# Low-dimensional topology

In

mathematics ,**low-dimensional topology**is the branch oftopology that studiesmanifold s of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds,knot theory , andbraid group s. It can be regarded as a part ofgeometric 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 thePoincaré 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 insurgery theory . Thurston'sgeometrization 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 forHaken manifold s utilized a variety of tools from previously only weakly linked areas of mathematics.Vaughan Jones ' discovery of theJones 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. In2002 Grigori Perelman announced a proof of the three-dimensional Poincaré conjecture, using Richard Hamilton'sRicci flow , an idea belonging to the field ofgeometric 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:

**Steenrod's theorem**states that an orientable 3-manifold has a trivial tangent bundle. Stated another way, the only characteristic class of a 3-manifold is the obstruction to orientability.Any closed 3-manifold is the boundary of a 4-manifold. This theorem is due independently to several people: it follows from the Dehn-Lickorish theorem via a Heegaard splitting of the 3-manifold. It also follows from Rene Thom's computation of the cobordism ring of closed manifolds.

The existence of exotic smooth structures on

**R**^{4}. This was originally observed byMichael Freedman , based on the work ofSimon Donaldson andAndrew Casson . It has since been elaborated by Freedman,Robert Gompf ,Clifford Taubes andLaurence Taylor to show there exists a continuum of non-diffeomorphic smooth structures on**R**^{4}. Meanwhile,**R**^{n}is known to have exactly one smooth structure up to diffeomorphism provided "n" ≠ 4.**ee also***

List of geometric topology topics **External links***Rob Kirby's [

*http://math.berkeley.edu/~kirby/problems.ps.gz Problems in Low-Dimensional Topology*] -gzipped postscript file (1.4MB)

*Mark Brittenham's [*http://www.math.unl.edu/~mbrittenham2/ldt/ldt.html links to low dimensional topology*] - lists of homepages, conferences, etc.

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