Inmathematics,low-dimensional topology is the branch oftopology that studiesmanifolds, or more generally topological spaces, of four or fewerdimensions. Representative topics are the theory of3-manifolds and4-manifolds,knot theory, andbraid groups. This can be regarded as a part ofgeometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part ofcontinuum theory.

A number of advances starting in the 1960s had the effect of emphasising low dimensions in topology. The solution byStephen Smale, in 1961, of thePoincaré 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 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 manifolds 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 andmathematical physics. In 2002,Grigori Perelman announced a proof of the three-dimensional Poincaré conjecture, usingRichard S. 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.

Asurface is atwo-dimensional,topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensionalEuclidean spaceR³—for example, the surface of aball. On the other hand, there are surfaces, such as theKlein bottle, that cannot beembedded in three-dimensional Euclidean space without introducingsingularities or self-intersections.

Theclassification theorem of closed surfaces states that anyconnectedclosed surface is homeomorphic to some member of one of these three families:

1. the sphere;
2. theconnected sum ofgtori, for${\displaystyle g\ge 1}$;
3. the connected sum ofkreal projective planes, for${\displaystyle k\ge 1}$.

The surfaces in the first two families areorientable. It is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The numberg of tori involved is called thegenus of the surface. The sphere and the torus haveEuler characteristics 2 and 0, respectively, and in general the Euler characteristic of the connected sum ofg tori is2 − 2g.

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 ofk of them is2 −k.

Inmathematics, theTeichmüller spaceT_X of a (real) topological surfaceX, is a space that parameterizescomplex structures onX up to the action ofhomeomorphisms that areisotopic to theidentity homeomorphism. Each point inT_X may be regarded as an isomorphism class of 'marked'Riemann surfaces where a 'marking' is an isotopy class of homeomorphisms fromX toX. The Teichmüller space is theuniversal covering orbifold of the (Riemann) moduli space.

Teichmüller space has a canonicalcomplexmanifold 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 byOswald Teichmüller (1940).^[1]

Inmathematics, theuniformization theorem says that everysimply connectedRiemann surface isconformally equivalent to one of the three domains: the openunit disk, thecomplex plane, or theRiemann sphere. In particular it admits aRiemannian metric ofconstant curvature. This classifies Riemannian surfaces as elliptic (positively curved—rather, admitting a constant positively curved metric), parabolic (flat), and hyperbolic (negatively curved) according to theiruniversal cover.

The uniformization theorem is a generalization of theRiemann mapping theorem from proper simply connectedopensubsets of the plane to arbitrary simply connected Riemann surfaces.

Atopological spaceX is a 3-manifold if every point inX has aneighbourhood that ishomeomorphic toEuclidean 3-space.

The topological,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 asknot theory,geometric group theory,hyperbolic geometry,number theory,Teichmüller theory,topological quantum field theory,gauge theory,Floer homology, andpartial differential equations. 3-manifold theory is considered a part of low-dimensional topology orgeometric topology.

Knot theory is the study ofmathematical knots. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined so that it cannot be undone. In mathematical language, a knot is anembedding of acircle in 3-dimensionalEuclidean space,R³ (since we're using topology, a circle isn't bound to the classical geometric concept, but to all of itshomeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation ofR³ upon itself (known as anambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

Knot complements are frequently-studied 3-manifolds. The knot complement of atame knotK is the three-dimensional space surrounding the knot. To make this precise, suppose thatK is a knot in a three-manifoldM (most often,M is the3-sphere). LetN be atubular neighborhood ofK; soN is asolid torus. The knot complement is then thecomplement ofN,

${\displaystyle X_K=M-\text{interior}(N).}$

A related topic isbraid theory. Braid theory is an abstractgeometrictheory studying the everydaybraid concept, and some generalizations. The idea is that braids can be organized intogroups, 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 explicitpresentations, as was shown byEmil Artin (1947).^[2] For an elementary treatment along these lines, see the article onbraid groups. Braid groups may also be given a deeper mathematical interpretation: as thefundamental group of certainconfiguration spaces.

Ahyperbolic 3-manifold is a3-manifold equipped with acompleteRiemannian metric of constantsectional curvature -1. In other words, it is the quotient of three-dimensionalhyperbolic space by a subgroup of hyperbolic isometries acting freely andproperly discontinuously. See alsoKleinian 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 calledcusps. Knot complements are the most commonly studied cusped manifolds.

Thurston's geometrization conjecture states that certain three-dimensionaltopological spaces each have a unique geometric structure that can be associated with them. It is an analogue of theuniformization theorem for two-dimensionalsurfaces, which states that everysimply-connectedRiemann surface can be given one of three geometries (Euclidean,spherical, orhyperbolic).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 closed3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed byWilliam Thurston (1982), and implies several other conjectures, such as thePoincaré conjecture and Thurston'selliptization conjecture.^[3]

A4-manifold is a 4-dimensionaltopological manifold. Asmooth 4-manifold is a 4-manifold with asmooth 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 arehomeomorphic but notdiffeomorphic).

4-manifolds are of importance in physics because, inGeneral Relativity,spacetime is modeled as apseudo-Riemannian 4-manifold.

AnexoticR⁴ is adifferentiable manifold that ishomeomorphic but notdiffeomorphic to theEuclidean spaceR⁴. The first examples were found in the early 1980s byMichael Freedman, by using the contrast between Freedman's theorems about topological 4-manifolds, andSimon Donaldson's theorems about smooth 4-manifolds.^[4] There is acontinuum of non-diffeomorphicdifferentiable structures ofR⁴, as was shown first byClifford Taubes.^[5]

Prior to this construction, non-diffeomorphicsmooth structures on spheres—exotic spheres—were already known to exist, although the question of the existence of such structures for the particular case of the4-sphere remained open (and still remains open to this day). For any positive integern other than 4, there are no exotic smooth structures onRⁿ; in other words, ifn ≠ 4 then any smooth manifold homeomorphic toRⁿ is diffeomorphic toRⁿ.^[6]

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, theKirby–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 H⁴(M,Z/2Z) 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 dimensionn for whichRⁿ can have an exotic smooth structure.R⁴ has an uncountable number of exotic smooth structures; seeexoticR⁴.
• The solution to the smoothPoincaré conjecture is known in all dimensions other than 4 (it is usually false in dimensions at least 7; seeexotic sphere). The Poincaré conjecture forPL manifolds 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 smoothh-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.

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 trivialtangent bundle. Stated another way, the onlycharacteristic 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 theDehn–Lickorish theorem via aHeegaard splitting of the 3-manifold. It also follows fromRené Thom's computation of thecobordism ring of closed manifolds.

The existence ofexotic smooth structures onR⁴. 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 onR⁴. Meanwhile,Rⁿ is known to have exactly one smooth structure up to diffeomorphism providedn ≠ 4.

• List of geometric topology topics

• Rob Kirby'sProblems in Low-Dimensional Topology – gzipped postscript file (1.4 MB)
• Mark Brittenham'slinks to low dimensional topology – lists of homepages, conferences, etc.

• Low-dimensional topology
• Geometric topology

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