morse homology also serves as a model for the various infinite-dimensional generalizations known as Floer homology theories...
10 KB (1,470 words) - 17:49, 29 January 2024
and Voevodsky (1996). It is sometimes called singular homology as it is analogous to the singular homology of topological spaces. By definition, given...
1 KB (216 words) - 01:19, 15 April 2020
The symplectic Floer homology of a Hamiltonian symplectomorphism of a compact manifold is isomorphic to the singular homology of the underlying manifold...
36 KB (4,649 words) - 00:59, 4 June 2024
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...
8 KB (1,181 words) - 13:40, 29 June 2023
1960. For reasonable compact spaces, Borel−Moore homology coincides with the usual singular homology. For non-compact spaces, each theory has its own...
14 KB (2,666 words) - 13:39, 22 July 2024
mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide...
13 KB (2,866 words) - 23:13, 26 October 2023
Homological algebra (redirect from Long exact sequence in homology)
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...
27 KB (3,857 words) - 20:15, 11 February 2024
Barycentric subdivision (category Simplicial homology)
isomorphism: Subdivision does not change the homology of the complex. To compute the singular homology groups of a topological space X {\displaystyle...
16 KB (2,530 words) - 15:21, 13 June 2024
Glossary of algebraic topology (redirect from Singular simplicial complex)
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...
52 KB (7,629 words) - 12:07, 26 July 2024
Euler characteristic Genus Riemann–Hurwitz formula Singular homology Cellular homology Relative homology Mayer–Vietoris sequence Excision theorem Universal...
4 KB (311 words) - 12:17, 30 October 2023
Excision theorem (category Homology theory)
{\displaystyle (X,A)} are isomorphic. This assists in computation of singular homology groups, as sometimes after excising an appropriately chosen subspace...
4 KB (663 words) - 18:59, 20 July 2024
Shape theory associates with the Čech homology theory while homotopy theory associates with the singular homology theory. Shape theory was invented and...
5 KB (650 words) - 21:44, 23 April 2024
Cap product (category Homology theory)
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...
7 KB (1,347 words) - 18:07, 30 May 2024
Algebraic topology (section Homology)
approach to basic algebraic topology, without needing a basis in singular homology, or the method of simplicial approximation. It contains a lot of material...
19 KB (2,081 words) - 18:42, 13 April 2024
Eilenberg–MacLane spaces. On simplicial complexes, these theories coincide with singular homology and cohomology. Spectrum: H (Eilenberg–MacLane spectrum of the integers...
14 KB (1,758 words) - 17:16, 6 December 2022
Orientability (section Orientability and homology)
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...
25 KB (3,515 words) - 01:43, 10 August 2024
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...
1 KB (127 words) - 18:29, 26 March 2024
Eilenberg–Steenrod axioms (category Homology theory)
that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homology, developed...
5 KB (750 words) - 02:45, 7 March 2024