Concept
Algebraic topology
Related concepts (56)
Combinatorial topology
In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes. After the proof of the simplicial approximation theorem this approach provided rigour. The change of name reflected the move to organise topological classes such as cycles-modulo-boundaries explicitly into abelian groups.
Hairy ball theorem
The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional n-spheres. For the ordinary sphere, or 2‐sphere, if f is a continuous function that assigns a vector in R3 to every point p on a sphere such that f(p) is always tangent to the sphere at p, then there is at least one pole, a point where the field vanishes (a p such that f(p) = 0).
René Thom
René Frédéric Thom (ʁəne tɔm; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as founder of catastrophe theory (later developed by Christopher Zeeman). René Thom grow up in a modest family in Montbéliard, Doubs and obtained a Baccalauréat in 1940.
Čech cohomology
In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech. Let X be a topological space, and let be an open cover of X. Let denote the nerve of the covering. The idea of Čech cohomology is that, for an open cover consisting of sufficiently small open sets, the resulting simplicial complex should be a good combinatorial model for the space X.
Universal coefficient theorem
In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space X, its integral homology groups: Hi(X; Z) completely determine 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. The usual proof of this result is a pure piece of homological algebra about chain complexes of free abelian groups.
Simplicial homology
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 a given dimension in the complex. This generalizes the number of connected components (the case of dimension 0). Simplicial homology arose as a way to study topological spaces whose building blocks are n-simplices, the n-dimensional analogs of triangles. This includes a point (0-simplex), a line segment (1-simplex), a triangle (2-simplex) and a tetrahedron (3-simplex).
Leopold Vietoris
Leopold Vietoris (viːˈtoʊrɪs; viːˈtoːʀɪs; 4 June 1891 – 9 April 2002) was an Austrian mathematician, World War I veteran and supercentenarian. He was born in Radkersburg and died in Innsbruck. He was known for his contributions to topology—notably the Mayer–Vietoris sequence—and other fields of mathematics, his interest in mathematical history and for being a keen alpinist. Vietoris studied mathematics and geometry at the Vienna University of Technology. He was drafted in 1914 in World War I and was wounded in September that same year.
Excision theorem
In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms. Given a topological space and subspaces and such that is also a subspace of , the theorem says that under certain circumstances, we can cut out (excise) from both spaces such that the relative homologies of the pairs into are isomorphic. This assists in computation of singular homology groups, as sometimes after excising an appropriately chosen subspace we obtain something easier to compute.
Weak equivalence (homotopy theory)
In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a . A model category is a with classes of morphisms called weak equivalences, fibrations, and cofibrations, satisfying several axioms. The associated of a model category has the same objects, but the morphisms are changed in order to make the weak equivalences into isomorphisms.
J. H. C. Whitehead
John Henry Constantine Whitehead FRS (11 November 1904 – 8 May 1960), known as Henry, was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai (then known as Madras), in India, and died in Princeton, New Jersey, in 1960. J. H. C. (Henry) Whitehead was the son of the Right Rev. Henry Whitehead, Bishop of Madras, who had studied mathematics at Oxford, and was the nephew of Alfred North Whitehead and Isobel Duncan. He was brought up in Oxford, went to Eton and read mathematics at Balliol College, Oxford.