Inmathematics,differential topology is the field dealing with thetopological properties andsmooth properties[a] ofsmooth manifolds. In this sense differential topology is distinct from the closely related field ofdifferential geometry, which concerns thegeometric 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, itshomotopy type, or the structure of itsdiffeomorphism group. Because many of these coarser properties may be capturedalgebraically, differential topology has strong links toalgebraic topology.[1]
The central goal of the field of differential topology is theclassification of all smooth manifolds up todiffeomorphism. Since dimension is an invariant of smooth manifolds up to diffeomorphism type, this classification is often studied by classifying the (connected) manifolds in each dimension separately:
Beginning in dimension 4, the classification becomes much more difficult for two reasons.[5][6] Firstly, everyfinitely presented group appears as thefundamental group of some4-manifold, and since thefundamental group is a diffeomorphism invariant, this makes the classification of 4-manifolds at least as difficult as the classification of finitely presented groups. By theword problem for groups, which is equivalent to thehalting 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-diffeomorphicsmooth structures. This is true even for the Euclidean space, which admits manyexotic 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 oftopological manifolds. One of the central open problems in differential topology is thefour-dimensional smooth Poincaré conjecture, which asks if every smooth 4-manifold that is homeomorphic to the4-sphere, is also diffeomorphic to it. That is, does the 4-sphere admit only onesmooth 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 theMilnor spheres.
Important tools in studying the differential topology of smooth manifolds include the construction of smoothtopological invariants of such manifolds, such asde Rham cohomology or theintersection form, as well as smoothable topological constructions, such as smoothsurgery theory or the construction ofcobordisms.Morse theory is an important tool which studies smooth manifolds by considering thecritical points ofdifferentiable functions on the manifold, demonstrating how the smooth structure of the manifold enters into the set of tools available.[7] Oftentimes more geometric or analytical techniques may be used, by equipping a smooth manifold with aRiemannian metric or by studying adifferential 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, theHodge theorem provides a geometric and analytical interpretation of the de Rham cohomology, andgauge theory was used bySimon Donaldson to prove facts about the intersection form of simply connected 4-manifolds.[8] In some cases techniques from contemporaryphysics may appear, such astopological quantum field theory, which can be used to compute topological invariants of smooth spaces.
Famous theorems in differential topology include theWhitney embedding theorem, thehairy ball theorem, theHopf theorem, thePoincaré–Hopf theorem,Donaldson's theorem, and thePoincaré conjecture.
Differential topology considers the properties and structures that require only asmooth structure 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 anddeformations that exist in differential topology. For instance, volume andRiemannian curvature areinvariants 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.[citation needed]
On the other hand, smooth manifolds are more rigid than thetopological manifolds.John Milnor discovered that some spheres have more than one smooth structure—seeExotic sphere andDonaldson's theorem.Michel Kervaire exhibited topological manifolds with no smooth structure at all.[9] Some constructions of smooth manifold theory, such as the existence oftangent 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, namelyimmersions andsubmersions, and the intersections of submanifolds viatransversality. More generally one is interested in properties and invariants of smooth manifolds that are carried over bydiffeomorphisms, 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 therank of theJacobian of a function.
For a list of differential topology topics, see the following reference:List of differential geometry topics.
Differential topology and differential geometry are first characterized by theirsimilarity. 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 areinherently 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 arethe 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 aredifferent 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 adiffeomorphism between two manifolds of the same dimension is inherently global sincelocally 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 betrivial 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,symplectic topology—a subbranch of differential topology—studies global properties ofsymplectic manifolds. Differential geometry concerns itself with problems—which may be localor global—that always have some non-trivial local properties. Thus differential geometry may study differentiable manifolds equipped with aconnection, ametric (which may beRiemannian,pseudo-Riemannian, orFinsler), a special sort ofdistribution (such as aCR 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 thetangent space at a point. Differential topology also deals with questions like these, which specifically pertain to the properties of differentiable mappings on (for example thetangent bundle,jet bundles, theWhitney extension theorem, and so forth).
The distinction is concise in abstract terms:
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