Differential topology
Encyclopedia
In mathematics
, differential topology is the field dealing with differentiable function
s on differentiable manifold
s. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.
on a manifold to be defined. Smooth manifolds are 'softer' than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold—that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume.
On the other hand, smooth manifolds are more rigid than the topological manifold
s. John Milnor
discovered that some spheres have more than one smooth structure—see exotic sphere
and Donaldson's theorem
. Kervaire exhibited topological manifolds with no smooth structure at all. Some constructions of smooth manifold theory, such as the existence of tangent bundles, can be done in the topological setting with much more work, and others cannot.
One of the main topics in differential topology is the study of special kinds of smooth mappings between manifolds, namely immersions and submersions
, and the intersections of submanifolds via transversality. More generally one is interested in properties and invariants of smooth manifolds which are carried over by diffeomorphisms, another special kind of smooth mapping. Morse theory
is another branch of differential topology, in which topological information about a manifold is deduced from changes in the rank of the Jacobian of a function.
For a list of differential topology topics, see the following reference: List of differential geometry topics.
One major difference lies in the nature of the problems that each subject tries to address. In one view, differential topology distinguishes itself from differential geometry by studying primarily those problems which are inherently global.
Consider the example of a coffee cup and a donut (see this example). From the point of view of differential topology, the donut and the coffee cup are the same (in a sense). This is an inherently global view, though, because there is no way for the differential topologist to tell whether the two objects are the same (in this sense) by looking at just a tiny (local) piece of either of them. He must have access to each entire (global) object.
From the point of view of differential geometry, the coffee cup and the donut are different because it is impossible to rotate the coffee cup in such a way that its configuration matches that of the donut. This is also a global way of thinking about the problem. But an important distinction is that the geometer does not need the entire object to decide this. By looking, for instance, at just a tiny piece of the handle, he can decide that the coffee cup is different from the donut because the handle is thinner (or more curved) than any piece of the donut.
To put it succinctly, differential topology studies structures on manifolds which, in a sense, have no interesting local structure. Differential geometry studies structures on manifolds which do have an interesting local (or sometimes even infinitesimal) structure.
More mathematically, for example, the problem of constructing a diffeomorphism
between two manifolds of the same dimension is inherently global since locally two such manifolds are always diffeomorphic. Likewise, the problem of computing a quantity on a manifold which is invariant under differentiable mappings is inherently global, since any local invariant will be trivial in the sense that it is already exhibited in the topology of Rn. Moreover, differential topology does not restrict itself necessarily to the study of diffeomorphism. For example, symplectic topology
— a subbranch of differential topology — studies global properties of symplectic manifold
s. Differential geometry concerns itself with problems — which may be local or global — that always have some non-trivial local properties. Thus differential geometry may study differentiable manifolds equipped with a connection
, a metric (which may be Riemannian, pseudo-Riemannian, or Finsler), a special sort of distribution (such as a CR structure), and so on.
This distinction between differential geometry and differential topology is blurred, however, in questions specifically pertaining to local diffeomorphism invariants such as the tangent space
at a point. Differential topology also deals with questions like these, which specifically pertain to the properties of differentiable mappings on Rn (for example the tangent bundle
, jet bundles
, the Whitney extension theorem
, and so forth).
Nevertheless, the distinction becomes clearer in abstract terms. Differential topology is the study of the (infinitesimal, local, and global) properties of structures on manifolds having no non-trivial local moduli, whereas differential geometry is the study of the (infinitesimal, local, and global) properties of structures on manifolds having non-trivial local moduli.
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...
, differential topology is the field dealing with differentiable function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s on differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
s. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.
Description
Differential topology considers the properties and structures that require only a smooth structureSmooth structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold....
on a manifold to be defined. Smooth manifolds are 'softer' than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold—that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume.
On the other hand, smooth manifolds are more rigid than the topological manifold
Topological manifold
In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
s. John Milnor
John Milnor
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...
discovered that some spheres have more than one smooth structure—see exotic sphere
Exotic sphere
In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...
and Donaldson's theorem
Donaldson's theorem
In mathematics, Donaldson's theorem states that a definite intersection form of a simply connected smooth manifold of dimension 4 is diagonalisable. If the intersection form is positive definite, it can be diagonalized to the identity matrix...
. Kervaire exhibited topological manifolds with no smooth structure at all. Some constructions of smooth manifold theory, such as the existence of tangent bundles, can be done in the topological setting with much more work, and others cannot.
One of the main topics in differential topology is the study of special kinds of smooth mappings between manifolds, namely immersions and submersions
Submersion (mathematics)
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology...
, and the intersections of submanifolds via transversality. More generally one is interested in properties and invariants of smooth manifolds which are carried over by diffeomorphisms, another special kind of smooth mapping. Morse theory
Morse theory
In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...
is another branch of differential topology, in which topological information about a manifold is deduced from changes in the rank of the Jacobian of a function.
For a list of differential topology topics, see the following reference: List of differential geometry topics.
Differential topology versus differential geometry
Differential topology and differential geometry are first characterized by their similarity. They both study primarily the properties of differentiable manifolds, sometimes with a variety of structures imposed on them.One major difference lies in the nature of the problems that each subject tries to address. In one view, differential topology distinguishes itself from differential geometry by studying primarily those problems which are inherently global.
Consider the example of a coffee cup and a donut (see this example). From the point of view of differential topology, the donut and the coffee cup are the same (in a sense). This is an inherently global view, though, because there is no way for the differential topologist to tell whether the two objects are the same (in this sense) by looking at just a tiny (local) piece of either of them. He must have access to each entire (global) object.
From the point of view of differential geometry, the coffee cup and the donut are different because it is impossible to rotate the coffee cup in such a way that its configuration matches that of the donut. This is also a global way of thinking about the problem. But an important distinction is that the geometer does not need the entire object to decide this. By looking, for instance, at just a tiny piece of the handle, he can decide that the coffee cup is different from the donut because the handle is thinner (or more curved) than any piece of the donut.
To put it succinctly, differential topology studies structures on manifolds which, in a sense, have no interesting local structure. Differential geometry studies structures on manifolds which do have an interesting local (or sometimes even infinitesimal) structure.
More mathematically, for example, the problem of constructing a diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
between two manifolds of the same dimension is inherently global since locally two such manifolds are always diffeomorphic. Likewise, the problem of computing a quantity on a manifold which is invariant under differentiable mappings is inherently global, since any local invariant will be trivial in the sense that it is already exhibited in the topology of Rn. Moreover, differential topology does not restrict itself necessarily to the study of diffeomorphism. For example, symplectic topology
Symplectic topology
Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form...
— a subbranch of differential topology — studies global properties of symplectic manifold
Symplectic manifold
In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...
s. Differential geometry concerns itself with problems — which may be local or global — that always have some non-trivial local properties. Thus differential geometry may study differentiable manifolds equipped with a connection
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
, a metric (which may be Riemannian, pseudo-Riemannian, or Finsler), a special sort of distribution (such as a CR structure), and so on.
This distinction between differential geometry and differential topology is blurred, however, in questions specifically pertaining to local diffeomorphism invariants such as the tangent space
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
at a point. Differential topology also deals with questions like these, which specifically pertain to the properties of differentiable mappings on Rn (for example the 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...
, jet bundles
Jet (mathematics)
In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain...
, the Whitney extension theorem
Whitney extension theorem
In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem. Roughly speaking, the theorem asserts that if A is a closed subset of a Euclidean space, then it is possible to extend a given function off A in such a way as to have...
, and so forth).
Nevertheless, the distinction becomes clearer in abstract terms. Differential topology is the study of the (infinitesimal, local, and global) properties of structures on manifolds having no non-trivial local moduli, whereas differential geometry is the study of the (infinitesimal, local, and global) properties of structures on manifolds having non-trivial local moduli.
See also
- List of differential geometry topics
- Glossary of differential geometry and topologyGlossary of differential geometry and topologyThis is a glossary of terms specific to differential geometry and differential topology.The following two glossaries are closely related:*Glossary of general topology*Glossary of Riemannian and metric geometry.See also:*List of differential geometry topics...
- Important publications in differential geometry
- Important publications in differential topology
- Basic introduction to the mathematics of curved spacetimeBasic introduction to the mathematics of curved spacetimeThe mathematics of general relativity are very complex. In Newton's theories of motions, an object's mass and length remain constant as it changes speed, and the rate of passage of time also remains unchanged. As a result, many problems in Newtonian mechanics can be solved with algebra alone...