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Monodromy

In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name

Regular homotopy

In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions. S

Thom space

In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associat

Dold manifold

In mathematics, a Dold manifold is one of the manifolds , where is the involution that acts as −1 on the m-sphere and as complex conjugation on the complex projective space . These manifolds were cons

Delta set

In mathematics, a Δ-set S, often called a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the computation of rela

Adams operation

In mathematics, an Adams operation, denoted ψk for natural numbers k, is a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic const

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 fro

Free product

In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated

One-relator group

In the mathematical subject of group theory, a one-relator group is a group given by a group presentation with a single defining relation. One-relator groups play an important role in geometric group

Symplectic frame bundle

In symplectic geometry, the symplectic frame bundle of a given symplectic manifold is the canonical principal -subbundle of the tangent frame bundle consisting of linear frames which are symplectic wi

James embedding

In mathematics, the James embedding is an embedding of a real, complex, or hyperbolic projective space into a sphere, introduced by Ioan James .

Complex cobordism

In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be qui

Lie algebra bundle

In mathematics, a weak Lie algebra bundle is a vector bundle over a base space X together with a morphism which induces a Lie algebra structure on each fibre . A Lie algebra bundle is a vector bundle

Plus construction

In mathematics, the plus construction is a method for simplifying the fundamental group of a space without changing its homology and cohomology groups. Explicitly, if is a based connected CW complex a

Nonholonomic system

A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differe

Duocylinder

The duocylinder, also called the double cylinder or the bidisc, is a geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two disks of respective radii r1 an

Murasugi sum

In algebraic topology, a Murasugi sum is a function that relates a finite sequence of surfaces over a disk, which is common to every parallel pair (adjacent), in such a way that it exists in the bound

House with two rooms

House with two rooms or Bing's house is a particular contractible, 2-dimensional simplicial complex that is not collapsible. The name was given by R. H. Bing. The house is made of 2-dimensional panels

Sheaf of spectra

In algebraic topology, a presheaf of spectra on a topological space X is a contravariant functor from the category of open subsets of X, where morphisms are inclusions, to the good category of commuta

G-fibration

In algebraic topology, a G-fibration or principal fibration is a generalization of a principal G-bundle, just as a fibration is a generalization of a fiber bundle. By definition, given a topological m

Torus knot

In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Eac

Cosheaf

In topology, a branch of mathematics, a cosheaf with values in an ∞-category C that admits colimits is a functor F from the category of open subsets of a topological space X (more precisely its nerve)

Milnor's sphere

In mathematics, specifically differential and algebraic topology, during the mid 1950's John Milnorpg 14 was trying to understand the structure of -connected manifolds of dimension (since -connected -

Irreducible ideal

In mathematics, a proper ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection of two strictly larger ideals.

Cocycle category

In category theory, a branch of mathematics, the cocycle category of objects X, Y in a model category is a category in which the objects are pairs of maps and the morphisms are obvious commutative dia

N-skeleton

In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace Xn that is the union of the simp

Induced homomorphism

In mathematics, especially in algebraic topology, an induced homomorphism is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space X to a t

G-spectrum

In algebraic topology, a G-spectrum is a spectrum with an action of a (finite) group. Let X be a spectrum with an action of a finite group G. The important notion is that of the homotopy fixed point s

Ramification (mathematics)

In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. The term is also used from the opposite

Simplicial map

A simplicial map (also called simplicial mapping) is a function between two simplicial complexes, with the property that the images of the vertices of a simplex always span a simplex. Simplicial maps

Glossary of topology

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topolo

Shelling (topology)

In mathematics, a shelling of a simplicial complex is a way of gluing it together from its maximal simplices (simplices that are not a face of another simplex) in a well-behaved way. A complex admitti

Borsuk–Ulam theorem

In mathematics, the Borsuk–Ulam theorem states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. Here, two points on a sphere

Size functor

Given a size pair where is a manifold of dimension and is an arbitrary real continuous function definedon it, the -th size functor, with , denoted by , is the functor in , where is the category of ord

Size theory

In mathematics, size theory studies the properties of topological spaces endowed with -valued functions, with respect to the change of these functions. More formally, the subject of size theory is the

Peripheral subgroup

In algebraic topology, a peripheral subgroup for a space-subspace pair X ⊃ Y is a certain subgroup of the fundamental group of the complementary space, π1(X − Y). Its conjugacy class is an invariant o

Fundamental group

In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records inf

Size homotopy group

The concept of size homotopy group is analogous in size theory of the classical concept of homotopy group. In order to give its definition, let us assume that a size pair is given, where is a closed m

Fibration

The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or

Intersection homology

In topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherso

Transgression map

In algebraic topology, a transgression map is a way to transfer cohomology classes.It occurs, for example in the inflation-restriction exact sequence in group cohomology, and in integration in fibers.

Projective bundle

In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces. By definition, a scheme X over a Noetherian scheme S is a Pn-bundle if it is locally a projective n-space; i.e

Alexander duality

In mathematics, Alexander duality refers to a duality theory presaged by a result of 1915 by J. W. Alexander, and subsequently further developed, particularly by Pavel Alexandrov and Lev Pontryagin. I

List of algebraic topology topics

This is a list of algebraic topology topics, by Wikipedia page. See also:
* Glossary of algebraic topology
* topology glossary
* List of topology topics
* List of general topology topics
* List o

Higher-dimensional algebra

In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract

James reduced product

In topology, a branch of mathematics, the James reduced product or James construction J(X) of a topological space X with given basepoint e is the quotient of the disjoint union of all powers X, X2, X3

Simplicial complex

In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with th

Cyclic cover

In algebraic topology and algebraic geometry, a cyclic cover or cyclic covering is a covering space for which the set of covering transformations forms a cyclic group. As with cyclic groups, there may

Generalized map

In mathematics, a generalized map is a topological model which allows one to represent and to handle subdivided objects. This model was defined starting from combinatorial maps in order to represent n

Path space fibration

In algebraic topology, the path space fibration over a based space is a fibration of the form where
* is the path space of X; i.e., equipped with the compact-open topology.
* is the fiber of over th

Homotopy group with coefficients

In topology, a branch of mathematics, for , the i-th homotopy group with coefficients in an abelian group G of a based space X is the pointed set of homotopy classes of based maps from the Moore space

Essential manifold

In geometry, an essential manifold is a special type of closed manifold. The notion was first introduced explicitly by Mikhail Gromov.

Topological modular forms

In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer n there is a topological space , and these

Simple space

In algebraic topology, a branch of mathematics, a simple space is a connected topological space that has a homotopy type of a CW complex and whose fundamental group is abelian and acts trivially on th

Shriek map

In category theory, a branch of mathematics, certain unusual functors are denoted and with the exclamation mark used to indicate that they are exceptional in some way. They are thus accordingly someti

Complex-oriented cohomology theory

In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map is surjective. An element of that restricts to the canonical generat

Microbundle

In mathematics, a microbundle is a generalization of the concept of vector bundle, introduced by the American mathematician John Milnor in 1964. It allows the creation of bundle-like objects in situat

Secondary cohomology operation

In mathematics, a secondary cohomology operation is a functorial correspondence between cohomology groups. More precisely, it is a natural transformation from the kernel of some primary cohomology ope

Mathai–Quillen formalism

In mathematics, the Mathai–Quillen formalism is an approach to topological quantum field theory introduced by Atiyah and Jeffrey, based on the Mathai–Quillen form constructed in Mathai and Quillen. In

Ganea conjecture

Ganea's conjecture is a claim in algebraic topology, now disproved. It states that for all , where is the Lusternik–Schnirelmann category of a topological space X, and Sn is the n-dimensional sphere.

Redshift conjecture

In mathematics, more specifically in chromatic homotopy theory, the redshift conjecture states, roughly, that algebraic K-theory has chromatic level one higher than that of a complex-oriented ring spe

Betti number

In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (

Associated bundle

In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to ,

Rose (topology)

In mathematics, a rose (also known as a bouquet of n circles) is a topological space obtained by gluing together a collection of circles along a single point. The circles of the rose are called petals

Symmetric spectrum

In algebraic topology, a symmetric spectrum X is a spectrum of pointed simplicial sets that comes with an action of the symmetric group on such that the composition of structure maps is equivariant wi

Comodule over a Hopf algebroid

In mathematics, at the intersection of algebraic topology and algebraic geometry, there is the notion of a Hopf algebroid which encodes the information of a presheaf of groupoids whose object sheaf an

Euler calculus

Euler calculus is a methodology from applied algebraic topology and integral geometry that integrates constructible functions and more recently by integrating with respect to the Euler characteristic

A∞-operad

In the theory of operads in algebra and algebraic topology, an A∞-operad is a parameter space for a multiplication map that is homotopy coherently associative. (An operad that describes a multiplicati

Classifying space for O(n)

In mathematics, the classifying space for the orthogonal group O(n) may be constructed as the Grassmannian of n-planes in an infinite-dimensional real space . It is analogous to the classifying space

Topological Hochschild homology

In mathematics, Topological Hochschild homology is a topological refinement of Hochschild homology which rectifies some technical issues with computations in characteristic . For instance, if we consi

Formal group law

In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by S. Bochner. The term formal group sometimes me

Inverse bundle

In mathematics, the inverse bundle of a fibre bundle is its inverse with respect to the Whitney sum operation. Let be a fibre bundle. A bundle is called the inverse bundle of if their Whitney sum is a

Locally constant sheaf

In algebraic topology, a locally constant sheaf on a topological space X is a sheaf on X such that for each x in X, there is an open neighborhood U of x such that the restriction is a constant sheaf o

Glossary of algebraic topology

This is a glossary of properties and concepts in algebraic topology in mathematics. See also: glossary of topology, list of algebraic topology topics, glossary of category theory, glossary of differen

Virtual knot

In knot theory, a virtual knot is a generalization of knots in 3-dimensional Euclidean space, R3, to knots in thickened surfaces modulo an equivalence relation called stabilization/destabilization. He

Eilenberg–Maclane spectrum

In mathematics, specifically algebraic topology, there is a distinguished class of spectra called Eilenberg–Maclane spectra for any Abelian group pg 134. Note, this construction can be generalized to

Direct limit of groups

In mathematics, a direct limit of groups is the direct limit of a direct system of groups. These are central objects of study in algebraic topology, especially stable homotopy theory and homological a

Hopf construction

In algebraic topology, the Hopf construction constructs a map from the join X*Y of two spaces X and Y to the suspension SZ of a space Z out of a map from X×Y to Z. It was introduced by Hopf in the cas

Join (topology)

In topology, a field of mathematics, the join of two topological spaces and , often denoted by or , is a topological space formed by taking the disjoint union of the two spaces, and attaching line seg

Topological pair

In mathematics, more specifically algebraic topology, a pair is shorthand for an inclusion of topological spaces . Sometimes is assumed to be a cofibration. A morphism from to is given by two maps and

Collapse (topology)

In topology, a branch of mathematics, a collapse reduces a simplicial complex (or more generally, a CW complex) to a homotopy-equivalent subcomplex. Collapses, like CW complexes themselves, were inven

Section (fiber bundle)

In the mathematical field of topology, a section (or cross section) of a fiber bundle is a continuous right inverse of the projection function . In other words, if is a fiber bundle over a base space,

Semi-locally simply connected

In mathematics, specifically algebraic topology, semi-locally simply connected is a certain local connectedness condition that arises in the theory of covering spaces. Roughly speaking, a topological

Chain (algebraic topology)

In algebraic topology, a k-chainis a formal linear combination of the k-cells in a cell complex. In simplicial complexes (respectively, cubical complexes), k-chains are combinations of k-simplices (re

Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For e

Cobordism

In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French bord, giving cobordism) of a m

Directed algebraic topology

In mathematics, directed algebraic topology is a refinement of algebraic topology for directed spaces, topological spaces and their combinatorial counterparts equipped with some notion of direction. S

Simplex category

In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps. It is used to define simplicia

Commutative ring spectrum

In the mathematical field of algebraic topology, a commutative ring spectrum, roughly equivalent to a -ring spectrum, is a commutative monoid in a good category of spectra. The category of commutative

Eckmann–Hilton duality

In the mathematical disciplines of algebraic topology and homotopy theory, Eckmann–Hilton duality in its most basic form, consists of taking a given diagram for a particular concept and reversing the

Lehrbuch der Topologie

In mathematics, Lehrbuch der Topologie (German for "textbook of topology") is a book by Herbert Seifert and William Threlfall, first published in 1934 and published in an English translation in 1980.

Topological monoid

In topology, a branch of mathematics, a topological monoid is a monoid object in the category of topological spaces. In other words, it is a monoid with a topology with respect to which the monoid's b

Covering space

A covering of a topological space is a continuous map with special properties.

Esquisse d'un Programme

"Esquisse d'un Programme" (Sketch of a Programme) is a famous proposal for long-term mathematical research made by the German-born, French mathematician Alexander Grothendieck in 1984. He pursued the

Line bundle

In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying lin

Cartan model

In mathematics, the Cartan model is a differential graded algebra that computes the equivariant cohomology of a space.

Serre spectral sequence

In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topolog

Cobordism ring

In mathematics, the oriented cobordism ring is a ring where elements are oriented cobordism classes of manifolds, the multiplication is given by the Cartesian product of manifolds and the addition is

Approximate fibration

In algebraic topology, a branch of mathematics, an approximate fibration is a sort of fibration such that the homotopy lifting property holds only approximately. The notion was introduced by Coram and

Poincaré complex

In mathematics, and especially topology, a Poincaré complex (named after the mathematician Henri Poincaré) is an abstraction of the singular chain complex of a closed, orientable manifold. The singula

Simplicial set

In mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories.

Invariance of domain

Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space . It states: If is an open subset of and is an injective continuous map, then is open in and is a homeomorph

Semi-s-cobordism

In mathematics, a cobordism (W, M, M−) of an (n + 1)-dimensional manifold (with boundary) W between its boundary components, two n-manifolds M and M−, is called a semi-s-cobordism if (and only if) the

3-sphere

In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to h

Postnikov square

In algebraic topology, a Postnikov square is a certain cohomology operation from a first cohomology group H1 to a third cohomology group H3, introduced by Postnikov. described a generalization taking

R-algebroid

In mathematics, R-algebroids are constructed starting from groupoids. These are more abstract concepts than the Lie algebroids that play a similar role in the theory of Lie groupoids to that of Lie al

Sphere bundle

In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres of some dimension n. Similarly, in a disk bundle, the fibers are disks . From a topological per

Using the Borsuk–Ulam Theorem

Using the Borsuk–Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry is a graduate-level mathematics textbook in topological combinatorics. It describes the use of results in t

Dunce hat (topology)

In topology, the dunce hat is a compact topological space formed by taking a solid triangle and gluing all three sides together, with the orientation of one side reversed. Simply gluing two sides orie

S-object

In algebraic topology, an -object (also called a symmetric sequence) is a sequence of objects such that each comes with an action of the symmetric group . The category of combinatorial species is equi

Matsushima's formula

In mathematics, Matsushima's formula, introduced by Matsushima, is a formula for the Betti numbers of a quotient of a symmetric space G/H by a discrete group, in terms of unitary representations of th

Čech complex

In algebraic topology and topological data analysis, the Čech complex is an abstract simplicial complex constructed from a point cloud in any metric space which is meant to capture topological informa

Cocycle

In mathematics a cocycle is a closed cochain. Cocycles are used in algebraic topology to express obstructions (for example, to integrating a differential equation on a closed manifold). They are likew

Quasi-isomorphism

In homological algebra, a branch of mathematics, a quasi-isomorphism or quism is a morphism A → B of chain complexes (respectively, cochain complexes) such that the induced morphisms of homology group

Gysin homomorphism

In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a sphere b

Barycentric subdivision

In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension on simplicial complexes is a canonical method to refine them. Therefore, the

Massey product

In algebraic topology, the Massey product is a cohomology operation of higher order introduced in, which generalizes the cup product. The Massey product was created by William S. Massey, an American a

Crossed module

In mathematics, and especially in homotopy theory, a crossed module consists of groups G and H, where G acts on H by automorphisms (which we will write on the left, , and a homomorphism of groups that

Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to h

Abstract simplicial complex

In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the

CW complex

A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the n

Acyclic space

In mathematics, an acyclic space is a nonempty topological space X in which cycles are always boundaries, in the sense of homology theory. This implies that integral homology groups in all dimensions

Moore space (algebraic topology)

In algebraic topology, a branch of mathematics, Moore space is the name given to a particular type of topological space that is the homology analogue of the Eilenberg–Maclane spaces of homotopy theory

Pseudocircle

The pseudocircle is the finite topological space X consisting of four distinct points {a,b,c,d } with the following non-Hausdorff topology: . This topology corresponds to the partial order where open

Surgery structure set

In mathematics, the surgery structure set is the basic object in the study of manifolds which are homotopy equivalent to a closed manifold X. It is a concept which helps to answer the question whether

4-polytope

In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal el

Highly structured ring spectrum

In mathematics, a highly structured ring spectrum or -ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory. A commutative version of an -ring

Timeline of bordism

This is a timeline of bordism, a topological theory based on the concept of the boundary of a manifold. For context see timeline of manifolds. Jean Dieudonné wrote that cobordism returns to the attemp

Volodin space

In mathematics, more specifically in topology, the Volodin space of a ring R is a subspace of the classifying space given by where is the subgroup of upper triangular matrices with 1's on the diagonal

Lusternik–Schnirelmann category

In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space is the homotopy invariant defined to be the smallest integer number such

Cone (topology)

In topology, especially algebraic topology, the cone of a topological space is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point. The cone of X i

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a num

Quasi-fibration

In algebraic topology, a quasifibration is a generalisation of fibre bundles and fibrations introduced by Albrecht Dold and René Thom. Roughly speaking, it is a continuous map p: E → B having the same

Derived algebraic geometry

Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded

Ring spectrum

In stable homotopy theory, a ring spectrum is a spectrum E together with a multiplication map μ: E ∧ E → E and a unit map η: S → E, where S is the sphere spectrum. These maps have to satisfy associati

Configuration space (mathematics)

In mathematics, a configuration space is a construction closely related to state spaces or phase spaces in physics. In physics, these are used to describe the state of a whole system as a single point

Cubical complex

In mathematics, a cubical complex (also called cubical set and Cartesian complex) is a set composed of points, line segments, squares, cubes, and their n-dimensional counterparts. They are used analog

Abelian 2-group

In mathematics, an Abelian 2-group is a higher dimensional analogue of an Abelian group, in the sense of higher algebra, which were originally introduced by Alexander Grothendieck while studying abstr

Spin structure

In differential geometry, a spin structure on an orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. S

Tesseract

In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hype

Cup product

In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative (an

Nonabelian algebraic topology

In mathematics, nonabelian algebraic topology studies an aspect of algebraic topology that involves (inevitably noncommutative) higher-dimensional algebras. Many of the higher-dimensional algebraic st

Real projective space

In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension n, and is a special case of a Grassman

Symmetric product (topology)

In algebraic topology, the nth symmetric product of a topological space consists of the unordered n-tuples of its elements. If one fixes a basepoint, there is a canonical way of embedding the lower-di

Chern–Simons form

In mathematics, the Chern–Simons forms are certain secondary characteristic classes. The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteris

Mapping cone (topology)

In mathematics, especially homotopy theory, the mapping cone is a construction of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated . Its dual, a fibrat

Timelike simply connected

Suppose a Lorentzian manifold contains a closed timelike curve (CTC). No CTC can be continuously deformed as a CTC (is timelike homotopic) to a point, as that point would not be causally well behaved.

Quasitoric manifold

In mathematics, a quasitoric manifold is a topological analogue of the nonsingular projective toric variety of algebraic geometry. A smooth -dimensional manifold is a quasitoric manifold if it admits

Good cover (algebraic topology)

In mathematics, an open cover of a topological space is a family of open subsets such that is the union of all of the open sets. A good cover is an open cover in which all sets and all non-empty inter

Metaplectic structure

In differential geometry, a metaplectic structure is the symplectic analog of spin structure on orientable Riemannian manifolds. A metaplectic structure on a symplectic manifold allows one to define t

Fundamental class

In mathematics, the fundamental class is a homology class [M] associated to a connected orientable compact manifold of dimension n, which corresponds to the generator of the homology group . The funda

Discrete calculus

Discrete calculus or the calculus of discrete functions, is the mathematical study of incremental change, in the same way that geometry is the study of shape and algebra is the study of generalization

Categorification

In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories, function

Local system

In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian

Products in algebraic topology

In algebraic topology, several types of products are defined on homological and cohomological theories.

Triangulation (topology)

In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic

Novikov–Shubin invariant

In mathematics, a Novikov–Shubin invariant, introduced by Sergei Novikov and Mikhail Shubin, is an invariant of a compact Riemannian manifold related to the spectrum of the Laplace operator acting on

Cap product

In algebraic topology the cap product is a method of adjoining a chain of degree p with a cochain of degree q, such that q ≤ p, to form a composite chain of degree p − q. It was introduced by Eduard Č

Fundamental groupoid

In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it capt

Degree of a continuous mapping

In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the r

Combinatorial map

A combinatorial map is a combinatorial representation of a graph on an orientable surface. A combinatorial map may also be called a combinatorial embedding, a rotation system, an orientable ribbon gra

Pontryagin cohomology operation

In mathematics, a Pontryagin cohomology operation is a cohomology operation taking cohomology classes in H2n(X,Z/prZ) to H2pn(X,Z/pr+1Z) for some prime number p. When p=2 these operations were introdu

Simply connected space

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitive

Steenrod algebra

In algebraic topology, a Steenrod algebra was defined by Henri Cartan to be the algebra of stable cohomology operations for mod cohomology. For a given prime number , the Steenrod algebra is the grade

Homeotopy

In algebraic topology, an area of mathematics, a homeotopy group of a topological space is a homotopy group of the group of self-homeomorphisms of that space.

Whitehead link

In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links. It can be drawn as an alternating link with five crossings, from the overlay of a circle and a figure-

Aspherical space

In topology, a branch of mathematics, an aspherical space is a topological space with all homotopy groups equal to 0 when . If one works with CW complexes, one can reformulate this condition: an asphe

Size function

Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane to the natural numbers, counting certain connected components of a topological space. T

Connective spectrum

In algebraic topology, a branch of mathematics, a connective spectrum is a spectrum whose homotopy sets of negative degrees are zero.

Adams resolution

In mathematics, specifically algebraic topology, there is a resolution analogous to free resolutions of spectra yielding a tool for constructing the Adams spectral sequence. Essentially, the idea is t

Fiber-homotopy equivalence

In algebraic topology, a fiber-homotopy equivalence is a map over a space B that has homotopy inverse over B (that is we require a homotopy be a map over B for each time t.) It is a relative analog of

H-space

In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed.

Spinor bundle

In differential geometry, given a spin structure on an -dimensional orientable Riemannian manifold one defines the spinor bundle to be the complex vector bundle associated to the corresponding princip

Algebraic topology (object)

In mathematics, the algebraic topology on the set of group representations from G to a topological group H is the topology of pointwise convergence, i.e. pi converges to p if the limit of pi(g) = p(g)

Timelike homotopy

On a Lorentzian manifold, certain curves are distinguished as timelike. A timelike homotopy between two timelike curves is a homotopy such that each intermediate curve is timelike. No closed timelike

Orientation sheaf

In the mathematical field of algebraic topology, the orientation sheaf on a manifold X of dimension n is a locally constant sheaf oX on X such that the stalk of oX at a point x is (in the integer coef

Homological stability

In mathematics, homological stability is any of a number of theorems asserting that the group homology of a series of groups is stable, i.e., is independent of n when n is large enough (depending on i

Lazard's universal ring

In mathematics, Lazard's universal ring is a ring introduced by Michel Lazard in over which the universal commutative one-dimensional formal group law is defined. There is a universal commutative one-

P-compact group

In mathematics, in particular algebraic topology, a p-compact group is a homotopical version of a compact Lie group, but with all the local structure concentrated at a single prime p. This concept was

Poincaré space

In algebraic topology, a Poincaré space is an n-dimensional topological space with a distinguished element µ of its nth homology group such that taking the cap product with an element of the kth cohom

Symplectic spinor bundle

In differential geometry, given a metaplectic structure on a -dimensional symplectic manifold the symplectic spinor bundle is the Hilbert space bundle associated to the metaplectic structure via the m

String topology

String topology, a branch of mathematics, is the study of algebraic structures on the homology of free loop spaces. The field was started by Moira Chas and Dennis Sullivan.

Simply connected at infinity

In topology, a branch of mathematics, a topological space X is said to be simply connected at infinity if for any compact subset C of X, there is a compact set D in X containing C so that the induced

Homotopical algebra

In mathematics, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra as well as possibly the abelian aspects as special cases. The homotopical nomen

Bockstein homomorphism

In homological algebra, the Bockstein homomorphism, introduced by Meyer Bockstein , is a connecting homomorphism associated with a short exact sequence of abelian groups, when they are introduced as c

Doomsday conjecture

In algebraic topology, the doomsday conjecture was a conjecture about Ext groups over the Steenrod algebra made by Joel Cohen, named by Michael Barratt, published by , conjecture 73) and disproved by

Sphere spectrum

In stable homotopy theory, a branch of mathematics, the sphere spectrum S is the monoidal unit in the category of spectra. It is the suspension spectrum of S0, i.e., a set of two points. Explicitly, t

Vietoris–Rips complex

In topology, the Vietoris–Rips complex, also called the Vietoris complex or Rips complex, is a way of forming a topological space from distances in a set of points. It is an abstract simplicial comple

Hopf fibration

In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles an

Classifying space

In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e. a topological space all of whose homotopy group

Mapping cylinder

In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function between topological spaces and is the quotient where the denotes the disjoint union, and ∼ is the equival

Bloch group

In mathematics, the Bloch group is a cohomology group of the Bloch–Suslin complex, named after Spencer Bloch and Andrei Suslin. It is closely related to polylogarithm, hyperbolic geometry and algebrai

Category of compactly generated weak Hausdorff spaces

In mathematics, the category of compactly generated weak Hausdorff spaces CGWH is one of typically used categories in algebraic topology as a substitute for the category of topological spaces, as the

Path space (algebraic topology)

In algebraic topology, a branch of mathematics, the path space of a based space is the space that consists of all maps from the interval to X such that , called paths. In other words, it is the mappin

Calculus of functors

In algebraic topology, a branch of mathematics, the calculus of functors or Goodwillie calculus is a technique for studying functors by approximating them by a sequence of simpler functors; it general

Genus (mathematics)

In mathematics, genus (plural genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1.

Twisted Poincaré duality

In mathematics, the twisted Poincaré duality is a theorem removing the restriction on Poincaré duality to oriented manifolds. The existence of a global orientation is replaced by carrying along local

Morava K-theory

In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early

Abstract polytope

In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points an

Equivariant cohomology

In mathematics, equivariant cohomology (or Borel cohomology) is a cohomology theory from algebraic topology which applies to topological spaces with a group action. It can be viewed as a common genera

Homotopy extension property

In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotop

Pair of spaces

No description available.

Mapping spectrum

In algebraic topology, the mapping spectrum of spectra X, Y is characterized by

Change of fiber

In algebraic topology, given a fibration p:E→B, the change of fiber is a map between the fibers induced by paths in B. Since a covering is a fibration, the construction generalizes the corresponding f

Whitehead conjecture

The Whitehead conjecture (also known as the Whitehead asphericity conjecture) is a claim in algebraic topology. It was formulated by J. H. C. Whitehead in 1941. It states that every connected subcompl

Presentation complex

In geometric group theory, a presentation complex is a 2-dimensional cell complex associated to any presentation of a group G. The complex has a single vertex, and one loop at the vertex for each gene

Homology manifold

In mathematics, a homology manifold (or generalized manifold)is a locally compact topological space X that looks locally like a topological manifold from the point of view of homology theory.

Homotopy fiber

In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration to an arbitrary continuous function of topolog

E∞-operad

In the theory of operads in algebra and algebraic topology, an E∞-operad is a parameter space for a multiplication map that is associative and commutative "up to all higher homotopies". (An operad tha

Gray's conjecture

In mathematics, Gray's conjecture is a conjecture made by Brayton Gray in 1984 about maps between loop spaces of spheres. It was later proved by John Harper.

Homotopy lifting property

In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a

Riemann–Hurwitz formula

In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of

Clique complex

Clique complexes, independence complexes, flag complexes, Whitney complexes and conformal hypergraphs are closely related mathematical objects in graph theory and geometric topology that each describe

Tautness (topology)

In mathematics, particularly in algebraic topology, taut pair is a topological pair whose direct limit of cohomology module of open neighborhood of that pair which is directed downward by inclusion is

Topological combinatorics

The mathematical discipline of topological combinatorics is the application of topological and algebro-topological methods to solving problems in combinatorics.

Vanishing cycle

In mathematics, vanishing cycles are studied in singularity theory and other parts of algebraic geometry. They are those homology cycles of a smooth fiber in a family which vanish in the . For example

Winding number

In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around

Solenoid (mathematics)

In mathematics, a solenoid is a compact connected topological space (i.e. a continuum) that may be obtained as the inverse limit of an inverse system of topological groups and continuous homomorphisms

Godement resolution

The Godement resolution of a sheaf is a construction in homological algebra that allows one to view global, cohomological information about the sheaf in terms of local information coming from its stal

Stabilization hypothesis

In mathematics, specifically in category theory and algebraic topology, the Baez–Dolan stabilization hypothesis, proposed in, states that suspension of a weak n-category has no more essential effect a

L-theory

In mathematics, algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall, with L being used as the letter after K. Algebraic L-theory, also known as "Hermitian K-the

Cohomology operation

In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functo

Topological degree theory

In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane. It can be used to estimate the number of solutions of an equation, and is closely c

Dual Steenrod algebra

In algebraic topology, through an algebraic operation (dualization), there is an associated commutative algebra from the noncommutative Steenrod algebras called the dual Steenrod algebra. This dual al

Algebraic cobordism

In mathematics, algebraic cobordism is an analogue of complex cobordism for smooth quasi-projective schemes over a field. It was introduced by Marc Levine and Fabien Morel . An oriented cohomology the

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