3-manifolds

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

Lie group of 3×3 upper triangular matrices under matrix multiplication

group of rotations in 3 dimensions

flow associated to the partial differential equation ∂𝑔/∂𝑡=−2Ric[𝑔] on a Riemannian manifold

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

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.

In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: \mbox{SL}(2,\mathbf{R}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \colon a,b,c,d \in \mathbf{R}\mbox{ and }ad-bc=1\right\}.

unique knot with a crossing number of four

3-manifold that is a quotient of S³ by ℤ/p actions: (z,w) ↦ (exp(2πi/p)z, exp(2πiq/p)w)

3-dimensional object

circle bundle over a 2 dimensional orbifold

symplectic topology tool

topological manifold whose homology coincides with that of a sphere, i.e. trivial except in the top and bottom degrees, where it has a single generator

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

discrete group of Möbius transformations

theorem in topology

area of mathematics that is a combination of algebraic number theory and topology

220px|right|thumb|A horosphere within the Poincaré disk model tangent to the edges of a [[hexagonal tiling cell of a hexagonal tiling honeycomb]] thumb|Apollonian sphere packing can be seen as showing horospheres that are tangent to an outer sphere of a [[Poincaré disk model]] In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic n-space. It is the boundary of a horoball, the limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of tangency. For n = 2 a horosphere is called a horocycle.

type of mathematical link

Decomposes compact, orientable 3-manifolds uniquely into finitely many prime 3-manifolds

link formed from a finite number of twisted sections (tangles)

mathematical concept