Editing Low-dimensional topology (section)

==History==
A number of advances starting in the 1960s had the effect of emphasising low dimensions in topology. The solution by [[Stephen Smale]], in 1961, of the [[Poincaré conjecture]] in five or more 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]].  [[William Thurston|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 manifold]]s 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 S. Hamilton]]'s [[Ricci flow]], an idea belonging to the field of [[geometric analysis]].

Overall, this progress has led to better integration of the field into the rest of mathematics.