Topics
Some examples of topics in geometric topology are orientability, handle decompositions, local flatness, and the planar and higher-dimensional Schönflies theorems.
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-manifolds
- 4-manifolds
each have their own theory, where there are some connections.
Knot theory is the study of the 3-dimensional embeddings 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.
2-dimensional topology can be studied as complex geometry in one variable (Riemann surfaces are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure.
Read more about this topic: Geometric Topology