---
date: 2021-04-26
---


Tags: #projects/review #expository #geomtop 

# What is the difference between low and high dimensional topology?

[High-dimensional topology](High-dimensional%20topology) refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. [Low-dimensional topology](Low-dimensional%20topology) is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2.

The distinction is because [surgery theory](surgery%20theory) works in dimension 5 and above (in fact, it works topologically in dimension 4, though this is very involved to prove), and thus the behavior of manifolds in dimension 5 and above is controlled algebraically by surgery theory. In dimension 4 and below (topologically, in dimension 3 and below), surgery theory does not work, and other phenomena occur.

The precise reason for the difference at dimension 5 is because the Whitney embedding theorem, the [key technical trick](Whitney%20trick.md) which underlies surgery theory, requires 2+1 dimensions.

Low-dimensional topology is strongly geometric, as reflected in the [uniformization](uniformization.md) 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](Geometrization.md) theorem in 3 dimensions - every [3-manifold](3-manifold) can be cut into pieces, each of which has one of 8 possible geometries.