Dimension
Manifolds differ radically in behavior in high and low dimension.
High-dimensional topology means manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above, while low-dimensional topology, concerning questions of dimensions up to four, 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 differentiable structures on R4. Thus the topological classification of 4-manifolds is in principle easy, 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 twists.
The distinction is because surgery theory 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. 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.
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, 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 handles – 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.
Read more about this topic: Geometric Topology
Famous quotes containing the word dimension:
“God cannot be seen: he is too bright for sight; nor grasped: he is too pure for touch; nor measured: for he is beyond all sense, infinite, measureless, his dimension known to himself alone.”
—Marcus Minucius Felix (2nd or 3rd cen. A.D.)
“Le Corbusier was the sort of relentlessly rational intellectual that only France loves wholeheartedly, the logician who flies higher and higher in ever-decreasing circles until, with one last, utterly inevitable induction, he disappears up his own fundamental aperture and emerges in the fourth dimension as a needle-thin umber bird.”
—Tom Wolfe (b. 1931)
“By intervening in the Vietnamese struggle the United States was attempting to fit its global strategies into a world of hillocks and hamlets, to reduce its majestic concerns for the containment of communism and the security of the Free World to a dimension where governments rose and fell as a result of arguments between two colonels’ wives.”
—Frances Fitzgerald (b. 1940)