Algebraic topology

branch of mathematics

topological invariant in mathematics

In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells, meeting at right angles. The tesseract is one of the six convex regular 4-polytopes.

mathematical group of the homotopy classes of loops in a topological space

collection of objects associated to subsets of a space in a manner admitting gluing and restriction

topological space which has no holes through it

topological property

{|class=wikitable style="float:right; margin-left:8px" |+ Graphs of the six convex regular 4-polytopes |- !{3,3,3} !{3,3,4} !{4,3,3} |- valign=top align=center |120px5-cellPentatope4-simplex |121px16-cellOrthoplex4-orthoplex |120px8-cellTesseract4-cube |- !{3,4,3} !{3,3,5} !{5,3,3} |- valign=top align=center |120px24-cellOctaplex |120px600-cellTetraplex |120px120-cellDodecaplex |}

a geometrical object (set composed of points, line segments, triangles, and their n-dimensional counterpart) useful to describe certain topological spaces

number of times a curve wraps around a point in the plane

type of topological space

type of continuous map in topology

thumb|The imaginary part of the complex logarithm. Trying to define the complex logarithm on \C-\{0\} gives different answers along different paths. This leads to an infinite cyclic monodromy group and a covering of \C-\{0\} by a [[helicoid (an example of a Riemann surface).]]

fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers

thumb|Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Because this projection is conformal, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect have infinite radius (= straight line). In this picture, the whole 3D space maps the surface of the hypersphere, whereas in the next picture the 3D space contained the shadow of the bulk hypersphere. thumb|Direct projection of 3-sphere into 3D space and covered with surface grid, showing structure as stack of 3D spher

thumb|A cobordism (W; M, N). 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 manifold. Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher.

used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes

knot which lies on the surface of a torus in 3-dimensional space

operation that takes two groups G and H and constructs a new group G ∗ H

right inverse of a fiber bundle map

generalization of winding number

small category, whose objects are sets of natural numbers of the form {0,1,…,n}, and whose morphisms are nondecreasing functions

topological construction

two interlinked loops with five structural crossings

construction in categorical homotopy theory; contravariant functor from the simplex category to the category of sets

algebraic partially ordered set or poset which captures the combinatorial properties of a traditional polytope, but not any purely geometric properties such as angles, edge lengths, etc

system with kinematic constraints that cannot be integrated in position-level constraints

secondary characteristic class defined for odd-dimensional manifolds with G-bundles with connection; in 2n−1 dimensions, defined as (formal) exterior antiderivative of tr(Fⁿ) where F is the curvature of the connection

integer-valued homotopy invariant of spaces; the size of the minimal open cover consisting of contractible sets

theorem in topology about homeomorphic subsets of Euclidean space

one-dimensional vector bundle

geometric structure

mathematics glossary

topological space obtained by gluing together a collection of circles along a single point

mathematical formula

homotopy theory in algebraic topology

mathematical subject

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.

Property in algebraic topology

method of adjoining two cocycles to form a composite cocycle

branching out of a mathematical structure

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

In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties. The term was coined by Louis Crane.

right|frame|Stereographic projection of the duocylinder's ridge (see below), as a [[flat torus. The ridge is rotating about the -plane.]]

mathematical subject

topological space equipped with a principal bundle with the property that any principal bundle (with the same fiber group) over a paracompact manifold is isomorphic to a pullback of the principal bundle over this topological space

proper ideal of a commutative ring that is not the intersection of two strictly larger ideals

method of adjoining a chain of with a cochain

topology term

thumb|This hypercube graph is the [[Graph of a polytope| of the tesseract.]]

lift of the SO(n) frame bundle of an oriented Riemannian manifold into Spin(n), the spin group

The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.

duality of the homology of the complement of a subspace

topological construction

way of dividing a simplicial complex

fiber bundle constructed by a representation of a group and a principal bundle

differential associative algebra with integer grading in which the differential has grading +1 (cohomological convention) or −1 (homological convention)

moduli space of n points on a space M; if M is a manifold, in general forms an orbifold

The pseudocircle is the finite topological space X consisting of four distinct points {a,b,c,d} with the following non-Hausdorff topology: \{\{a,b,c,d\}, \{a,b,c\}, \{a,b,d\}, \{a,b\}, \{a\}, \{b\}, \varnothing\}.

topological construction