Low-dimensional topology
In
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. 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 manifolds 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
A
Classification of surfaces
The
- the sphere;
- the connected sum of g tori, for ;
- the connected sum of k real projective planes, for .
The surfaces in the first two families are 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 characteristics 2 and 0, respectively, and in general the Euler characteristic of the connected sum of g tori is 2 − 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 of k of them is 2 − k.
Teichmüller space
In mathematics, the Teichmüller space TX of a (real) topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Each point in TX may be regarded as an isomorphism class of 'marked' Riemann surfaces where a 'marking' is an isotopy class of homeomorphisms from X to X. The Teichmüller space is the universal covering orbifold of the (Riemann) moduli space.
Teichmüller space has a canonical 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 Oswald Teichmüller (1940).[1]
Uniformization theorem
In
The uniformization theorem is a generalization of the Riemann mapping theorem from proper simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces.
Three dimensions
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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 as
Knot and braid theory
Knot theory is the study of mathematical 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 together so that it cannot be undone. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3 (since we're using topology, a circle isn't bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 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.
A related topic is
Hyperbolic 3-manifolds
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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
Four dimensions
A 4-manifold is a 4-dimensional
4-manifolds are of importance in physics because, in
Exotic R4
An exotic R4 is a
Prior to this construction, non-diffeomorphic
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 invariantprovides 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 dimension n for which Rn can have an exotic smooth structure. R4 has an uncountable number of exotic smooth structures; see exotic R4.
- The solution to the smooth PL manifoldshas 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 theoremholds 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 the 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 Dehn–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 smooth structures on R4. 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 to show there exists a continuum of non-diffeomorphic smooth structures on R4. Meanwhile, Rn is known to have exactly one smooth structure up to diffeomorphism provided n ≠ 4.
See also
References
External links
- Rob Kirby's Problems in Low-Dimensional Topology – gzipped postscript file (1.4 MB)
- Mark Brittenham's links to low dimensional topology – lists of homepages, conferences, etc.