Differential topology
In
The central goal of the field of differential topology is the
- In dimension 1, the only smooth manifolds up to diffeomorphism are the and fully closed interval .[2]
- In dimension 2, every orientable. This is the famous classification of closed surfaces.[3][4] Already in dimension two the classification of non-compact surfaces becomes difficult, due to the existence of exotic spaces such as Jacob's ladder.
- In dimension 3, homeomorphic (and in fact diffeomorphic) to the 3-sphere.
Beginning in dimension 4, the classification becomes much more difficult for two reasons.
Important tools in studying the differential topology of smooth manifolds include the construction of smooth
Famous theorems in differential topology include the Whitney embedding theorem, the hairy ball theorem, the Hopf theorem, the Poincaré–Hopf theorem, Donaldson's theorem, and the Poincaré conjecture.
Description
Differential topology considers the properties and structures that require only a
On the other hand, smooth manifolds are more rigid than the topological manifolds. John Milnor discovered that some spheres have more than one smooth structure—see Exotic sphere and Donaldson's theorem. Michel Kervaire exhibited topological manifolds with no smooth structure at all.[9] Some constructions of smooth manifold theory, such as the existence of tangent bundles,[10] 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
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,[4] differential topology distinguishes itself from differential geometry by studying primarily those problems that are inherently global. Consider the example of a coffee cup and a donut. 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. They 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, they 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 that, in a sense, have no interesting local structure. Differential geometry studies structures on manifolds that 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 that 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 . Moreover, differential topology does not restrict itself necessarily to the study of diffeomorphism. For example,
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 (for example the tangent bundle, jet bundles, the Whitney extension theorem, and so forth).
The distinction is concise in abstract terms:
- Differential topology is the study of the (infinitesimal, local, and global) properties of structures on manifolds that have only trivial local moduli.
- Differential geometry is such a study of structures on manifolds that have one or more non-trivial local moduli.
See also
- List of differential geometry topics
- Glossary of differential geometry and topology
- Important publications in differential geometry
- Important publications in differential topology
- Basic introduction to the mathematics of curved spacetime
Notes
- ^ Bott, R. and Tu, L.W., 1982. Differential forms in algebraic topology (Vol. 82, pp. xiv+-331). New York: Springer.
- ^ Milnor, J. and Weaver, D.W., 1997. Topology from the differentiable viewpoint. Princeton university press.
- ^ Lee, J., 2010. Introduction to topological manifolds (Vol. 202). Springer Science & Business Media.
- ^ ISBN 978-0-387-90148-0.
- ^ Scorpan, A., 2005. The wild world of 4-manifolds. American Mathematical Soc.
- ^ Freed, D.S. and Uhlenbeck, K.K., 2012. Instantons and four-manifolds (Vol. 1). Springer Science & Business Media.
- ^ Milnor, J., 2016. Morse Theory.(AM-51), Volume 51. Princeton university press.
- ^ Donaldson, S.K., Donaldson, S.K. and Kronheimer, P.B., 1997. The geometry of four-manifolds. Oxford university press.
- ^ Kervaire 1960
- ^ Lashof 1972
- Riemannian metric and are only invariant up to isometry.
References
- Bloch, Ethan D. (1996). A First Course in Geometric Topology and Differential Geometry. Boston: Birkhäuser. ISBN 978-0-8176-3840-5.
- ISBN 978-0-387-90148-0.
- JSTOR 2317423.
- .
External links
- "Differential topology", Encyclopedia of Mathematics, EMS Press, 2001 [1994]