Editing Low-dimensional topology

{{short description|Branch of topology}}
[[Image:Trefoil knot arb.png|thumb|A three-dimensional depiction of a thickened [[trefoil knot]], the simplest non-[[trivial knot]]. [[Knot theory]] is an important part of low-dimensional topology.]]

In [[mathematics]], '''low-dimensional topology''' is the branch of [[topology]] that studies [[manifold]]s, or more generally topological spaces, of four or fewer [[dimension]]s. Representative topics are the structure theory of [[3-manifold]]s and [[4-manifold]]s, [[knot theory]], and [[braid group]]s. This can be regarded as a part of [[geometric topology]]. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of [[continuum theory]].

==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.

==Two dimensions==
{{Main|surface (topology)}}

A [[Surface (topology)|surface]] is a [[two-dimensional]], [[topological manifold]]. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional [[Euclidean space]] '''R'''<sup>3</sup>—for example, the surface of a [[ball (mathematics)|ball]]. On the other hand, there are surfaces, such as the [[Klein bottle]], that cannot be [[embedding|embedded]] in three-dimensional Euclidean space without introducing [[singularity theory|singularities]] or self-intersections.

===Classification of surfaces===
The ''[[classification theorem]] of closed surfaces'' states that any [[connected (topology)|connected]] [[closed manifold|closed]] surface is homeomorphic to some member of one of these three families:

# the sphere;
# the [[connected sum]] of ''g'' [[torus|tori]], for <math>g \geq 1</math>;
# the connected sum of ''k'' [[real projective plane]]s, for <math>k \geq 1</math>.

The surfaces in the first two families are [[orientability|orientable]]. It is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The number ''g'' of tori involved is called the ''genus'' of the surface. The sphere and the torus have [[Euler characteristic]]s 2 and 0, respectively, and in general the Euler characteristic of the connected sum of ''g'' tori is {{nowrap|2 &minus; 2''g''}}.

The surfaces in the third family are nonorientable. The Euler characteristic of the real projective plane is 1, and in general the Euler characteristic of the connected sum of ''k'' of them is {{nowrap|2 &minus; ''k''}}.

===Teichmüller space===
{{Main|Teichmüller space}}

In [[mathematics]], the '''Teichmüller space''' ''T<sub>X</sub>'' of a (real) topological surface ''X'', is a space that parameterizes [[complex manifold|complex structures]] on ''X'' up to the action of [[homeomorphism]]s that are [[Homotopy#Isotopy|isotopic]] to the [[identity function|identity homeomorphism]]. Each point in ''T<sub>X</sub>'' may be regarded as an isomorphism class of 'marked' [[Riemann surface]]s where a 'marking' is an isotopy class of homeomorphisms from ''X'' to ''X''. 
The Teichmüller space is the [[orbifold|universal covering orbifold]] of the (Riemann) moduli space.

Teichmüller space has a canonical [[complex number|complex]] [[manifold]] structure and a wealth of natural metrics. The underlying topological space of Teichmüller space was studied by Fricke, and the Teichmüller metric on it  was introduced by {{harvs|txt|authorlink=Oswald Teichmüller|first=Oswald |last=Teichmüller|year=1940}}.<ref>{{citation
 | last = Teichmüller | first = Oswald | authorlink = Oswald Teichmüller
 | issue = 22
 | journal = Abh. Preuss. Akad. Wiss. Math.-Nat. Kl.
 | mr = 0003242
 | page = 197
 | title = Extremale quasikonforme Abbildungen und quadratische Differentiale
 | volume = 1939
 | year = 1940}}.</ref>

===Uniformization theorem===
{{Main|Uniformization theorem}}

In [[mathematics]], the '''uniformization theorem'''  says that every [[simply connected]] [[Riemann surface]] is [[Conformal equivalence|conformally equivalent]] to one of the three domains: the open [[unit disk]], the [[complex plane]], or the [[Riemann sphere]]. In particular it admits a [[Riemannian metric]] of [[constant curvature]]. This classifies Riemannian surfaces as elliptic (positively curved—rather, admitting a constant positively curved metric), parabolic (flat), and hyperbolic (negatively curved) according to their [[universal cover]].

The uniformization theorem is a generalization of the [[Riemann mapping theorem]] from proper simply connected [[open set|open]] [[subset]]s of the plane to arbitrary simply connected Riemann surfaces.

==Three dimensions==
{{Main|3-manifold}}

A [[topological space]] ''X'' is a 3-manifold if every point in ''X'' has a [[neighbourhood (mathematics)|neighbourhood]] that is [[homeomorphic]] to [[Euclidean 3-space]].

The topological, [[Piecewise linear manifold|piecewise-linear]], and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.

Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions greater than three.  This special role has led to the discovery of close connections to a diversity of other fields, such as [[knot theory]], [[geometric group theory]], [[hyperbolic geometry]],  [[number theory]], [[Teichmüller space|Teichmüller theory]], [[topological quantum field theory]], [[gauge theory]], [[Floer homology]], and [[partial differential equations]].  3-manifold theory is considered a part of low-dimensional topology or [[geometric topology]].

===Knot and braid theory===
{{Main|Knot theory|Braid theory}}

[[Knot theory]] is the study of [[knot (mathematics)|mathematical knot]]s. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an [[embedding]] of a [[circle]] in 3-dimensional [[Euclidean space]], '''R'''<sup>3</sup> (since we're using topology, a circle isn't bound to the classical geometric concept, but to all of its [[homeomorphism]]s). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of '''R'''<sup>3</sup> upon itself (known as an [[ambient isotopy]]); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

[[Knot complement]]s are frequently-studied 3-manifolds. The knot complement of a [[tame knot]] ''K'' is the three-dimensional space surrounding the knot. To make this precise, suppose that ''K'' is a knot in a three-manifold ''M'' (most often, ''M'' is the  [[3-sphere]]).  Let ''N'' be a [[tubular neighborhood]] of ''K''; so ''N'' is a [[solid torus]].  The knot complement is then the [[complement (set theory)|complement]] of ''N'',
:<math>X_K = M - \mbox{interior}(N).</math>

A related topic is [[braid theory]]. Braid theory is an abstract [[geometry|geometric]] [[theory]] studying the everyday [[braid]] concept, and some generalizations. The idea is that braids can be organized into [[group (mathematics)|group]]s, in which the group operation is 'do the first braid on a set of strings, and then follow it with a second on the twisted strings'. Such groups may be described by explicit [[presentation of a group|presentation]]s, as was shown by {{harvs|first=Emil|last=Artin|authorlink=Emil Artin|year=1947|txt}}.<ref>{{citation
 | last = Artin | first = E. | authorlink = Emil Artin
 | doi = 10.2307/1969218
 | journal = [[Annals of Mathematics]]
 | mr = 0019087
 | pages = 101–126
 | series = Second Series
 | title = Theory of braids
 | volume = 48
 | year = 1947}}.</ref> For an elementary treatment along these lines, see the article on [[braid group]]s. Braid groups may also be given a deeper mathematical interpretation: as the [[fundamental group]] of certain [[Configuration space (mathematics)|configuration space]]s.

===Hyperbolic 3-manifolds===
{{Main|Hyperbolic 3-manifold}}

A [[hyperbolic 3-manifold]] is a [[3-manifold]] equipped with a [[complete space|complete]] [[Riemannian metric]] of constant [[sectional curvature]] -1.  In other words, it is the quotient of three-dimensional [[hyperbolic space]] by a subgroup of hyperbolic isometries acting freely and [[Properly discontinuous action|properly discontinuously]].  See also [[Kleinian model]].

Its thick-thin decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and/or ends that are the product of a Euclidean surface and the closed half-ray.  The manifold is of finite volume if and only if its thick part is compact.  In this case, the ends are of the form torus cross the closed half-ray and are called '''cusps'''. Knot complements are the most commonly studied cusped manifolds.

===Poincaré conjecture and geometrization===
{{Main|Geometrization conjecture}}

[[Thurston's geometrization conjecture]] states that certain three-dimensional [[topological space]]s each have a unique geometric structure that can be associated with them. It is an analogue of the [[uniformization theorem]] for two-dimensional [[surface (topology)|surface]]s, which states that every [[simply connected|simply-connected]] [[Riemann surface]] can be given one of three geometries ([[Euclidean geometry|Euclidean]], [[Spherical geometry|spherical]], or [[hyperbolic geometry|hyperbolic]]).
In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed [[3-manifold]] can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure.  The conjecture was proposed by {{harvs|txt|authorlink=William Thurston|first=William|last= Thurston|year= 1982}}, and implies several other conjectures, such as the [[Poincaré conjecture]] and Thurston's [[elliptization conjecture]].<ref>{{citation
 | last = Thurston | first = William P. | authorlink = William Thurston
 | doi = 10.1090/S0273-0979-1982-15003-0
 | issue = 3
 | journal = [[Bulletin of the American Mathematical Society]]
 | mr = 648524
 | pages = 357–381
 | series = New Series
 | title = Three-dimensional manifolds, Kleinian groups and hyperbolic geometry
 | volume = 6
 | year = 1982| doi-access = free
 }}.</ref>

==Four dimensions==
{{Main|4-manifold}}

A '''4-manifold''' is a 4-dimensional [[topological manifold]]. A '''smooth 4-manifold''' is a 4-manifold with a [[smooth structure]]. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds  are quite different. There exist some topological 4-manifolds that admit no smooth structure and even if there exists a smooth structure it need not be unique (i.e. there are smooth 4-manifolds that are [[homeomorphic]] but not [[diffeomorphic]]).

4-manifolds are of importance in physics because, in [[General Relativity]], [[spacetime]] is modeled as a [[pseudo-Riemannian]] 4-manifold.

===Exotic R<sup>4</sup>===
{{main|Exotic R4|l1=Exotic '''R'''<sup>4</sup>}}

An '''exotic''' '''R'''<sup>4</sup> is a [[differentiable manifold]] that is [[homeomorphic]] but not [[diffeomorphism|diffeomorphic]] to the [[Euclidean space]] '''R'''<sup>4</sup>. The first examples were found in the early 1980s by [[Michael Freedman]], by using the contrast between Freedman's theorems about topological 4-manifolds, and [[Simon Donaldson]]'s theorems about smooth 4-manifolds.<ref>{{citation
 | last = Gompf | first = Robert E. | authorlink = Robert Gompf
 | issue = 2
 | journal = [[Journal of Differential Geometry]]
 | mr = 710057
 | pages = 317–328
 | title = Three exotic '''R'''<sup>4</sup>'s and other anomalies
 | url = http://projecteuclid.org/euclid.jdg/1214437666
 | volume = 18
 | year = 1983}}.</ref>  There is a [[cardinality of the continuum|continuum]] of non-diffeomorphic [[differentiable structure]]s of '''R'''<sup>4</sup>, as was shown first by [[Clifford Taubes]].<ref>Theorem 1.1 of {{citation
 | last = Taubes | first = Clifford Henry | authorlink = Clifford Taubes
 | issue = 3
 | journal = [[Journal of Differential Geometry]]
 | mr = 882829
 | pages = 363–430
 | title = Gauge theory on asymptotically periodic 4-manifolds
 | url = http://projecteuclid.org/euclid.jdg/1214440981
 | volume = 25
 | year = 1987}}</ref>

Prior to this construction, non-diffeomorphic [[smooth structure]]s on spheres—[[exotic sphere]]s—were already known to exist, although the question of the existence of such structures for the particular case of the [[4-sphere]] remained open (and still remains open to this day).  For any positive integer ''n'' other than 4, there are no exotic smooth structures on '''R'''<sup>''n''</sup>; in other words, if ''n'' ≠ 4 then any smooth manifold homeomorphic to '''R'''<sup>''n''</sup> is diffeomorphic to '''R'''<sup>''n''</sup>.<ref>Corollary 5.2 of {{citation
 | last = Stallings | first = John | authorlink = John R. Stallings
 | doi = 10.1017/S0305004100036756
 | journal = [[Mathematical Proceedings of the Cambridge Philosophical Society]]
 | mr = 0149457
 | pages = 481–488
 | title = The piecewise-linear structure of Euclidean space
 | volume = 58
 | issue = 3
 | year = 1962}}.</ref>

===Other special phenomena in four dimensions===
There are several fundamental theorems about manifolds that can be proved by low-dimensional methods in dimensions at most 3, and by completely different high-dimensional methods in dimension at least 5, but which are false in four dimensions. Here are some examples:

* In dimensions other than 4, the [[Kirby–Siebenmann invariant]] provides the obstruction to the existence of a PL structure; in other words a compact topological manifold has a PL structure if and only if its Kirby–Siebenmann invariant in H<sup>4</sup>(''M'','''Z'''/2'''Z''') vanishes. In dimension 3 and lower, every topological manifold admits an essentially unique PL structure. In dimension 4 there are many examples with vanishing Kirby–Siebenmann invariant but no PL structure.
* In any dimension other than 4, a compact topological manifold has only a finite number of essentially distinct PL or smooth structures. In dimension 4, compact manifolds can have a countable infinite number of non-diffeomorphic smooth structures.
* Four is the only dimension ''n'' for which '''R'''<sup>''n''</sup> can have an exotic smooth structure. '''R'''<sup>4</sup> has an uncountable number of exotic smooth structures; see [[exotic R4|exotic '''R'''<sup>4</sup>]].
* The solution to the smooth [[Poincaré conjecture]] is known in all dimensions other than 4 (it is usually false in dimensions at least 7; see [[exotic sphere]]). The Poincaré conjecture for [[PL manifold]]s has been proved for all dimensions other than 4, but it is not known whether it is true in 4 dimensions (it is equivalent to the smooth Poincaré conjecture in 4 dimensions).
* The smooth [[h-cobordism theorem]] holds for cobordisms provided that neither the cobordism nor its boundary has dimension 4. It can fail if the boundary of the cobordism has dimension 4 (as shown by Donaldson). If the cobordism has dimension  4, then it is unknown whether the h-cobordism theorem holds.
* A topological manifold of dimension not equal to 4 has a handlebody decomposition. Manifolds of dimension 4 have a handlebody decomposition if and only if they are smoothable.
* There are compact 4-dimensional topological manifolds that are not homeomorphic to any simplicial complex. In dimension at least 5 the existence of topological manifolds not homeomorphic to a simplicial complex was an open problem. In 2013, Ciprian Manolescu posted a preprint on ArXiv showing that there are manifolds in each dimension greater than or equal to 5, that are not homeomorphic to a simplicial complex.

==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 [[Max Dehn|Dehn]]&ndash;[[W. B. R. Lickorish|Lickorish]] theorem via a [[Heegaard splitting]] of the 3-manifold.  It also follows from [[René Thom]]'s computation of the [[cobordism]] ring of closed manifolds.

The existence of [[Exotic R4|exotic smooth structures on '''R'''<sup>4</sup>]]. This was originally observed by [[Michael Freedman]], based on the work of [[Simon Donaldson]] and [[Andrew Casson]]. It has since been elaborated by Freedman, [[Robert Gompf]], [[Clifford Taubes]] and [[Laurence Taylor (mathematician)|Laurence Taylor]] to show there exists a continuum of non-diffeomorphic smooth structures on '''R'''<sup>4</sup>. Meanwhile, '''R'''<sup>n</sup> is known to have exactly one smooth structure up to diffeomorphism provided ''n'' ≠ 4.

==See also==
* [[List of geometric topology topics]]

==References==
{{Reflist}}

==External links==
* [[Robion Kirby|Rob Kirby]]'s [http://math.berkeley.edu/~kirby/problems.ps.gz Problems in Low-Dimensional Topology]{{snd}}gzipped postscript file (1.4&nbsp;MB)
* Mark Brittenham's [http://www.math.unl.edu/~mbrittenham2/ldt/ldt.html links to low dimensional topology]{{snd}}lists of homepages, conferences, etc.

{{Topology}}

[[Category:Low-dimensional topology| ]]
[[Category:Geometric topology]]