Low-dimensional topology

In mathematics, low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. It can be regarded as a part of geometric topology.

History

A number of advances starting in the 1960s had the effect of emphasising low dimensions in topology. The solution by Smale, in 1961, of the Poincaré conjecture in higher dimensions made dimensions three and four seem the hardest; and indeed they required new methods, while the freedom of higher dimensions meant that questions could be reduced to computational methods available in surgery theory. Thurston's geometrization conjecture, formulated in the late 1970s, offered a framework that suggested geometry and topology were closely intertwined in low dimensions, and Thurston's proof of geometrization for Haken manifolds utilized a variety of tools from previously only weakly linked areas of mathematics. Vaughan Jones' discovery of the Jones polynomial in the early 1980s not only led knot theory in new directions but gave rise to still mysterious connections between low-dimensional topology and mathematical physics. In 2002 Grigori Perelman announced a proof of the three-dimensional Poincaré conjecture, using Richard Hamilton's Ricci flow, an idea belonging to the field of geometric analysis.