Editing Geometric topology (section)

== Differences between low-dimensional and high-dimensional topology ==
Manifolds differ radically in behavior in high and low dimension.

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

Dimension 4 is special, in that in some respects (topologically), dimension 4 is high-dimensional, while in other respects (differentiably), dimension 4 is low-dimensional; this overlap yields phenomena exceptional to dimension 4, such as [[exotic R4|exotic differentiable structures on '''R'''<sup>4</sup>]]. Thus the topological classification of 4-manifolds is in principle tractable, and the key questions are: does a topological manifold admit a differentiable structure, and if so, how many? Notably, the smooth case of dimension 4 is the last open case of the [[generalized Poincaré conjecture]]; see [[Gluck twist]]s.

The distinction is because [[surgery theory]] works in dimension 5 and above (in fact, in many cases, it works topologically in dimension 4, though this is very involved to prove), and thus the behavior of manifolds in dimension 5 and above may be studied using the surgery theory program. In dimension 4 and below (topologically, in dimension 3 and below), surgery theory does not work.
Indeed, one approach to discussing low-dimensional manifolds is to ask "what would surgery theory predict to be true, were it to work?" – and then understand low-dimensional phenomena as deviations from this.

[[File:Whitneytrickstep2.svg|thumb|The [[Whitney trick]] requires 2+1 dimensions, hence surgery theory requires 5 dimensions.]]
The precise reason for the difference at dimension 5 is because the [[Whitney embedding theorem]], the key technical trick which underlies surgery theory, requires 2+1 dimensions. Roughly, the Whitney trick allows one to "unknot" knotted spheres – more precisely, remove self-intersections of immersions; it does this via a [[homotopy]] of a disk – the disk has 2 dimensions, and the homotopy adds 1 more – and thus in codimension greater than 2, this can be done without intersecting itself; hence embeddings in codimension greater than 2 can be understood by surgery. In surgery theory, the key step is in the middle dimension, and thus when the middle dimension has codimension more than 2 (loosely, 2½ is enough, hence total dimension 5 is enough), the Whitney trick works. The key consequence of this is Smale's [[h-cobordism theorem|''h''-cobordism theorem]], which works in dimension 5 and above, and forms the basis for surgery theory.

A modification of the Whitney trick can work in 4 dimensions, and is called [[Casson handle]]s – because there are not enough dimensions, a Whitney disk introduces new kinks, which can be resolved by another Whitney disk, leading to a sequence ("tower") of disks. The limit of this tower yields a topological but not differentiable map, hence surgery works topologically but not differentiably in dimension 4.