Geometric topology

two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space

theorem in geometric topology that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere

type of non-orientable surface

topological property

link of three loops; simplest Brunnian link

a compact non-orientable two-dimensional manifold

study of manifolds and maps between them, particularly embeddings of one manifold into another

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

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

theorem that closed 3-manifolds uniquely decompose into pieces with 1 of 8 types of geometric structure

In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.

topological embedding of a 2-dimensional sphere in 3-space whose interior is a 3-ball but whose exterior is not simply connected

geometric modification on manifolds

right|thumb|250px| An image from inside a Three-torus|3-torus. All of the cubes in the image are the same cube, since light in the manifold wraps around into closed loops, the effect is that the cube is tiling all of space. This space has finite volume and no boundary.

branch of topology that studies topological spaces of four or fewer dimensions

mathematical concept of a compact manifold without boundary

three-dimensional self-intersecting surface, an immersion of the real projective plane

neighborhood of a submanifold homeomorphic to that submanifold’s normal bundle

right|thumb|A genus three handlebody. In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surgery theory of high-dimensional manifolds. Handles are used to particularly study 3-manifolds.

two interlinked loops with five structural crossings

surface whose boundary is a knot or a link

four-dimensional geometrical object

In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic).

mathematics glossary

circle bundle over a 2 dimensional orbifold

problem in geometric topology

Group of isotopy classes of a topological automorphism group

open 3-manifold that is contractible, but not homeomorphic to R³

Manifold union

Study in mathematical gauge theory

type of spatial relationship

A simply connected 4-manifold with intersection form the E8 lattice

polygon associated with a compact Riemann surface

geodesically complete metric space of non-positive curvature

Cohomology class of topological structures

way of dividing a simplicial complex

steepest slope on a surface