Differential structure

Differential structure

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In mathematics
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...

, an n-dimensional differential structure (or differentiable structure) on a set M makes M into an n-dimensional differential manifold, which is a topological manifold
Topological manifold
In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...

with some additional structure that allows us to do differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

on the manifold. If M is already a topological manifold, we require that the new topology be identical to the existing one.

Definition

For a natural number n and some k which may be a non-negative integer or infinity, an n-dimensional Ck differential structure is defined using a Ck-atlas, which is a set of bijections called charts between a collection of subsets of M (whose union is the whole of M), and a set of open subsets of :

which are Ck-compatible (in the sense defined below):

Each such map provides a way in which certain subsets of the manifold may be viewed as being like open subsets of but the usefulness of this notion depends on to what extent these notions agree when the domains of two such maps overlap.

Consider two charts:

The intersection of the domains of these two functions is:

and is mapped to two images

by the two chart maps.

The transition map between the two charts is the map between the two images of this intersection under the two chart maps.

Two charts are Ck-compatible if

are open, and the transition maps

have continuous derivatives of order k. If k = 0, we only require that the transition maps are continuous, consequently a C0-atlas is simply another way to define a topological manifold. If k = ∞, derivatives of all orders must be continuous. A family of Ck-compatible charts covering the whole manifold is a Ck-atlas defining a Ck differential manifold. Two atlases are Ck-equivalent if the union of their sets of charts forms a Ck-atlas. In particular, a Ck-atlas that is C0-compatible with a C0-atlas that defines a topological manifold is said to determine a Ck differential structure on the topological manifold. The Ck equivalence classes of such atlases are the distinct Ck differential structures of the 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....

. Each distinct differential structure is determined by a unique maximal atlas, which is simply the union of all atlases in the equivalence class.

For simplification of language, without any loss of precision, one might just call a maximal Ck−atlas on a given set a Ck−manifold. This maximal atlas then uniquely determines both the topology and the underlying set, the latter being the union of the domains of all charts, and the former having the set of all these domains as a basis.

Existence and uniqueness theorems

For 0 < k < ∞ and any n−dimensional Ck−manifold, the maximal atlas contains a C−atlas on the same underlying set by a theorem due to Whitney
Hassler Whitney
Hassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...

. However, a given maximal Ck−atlas contains distinct maximal C−atlases whenever n > 0. Anyway, there is a C−diffeomorphism between any two of these distinct C−atlases. Thus there is only one class of pairwise smoothly diffeomorphic smooth, i.e. C−structures in a Ck−manifold. A bit loosely, one might express this by saying that the smooth structure is (essentially) unique. The case for k = 0 is different. Namely, there exist 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 which admit no C1−structure, a result proved by , and later explained in the context of 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...

(compare Hilbert's fifth problem
Hilbert's fifth problem
Hilbert's fifth problem, is the fifth mathematical problem from the problem-list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups. The theory of Lie groups describes continuous symmetry in mathematics; its importance there and in theoretical physics...

).

Smooth structures on a orientable manifold are usually counted modulo orientation-preserving smooth homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

s. There then arises the question whether orientation-reversing diffeomorphisms exist. There is an "essentially unique" smooth structure for any topological manifold of dimension smaller than 4. For compact manifolds of dimension greater than 4, there is a finite number of "smooth types", i.e. equivalence classes of pairwise smoothly diffeomorphic smooth structures. In the case of Rn with n ≠ 4, the number of these types in one, whereas for n = 4, there are uncountably many such types. One refers to these by exotic 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...

.

Differential structures on spheres of dimension from 1 to 20

The following table lists the number of smooth types of the topological m−sphere Sm for the values of the dimension m from 1 up to 20. Spheres with a smooth, i.e. C−differential structure not smoothly diffeomorphic to the usual one are known as 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...

s.
Dimension 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Smooth types 1 1 1 ? 1 1 28 2 8 6 992 1 3 2 16256 2 16 16 523264 24

It is not currently known how many smooth types the topological 4-sphere S4 has, except that there is at least one. There may be one, a finite number, or an infinite number. The claim that there is just one is known as the smooth Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...

(see 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...

). Most mathematicians believe that this conjecture is false, i.e. that S4 has more than one smooth type. The problem is connected with the existence of more than one smooth type of the topological 4-disk (or 4-ball).

Differential structures on topological manifolds

As mentioned above, in dimensions smaller than 4, there is only one differential structure for each topological manifold. That was proved by Johann Radon
Johann Radon
Johann Karl August Radon was an Austrian mathematician. His doctoral dissertation was on calculus of variations .- Life :...

for dimension 1 and 2, and by Edwin E. Moise
Edwin E. Moise
Edwin Evariste Moise was an American mathematician and mathematics education reformer. After his retirement from mathematics he became a literary critic of 19th century English poetry and had several notes published in that field.-Early life and education:...

in dimension 3. By using Obstruction theory
Obstruction theory
In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.-In homotopy theory:...

, Robion Kirby
Robion Kirby
Robion Cromwell Kirby is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology...

and Laurent Siebenmann were able to show that the number of PL structures for compact topological manifolds of dimension greater than 4 is finite. 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...

, Michel Kervaire
Michel Kervaire
Michel André Kervaire was a French mathematician who made significant contributions to topology and algebra. He was the first to show the existence of topological n-manifolds with no differentiable structure , and computed the number of exotic spheres in dimensions greater than four...

, and Morris Hirsch
Morris Hirsch
Morris William Hirsch is an American mathematician, formerly at the University of California, Berkeley.A native of Chicago, Illinois, Hirsch attained his doctorate from the University of Chicago in 1958, under supervision of Edwin Spanier and Stephen Smale. His thesis was entitled Immersions of...

proved that the number of smooth structures on a compact PL manifold is finite and agrees with the number of differential structures on the sphere for the same dimension (see the book Asselmeyer-Maluga, Brans chapter 7) By combining these results, the number of smooth structures on a compact topological manifold of dimension not equal to 4 is finite.

Dimension 4
4-manifold
In 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...

is more complicated. For compact manifolds, results depend on the complexity of the manifold as measured by the second Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....

. For large Betti numbers in a simply connected 4-manifold, one can use a surgery along a knot or link to produce a new differential structure. With the help of this procedure one can produce countably infinite many differential structures. But even for simple spaces like one doesn't know the construction of other differential structures. For non-compact 4-manifolds there are many examples like having uncountably many differential structures.

In 2010, Akhmedov and Park constructed infinitely many non-diffeomorphic smooth structures on .