Editing Differential topology (section)

{{Short description|Branch of mathematics}}
{{Use American English|date = March 2019}}
In [[mathematics]], '''differential topology''' is the field dealing with the [[topology|topological properties]] and [[smooth structure|smooth properties]]{{efn|A ''smooth property'' of a manifold is any property preserved up to [[diffeomorphism]]. This does not include certain [[geometry|geometric]] properties such as distances between points or volume, which depend on a further choice of [[Riemannian metric]] and are only invariant up to [[isometry]].}} of [[smooth manifold]]s. In this sense differential topology is distinct from the closely related field of [[differential geometry]], which concerns the ''geometric'' properties of smooth manifolds, including notions of size, distance, and rigid shape. By comparison differential topology is concerned with coarser properties, such as the number of holes in a manifold, its [[homotopy type]], or the structure of its [[diffeomorphism group]]. Because many of these coarser properties may be captured [[abstract algebra|algebraically]], differential topology has strong links to [[algebraic topology]].<ref>[[Raoul Bott|Bott, R.]] and Tu, L.W., 1982. Differential forms in algebraic topology (Vol. 82, pp. xiv+-331). New York: Springer.</ref>
[[Image:3D-Cylinder with handle and torus with hole.png|thumb|right|The [[Morse theory]] of the height function on a [[torus]] can describe its [[homotopy type]].]]
The central goal of the field of differential topology is the [[classification theorem|classification]] of all smooth manifolds up to [[diffeomorphism]]. Since dimension is an invariant of smooth manifolds up to diffeomorphism type, this classification is often studied by classifying the ([[connected (topology)|connected]]) manifolds in each dimension separately:

* In dimension 1, the only smooth manifolds up to diffeomorphism are the [[circle]], the [[real number line]], and allowing a [[boundary (topology)|boundary]], the half-closed [[interval (mathematics)|interval]] <math>[0,1)</math> and fully closed interval <math>[0,1]</math>.<ref name="milnor">Milnor, J. and Weaver, D.W., 1997. Topology from the differentiable viewpoint. Princeton university press.</ref>
* In dimension 2, every [[closed surface]] is classified up to diffeomorphism by its [[genus (topology)|genus]], the number of holes (or equivalently its [[Euler characteristic]]), and whether or not it is [[orientable]]. This is the famous [[Surface (topology)#Classification of closed surfaces|classification of closed surfaces]].<ref name="leetopological">Lee, J., 2010. Introduction to topological manifolds (Vol. 202). Springer Science & Business Media.</ref><ref name="hirsch">{{cite book |title = Differential Topology|author-link=Morris Hirsch|first = Morris|last = Hirsch|publisher=Springer-Verlag|year=1997|isbn = 978-0-387-90148-0}}</ref> Already in dimension two the classification of non-[[compact space|compact]] surfaces becomes difficult, due to the existence of exotic spaces such as [[Jacob's ladder surface|Jacob's ladder]].
* In dimension 3, [[William Thurston]]'s [[geometrization conjecture]], proven by [[Grigori Perelman]], gives a partial classification of compact three-manifolds. Included in this theorem is the [[Poincaré conjecture]], which states that any closed, [[simply connected]] three-manifold is [[homeomorphic]] (and in fact diffeomorphic) to the [[3-sphere]].

[[File:Cobordism.svg|thumb|A [[cobordism]] (''W''; ''M'', ''N''), which generalises the notion of a diffeomorphism.]]

Beginning in dimension 4, the classification becomes much more difficult for two reasons.<ref>Scorpan, A., 2005. The wild world of 4-manifolds. American Mathematical Soc.</ref><ref>Freed, D.S. and Uhlenbeck, K.K., 2012. Instantons and four-manifolds (Vol. 1). Springer Science & Business Media.</ref> Firstly, every [[finitely presented group|finitely presented]] group appears as the [[fundamental group]] of some [[4-manifold]], and since the [[fundamental group]] is a diffeomorphism invariant, this makes the classification of 4-manifolds at least as difficult as the classification of finitely presented groups. By the [[word problem for groups]], which is equivalent to the [[halting problem]], it is impossible to classify such groups, so a full topological classification is impossible. Secondly, beginning in dimension four it is possible to have smooth manifolds that are homeomorphic, but with distinct, non-diffeomorphic [[smooth structure]]s. This is true even for the Euclidean space <math>\mathbb{R}^4</math>, which admits many [[exotic R4|exotic <math>\mathbb{R}^4</math>]] structures. This means that the study of differential topology in dimensions 4 and higher must use tools genuinely outside the realm of the regular continuous topology of [[topological manifold]]s. One of the central open problems in differential topology is the [[generalized Poincaré conjecture|four-dimensional smooth Poincaré conjecture]], which asks if every smooth 4-manifold that is homeomorphic to the [[4-sphere]], is also diffeomorphic to it. That is, does the 4-sphere admit only one [[exotic sphere|smooth structure]]? This conjecture is true in dimensions 1, 2, and 3, by the above classification results, but is known to be false in dimension 7 due to the [[Milnor sphere]]s.

Important tools in studying the differential topology of smooth manifolds include the construction of smooth [[topological invariant]]s of such manifolds, such as [[de Rham cohomology]] or the [[intersection form (4-manifold)|intersection form]], as well as smoothable topological constructions, such as smooth [[surgery theory]] or the construction of [[cobordism]]s. [[Morse theory]] is an important tool which studies smooth manifolds by considering the [[critical point (mathematics)|critical point]]s of [[differentiable function]]s on the manifold, demonstrating how the smooth structure of the manifold enters into the set of tools available.<ref>Milnor, J., 2016. Morse Theory.(AM-51), Volume 51. Princeton university press.</ref> Oftentimes more geometric or analytical techniques may be used, by equipping a smooth manifold with a [[Riemannian metric]] or by studying a [[differential equation]] on it. Care must be taken to ensure that the resulting information is insensitive to this choice of extra structure, and so genuinely reflects only the topological properties of the underlying smooth manifold. For example, the [[Hodge theory|Hodge theorem]] provides a geometric and analytical interpretation of the de Rham cohomology, and [[gauge theory (mathematics)|gauge theory]] was used by [[Simon Donaldson]] to prove facts about the intersection form of simply connected 4-manifolds.<ref>Donaldson, S.K., Donaldson, S.K. and Kronheimer, P.B., 1997. The geometry of four-manifolds. Oxford university press.</ref> In some cases techniques from contemporary [[physics]] may appear, such as [[topological quantum field theory]], which can be used to compute topological invariants of smooth spaces.

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]].