Geometric topology
Encyclopedia
In mathematics
, geometric topology is the study of manifold
s and maps between them, particularly embedding
s of one manifold into another.
, and the planar and higher-dimensional Schönflies theorems.
In all dimensions, the fundamental group
of a manifold is a very important invariant, and determines much of the structure; in dimensions 1, 2 and 3, the possible fundamental groups are restricted, while in every dimension 4 and above every finitely presented group is the fundamental group of a manifold (note that it is sufficient to show this for 4 and 5-dimensional manifolds, and then to take products with spheres to get higher ones).
In low-dimensional topology:
each have their own theory, where there are some connections.
Knot theory
is the study of the 3-dimensional
embedding
s of circles: 1 dimension into 3.
In high-dimensional topology, characteristic classes are a basic invariant, and surgery theory
is a key theory.
Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem
in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature/spherical, zero curvature/flat, negative curvature/hyperbolic – and the geometrization conjecture
(now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of 8 possible geometries.
2-dimensional topology can be studied as complex geometry in one variable (Riemann surfaces are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure.
High-dimensional topology means manifolds of dimension 5 and above, or in relative terms, embeddings in codimension
3 and above, while low-dimensional topology
, concerning questions of dimensions up to four, or embeddings in codimension up to 2.
Dimension 4 is special, in that in some respects (topologically), dimension 4 is high-dimensional, while in other respects (differentiably), dimension 4 is low-dimensional; this overlap yields phenomena exceptional to dimension 4, such as exotic differentiable structures on R4
. Thus the topological classification of 4-manifolds is in principle easy, and the key questions are: does a topological manifold admit a differentiable structure, and if so, how many? Notably, the smooth case of dimension 4 is the last open case of the generalized Poincaré conjecture
; see Gluck twists.
The distinction is because surgery theory
works in dimension 5 and above (in fact, it works topologically in dimension 4, though this is very involved to prove), and thus the behavior of manifolds in dimension 5 and above is controlled algebraically by surgery theory. In dimension 4 and below (topologically, in dimension 3 and below), surgery theory does not work, and other phenomena occur.
Indeed, one approach to discussing low-dimensional manifolds is to ask "what would surgery theory predict to be true, were it to work?" – and then understand low-dimensional phenomena as deviations from this.
The precise reason for the difference at dimension 5 is because the Whitney embedding theorem, the key technical trick which underlies surgery theory, requires 2+1 dimensions. Roughly, the Whitney trick allows one to "unknot" knotted spheres – more precisely, remove self-intersections of immersions;
it does this via a homotopy
of a disk – the disk has 2 dimensions, and the homotopy adds 1 more – and thus in codimension greater than 2, this can be done without intersecting itself; hence embeddings in codimension greater than 2 can be understood by surgery. In surgery theory, the key step is in the middle dimension, and thus when the middle dimension has codimension more than 2 (loosely, 2½ is enough, hence total dimension 5 is enough), the Whitney trick works. The key consequence of this is Smale's h-cobordism theorem, which works in dimension 5 and above, and forms the basis for surgery theory.
A modification of the Whitney trick can work in 4 dimensions, and is called Casson handle
s – because there are not enough dimensions, a Whitney disk introduces new kinks, which can be resolved by another Whitney disk, leading to a sequence ("tower") of disks. The limit of this tower yields a topological but not differentiable map, hence surgery works topologically but not differentiably in dimension 4.
may be said to have originated in the 1935 classification of lens spaces by Reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic. This was the origin of simple homotopy theory.
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...
, geometric topology is the study of 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 and maps between them, particularly embedding
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
s of one manifold into another.
Topics
Some examples of topics in geometric topology are orientability, handle decompositions, local flatnessLocal flatness
In topology, a branch of mathematics, local flatness is a property of a submanifold in a topological manifold of larger dimension.Suppose a d dimensional manifold N is embedded in an n dimensional manifold M...
, and the planar and higher-dimensional Schönflies theorems.
In all dimensions, the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of a manifold is a very important invariant, and determines much of the structure; in dimensions 1, 2 and 3, the possible fundamental groups are restricted, while in every dimension 4 and above every finitely presented group is the fundamental group of a manifold (note that it is sufficient to show this for 4 and 5-dimensional manifolds, and then to take products with spheres to get higher ones).
In low-dimensional topology:
- Surfaces (2-manifolds)
- 3-manifold3-manifoldIn 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 - 4-manifold4-manifoldIn mathematics, 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...
s
each have their own theory, where there are some connections.
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...
is the study of the 3-dimensional
Three-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...
embedding
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
s of circles: 1 dimension into 3.
In high-dimensional topology, characteristic classes are a basic invariant, and 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...
is a key theory.
Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem
Uniformization theorem
In mathematics, the uniformization theorem says that any simply connected Riemann surface is 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...
in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature/spherical, zero curvature/flat, negative curvature/hyperbolic – and the 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...
(now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of 8 possible geometries.
2-dimensional topology can be studied as complex geometry in one variable (Riemann surfaces are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure.
Dimension
Manifolds differ radically in behavior in high and low dimension.High-dimensional topology means manifolds of dimension 5 and above, or in relative terms, embeddings in codimension
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...
3 and above, while low-dimensional topology
Low-dimensional topology
In mathematics, low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. It can be regarded as a part of geometric topology.A number of...
, concerning questions of dimensions up to four, or embeddings in codimension up to 2.
Dimension 4 is special, in that in some respects (topologically), dimension 4 is high-dimensional, while in other respects (differentiably), dimension 4 is low-dimensional; this overlap yields phenomena exceptional to dimension 4, such as exotic differentiable 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...
. Thus the topological classification of 4-manifolds is in principle easy, and the key questions are: does a topological manifold admit a differentiable structure, and if so, how many? Notably, the smooth case of dimension 4 is the last open case of the generalized Poincaré conjecture
Generalized Poincaré conjecture
In the mathematical area of topology, the term Generalized Poincaré conjecture refers to a statement that a manifold which is a homotopy sphere 'is' a sphere. More precisely, one fixes a...
; see Gluck twists.
The distinction is because 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...
works in dimension 5 and above (in fact, it works topologically in dimension 4, though this is very involved to prove), and thus the behavior of manifolds in dimension 5 and above is controlled algebraically by surgery theory. In dimension 4 and below (topologically, in dimension 3 and below), surgery theory does not work, and other phenomena occur.
Indeed, one approach to discussing low-dimensional manifolds is to ask "what would surgery theory predict to be true, were it to work?" – and then understand low-dimensional phenomena as deviations from this.
it does this via a homotopy
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
of a disk – the disk has 2 dimensions, and the homotopy adds 1 more – and thus in codimension greater than 2, this can be done without intersecting itself; hence embeddings in codimension greater than 2 can be understood by surgery. In surgery theory, the key step is in the middle dimension, and thus when the middle dimension has codimension more than 2 (loosely, 2½ is enough, hence total dimension 5 is enough), the Whitney trick works. The key consequence of this is Smale's h-cobordism theorem, which works in dimension 5 and above, and forms the basis for surgery theory.
A modification of the Whitney trick can work in 4 dimensions, and is called Casson handle
Casson handle
In 4-dimensional topology, a branch of mathematics, a Casson handle is a 4-dimensional topological 2-handle constructed by an infinite procedure. They are named for Andrew Casson, who introduced them in about 1973. They were originally called "flexible handles" by Casson himself, and introduced...
s – because there are not enough dimensions, a Whitney disk introduces new kinks, which can be resolved by another Whitney disk, leading to a sequence ("tower") of disks. The limit of this tower yields a topological but not differentiable map, hence surgery works topologically but not differentiably in dimension 4.
History
Geometric topology as an area distinct from algebraic topologyAlgebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
may be said to have originated in the 1935 classification of lens spaces by Reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic. This was the origin of simple homotopy theory.