geometric topology

In [mathematics] , geometric topology is the study of [manifold] s and maps between them, particularly [embedding] s of one manifold into another.

Topics

Some examples of topics in geometric topology are [orientability] , [handle decomposition] s, [local flatness] , and the planar and higher-dimensional [Schönflies theorem] s.

In all dimensions, the [fundamental group] of a manifold is a very important invariant, and determines much of the structure; in dimensions 1, 2 and 3, the possible fundamental groups are restricted, while in every dimension 4 and above every [finitely presented group] is the fundamental group of a manifold (note that it is sufficient to show this for 4 and 5-dimensional manifolds, and then to take products with spheres to get higher ones).

In low-dimensional topology:
- [Surfaces] (2-manifolds)
- [3-manifold] s
- [4-manifold] s
each have their own theory, where there are some connections.

[Knot theory] is the study of the [3-dimensional] [embedding] s of [circles] : 1 dimension into 3.

In high-dimensional topology, [characteristic classes] are a basic invariant, and [surgery theory] is a key theory.

Low-dimensional topology is strongly geometric, as reflected in the [uniformization theorem] in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature/spherical, zero curvature/flat, negative curvature/hyperbolic – and the [geometrization conjecture] (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of 8 possible geometries.