Category: Theorems in algebraic topology

Seifert–Van Kampen theorem
In mathematics, the Seifert–Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called Van Kampen's theorem, expresses the structure of the fun
Borel's theorem
In topology, a branch of mathematics, Borel's theorem, due to Armand Borel, says the cohomology ring of a classifying space or a classifying stack is a polynomial ring.
Hurewicz theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after W
Hattori–Stong theorem
In algebraic topology, the Hattori–Stong theorem, proved by Robert Evert Stong and Akio Hattori, gives an isomorphism between the stable homotopy of a Thom spectrum and the primitive elements of its K
Eilenberg–Zilber theorem
In mathematics, specifically in algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the homology groups of a product space and those of the spaces
Zeeman's comparison theorem
In homological algebra, Zeeman's comparison theorem, introduced by Zeeman, gives conditions for a morphism of spectral sequences to be an isomorphism.
Alexander's theorem
In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid; that is, a braid in which the corresponding ends of the strings are connected in pairs. The theo
Nilpotence theorem
In algebraic topology, the nilpotence theorem gives a condition for an element in the homotopy groups of a ring spectrum to be nilpotent, in terms of the complex cobordism spectrum . More precisely, i
Snaith's theorem
In algebraic topology, a branch of mathematics, Snaith's theorem, introduced by , identifies the complex K-theory spectrum with the of the suspension spectrum of away from the Bott element.
Cellular approximation theorem
In algebraic topology, the cellular approximation theorem states that a map between CW-complexes can always be taken to be of a specific type. Concretely, if X and Y are CW-complexes, and f : X → Y is
Norm residue isomorphism theorem
In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor K-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time rep
Hirzebruch signature theorem
In differential topology, an area of mathematics, the Hirzebruch signature theorem (sometimes called the Hirzebruch index theorem)is Friedrich Hirzebruch's 1954 result expressing the signatureof a smo
Lefschetz fixed-point theorem
In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space to itself by means of traces of the induced mapping
Leray's theorem
In algebraic topology and algebraic geometry, Leray's theorem (so named after Jean Leray) relates abstract sheaf cohomology with Čech cohomology. Let be a sheaf on a topological space and an open cove
Almgren isomorphism theorem
Almgren isomorphism theorem is a result in geometric measure theory and algebraic topology about the topology of the space of flat cycles in a Riemannian manifold. The theorem plays a fundamental role
Landweber exact functor theorem
In mathematics, the Landweber exact functor theorem, named after Peter Landweber, is a theorem in algebraic topology. It is known that a complex orientation of a homology theory leads to a formal grou
Serre–Swan theorem
In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules a
Milnor conjecture
In mathematics, the Milnor conjecture was a proposal by John Milnor of a description of the Milnor K-theory (mod 2) of a general field F with characteristic different from 2, by means of the Galois (o
Basic theorems in algebraic K-theory
In mathematics, there are several theorems basic to algebraic K-theory. Throughout, for simplicity, we assume when an exact category is a subcategory of another exact category, we mean it is strictly
Künneth theorem
In mathematics, especially in homological algebra and algebraic topology, a Künneth theorem, also called a Künneth formula, is a statement relating the homology of two objects to the homology of their
Blakers–Massey theorem
In mathematics, the first Blakers–Massey theorem, named after and William S. Massey, gave vanishing conditions for certain triad homotopy groups of spaces.
Peterson–Stein formula
In mathematics, the Peterson–Stein formula, introduced by Franklin P. Peterson and Norman Stein, describes the Spanier–Whitehead dual of a secondary cohomology operation.
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 t
Stallings–Zeeman theorem
In mathematics, the Stallings–Zeeman theorem is a result in algebraic topology, used in the proof of the Poincaré conjecture for dimension greater than or equal to five. It is named after the mathemat
Hilton's theorem
In algebraic topology, Hilton's theorem, proved by Peter Hilton, states that the loop space of a wedge of spheres is homotopy-equivalent to a product of loop spaces of spheres. John Milnor showed more
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, i
Dold–Thom theorem
In algebraic topology, the Dold-Thom theorem states that the homotopy groups of the infinite symmetric product of a connected CW complex are the same as its reduced homology groups. The most common ve
Karoubi conjecture
In mathematics, the Karoubi conjecture is a conjecture by Max Karoubi that the algebraic and topological K-theories coincide on C* algebras spatially tensored with the algebra of compact operators. It
Simplicial approximation theorem
In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by a slight deformation) approximated by ones that a
Eilenberg–Ganea theorem
In mathematics, particularly in homological algebra and algebraic topology, the Eilenberg–Ganea theorem states for every finitely generated group G with certain conditions on its cohomological dimensi
Acyclic model
In algebraic topology, a discipline within mathematics, the acyclic models theorem can be used to show that two homology theories are isomorphic. The theorem was developed by topologists Samuel Eilenb
Bloch's formula
In algebraic K-theory, a branch of mathematics, Bloch's formula, introduced by Spencer Bloch for , states that the Chow group of a smooth variety X over a field is isomorphic to the cohomology of X wi
Eilenberg–Ganea conjecture
The Eilenberg–Ganea conjecture is a claim in algebraic topology. It was formulated by Samuel Eilenberg and Tudor Ganea in 1957, in a short, but influential paper. It states that if a group G has cohom
Fundamental theorem of algebraic K-theory
In algebra, the fundamental theorem of algebraic K-theory describes the effects of changing the ring of K-groups from a ring R to or . The theorem was first proved by Hyman Bass for and was later exte
Markov theorem
In mathematics the Markov theorem gives necessary and sufficient conditions for two braids to have closures that are equivalent knots or links. The conditions are stated in terms of the group structur
Vietoris–Begle mapping theorem
The Vietoris–Begle mapping theorem is a result in the mathematical field of algebraic topology. It is named for Leopold Vietoris and Edward G. Begle. The statement of the theorem, below, is as formula
De Franchis theorem
In mathematics, the de Franchis theorem is one of a number of closely related statements applying to compact Riemann surfaces, or, more generally, algebraic curves, X and Y, in the case of genus g > 1
Lefschetz hyperplane theorem
In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and th
Leray–Hirsch theorem
In mathematics, the Leray–Hirsch theorem is a basic result on the algebraic topology of fiber bundles. It is named after Jean Leray and Guy Hirsch, who independently proved it in the late 1940s. It ca