Low-dimensional topology

It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.

A number of advances starting in the 1960s had the effect of emphasising low dimensions in topology.

Thurston's geometrization conjecture, formulated in the late 1970s, offered a framework that suggested geometry and topology were closely intertwined in low dimensions, and Thurston's proof of geometrization for Haken manifolds utilized a variety of tools from previously only weakly linked areas of mathematics.

In 2002, Grigori Perelman announced a proof of the three-dimensional Poincaré conjecture, using Richard S. Hamilton's Ricci flow, an idea belonging to the field of geometric analysis.

On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in three-dimensional Euclidean space without introducing singularities or self-intersections.

It is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori.

The Euler characteristic of the real projective plane is 1, and in general the Euler characteristic of the connected sum of k of them is 2 − k. In mathematics, the Teichmüller space TX of a (real) topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism.

Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics.

The uniformization theorem is a generalization of the Riemann mapping theorem from proper simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces.

A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1.

In this case, the ends are of the form torus cross the closed half-ray and are called cusps.

Here are some examples: There are several theorems that in effect state that many of the most basic tools used to study high-dimensional manifolds do not apply to low-dimensional manifolds, such as: Steenrod's theorem states that an orientable 3-manifold has a trivial tangent bundle.

It also follows from René Thom's computation of the cobordism ring of closed manifolds.

This was originally observed by Michael Freedman, based on the work of Simon Donaldson and Andrew Casson.

It has since been elaborated by Freedman, Robert Gompf, Clifford Taubes and Laurence Taylor to show there exists a continuum of non-diffeomorphic smooth structures on R4.