Low-dimensional topology
Encyclopedia
In mathematics
, low-dimensional topology is the branch of topology
that studies manifold
s of four or fewer dimension
s. 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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, low-dimensional topology is the branch of topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
that studies manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s of four or fewer dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
s. Representative topics are the structure theory of 3-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. 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...
s and 4-manifolds, knot theory
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which 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 precise mathematical language, a knot is an embedding of a...
, and braid group
Braid group
In mathematics, the braid group on n strands, denoted by Bn, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group Sn. Here, n is a natural number; if n > 1, then Bn is an infinite group...
s. It can be regarded as a part of geometric topology
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
.
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
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
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
Surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...
. Thurston's
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
geometrization conjecture
Geometrization conjecture
Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
, 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
Haken manifold
In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface...
s utilized a variety of tools from previously only weakly linked areas of mathematics. Vaughan Jones
Vaughan Jones
Sir Vaughan Frederick Randal Jones, KNZM, FRS, FRSNZ is a New Zealand mathematician, known for his work on von Neumann algebras, knot polynomials and conformal field theory. He was awarded a Fields Medal in 1990, and famously wore a New Zealand rugby jersey when he accepted the prize...
' 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
Grigori Perelman
Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...
announced a proof of the three-dimensional Poincaré conjecture, using Richard Hamilton
Richard Hamilton (professor)
Richard Streit Hamilton is Davies Professor of mathematics at Columbia University.He received his B.A in 1963 from Yale University and Ph.D. in 1966 from Princeton University. Robert Gunning supervised his thesis...
's Ricci flow
Ricci flow
In differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric....
, an idea belonging to the field of geometric analysis
Geometric analysis
Geometric analysis is a mathematical discipline at the interface of differential geometry and differential equations. It includes both the use of geometrical methods in the study of partial differential equations , and the application of the theory of partial differential equations to geometry...
.
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
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
. Stated another way, the only characteristic class
Characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...
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
Max Dehn
Max Dehn was a German American mathematician and a student of David Hilbert. He is most famous for his work in geometry, topology and geometric group theory...
–Lickorish theorem via a Heegaard splitting
Heegaard splitting
In the mathematical field of geometric topology, a Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.-Definitions:...
of the 3-manifold. It also follows from René Thom
René Thom
René Frédéric Thom was a French mathematician. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as...
's computation of the cobordism
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes...
ring of closed manifolds.
The existence of exotic smooth structures on R4
Exotic R4
In mathematics, an exotic R4 is a differentiable manifold that is homeomorphic to the Euclidean space R4, but not diffeomorphic.The first examples were found by Robion Kirby and Michael Freedman, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's...
. This was originally observed by Michael Freedman
Michael Freedman
Michael Hartley Freedman is a mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...
, based on the work of Simon Donaldson
Simon Donaldson
Simon Kirwan Donaldson FRS , is an English mathematician known for his work on the topology of smooth four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London...
and Andrew Casson
Andrew Casson
Andrew John Casson FRS is a mathematician, an expert on geometric topology.Casson is the Philip Schuyler Beebe Professor of Mathematics at Yale University in the United States where he served as department chair between 2004 and 2007. His Ph.D. advisor at the University of Liverpool was C. T. C...
. It has since been elaborated by Freedman, Robert Gompf, Clifford Taubes
Clifford Taubes
Clifford Henry Taubes is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology.-Early career:Taubes received his Ph.D...
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.
External links
- Rob KirbyRobion KirbyRobion Cromwell Kirby is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology...
's Problems in Low-Dimensional Topology -gzipped postscript file (1.4MB) - Mark Brittenham's links to low dimensional topology - lists of homepages, conferences, etc.