About

Low dimensional topology generally refers to the study of 3 or 4 dimensional topological manifolds (which, as it turns out, is highly related to the study of knot theory: a knot is an embedding of the circle into the 3-sphere, and the property of knots can be completely classified by the topology of the 3-manifold formed from removing the knot from the 3-sphere).

That topologists are interested in low dimensional topology has largely to do with the set of tools available to them. In dimensions 1 and 2, the study of topological manifolds is completely equivalent to the study of Riemannian manifolds, and topological surfaces have long been completely classified. In dimensions 5 and higher, topological manifolds become very pliable: on the one hand this allows for a lot of pretty bad behaviour, on the other one also gets some really powerful tools (h-cobordism theorem, for example). In 3 and 4 dimensions, the study of topological manifolds becomes "just right": the manifolds are floppy enough that (Riemannian/differential) geometry doesn't completely determine topology (existence of exotic 4-manifolds; any 3 (or higher) dimensional smooth manifold admits a negative Ricci curvature metric), but rigid enough that some tools from geometry can be used (Perelman's proof of the Poincare conjecture using Ricci flow, application of Yang-Mills theory to the topology of 4-manifolds).

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