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Low-dimensional topology

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

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 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 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.

A few typical theorems that distinguish low-dimensional topology
There are several theorems that in effect state that many of the most basic tools used to study high-dimensional manifolds do not apply to low-dimensional manifolds, such as:

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