Smooth structure

In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows mathematical analysis to be performed on the manifold.^[1]

Definition

A smooth structure on a manifold $M$ is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold $M$ is an atlas for $M$ such that each transition function is a smooth map, and two smooth atlases for $M$ are smoothly equivalent provided their union is again a smooth atlas for $M .$ This gives a natural equivalence relation on the set of smooth atlases.

A smooth manifold is a topological manifold $M$ together with a smooth structure on $M .$

Maximal smooth atlases

By taking the union of all atlases belonging to a smooth structure, we obtain a maximal smooth atlas. This atlas contains every chart that is compatible with the smooth structure. There is a natural one-to-one correspondence between smooth structures and maximal smooth atlases. Thus, we may regard a smooth structure as a maximal smooth atlas and vice versa.

In general, computations with the maximal atlas of a manifold are rather unwieldy. For most applications, it suffices to choose a smaller atlas. For example, if the manifold is compact, then one can find an atlas with only finitely many charts.

Equivalence of smooth structures

If $μ$ and $ν$ are two maximal smooth atlases on $M$ then these two maximal smooth atlases $μ$ and $ν$ are said to be equivalent if there is a bijection $f : M \to M$ satisfying the property that for all $p \in M$ there exists some $(U, φ) \in μ$ with $p \in U$ and there exists some $(V, ψ) \in ν$ with $f (U) \subseteq V$ such that the transition function $ψ \circ f \circ φ^{- 1} : φ (U \cap V) \to ψ (U \cap V)$ is a smooth mapping, and satisfying the property that for all $q \in M$ there exists some $(W, ϕ) \in ν$ with $q \in W$ as well as some $(Z, χ) \in μ$ with $f^{- 1} (W) \subseteq Z$ such that the transition function $χ \circ f^{- 1} \circ ϕ^{- 1} : ϕ (W \cap Z) \to χ (W \cap Z)$ is a smooth mapping; this is the sense in which the bijection $f : M \to M$ is required to be a diffeomorphism; the function $f : M \to M$ and its inverse $f^{- 1} : M \to M$ must be smooth relative to both maximal smooth atlases on $M$ . ^{[citation needed]}

Exotic spheres

John Milnor showed in 1956 that the 7-dimensional sphere admits a smooth structure that is not equivalent to the standard smooth structure. A sphere equipped with a nonstandard smooth structure is called an exotic sphere.

E8 manifold

The E8 manifold is an example of a topological manifold that does not admit a smooth structure. This essentially demonstrates that Rokhlin's theorem holds only for smooth structures, and not topological manifolds in general.

Related structures

The smoothness requirements on the transition functions can be weakened, so that the transition maps are only required to be $k$ -times continuously differentiable; or strengthened, so that the transition maps are required to be real-analytic. Accordingly, this gives a $C^{k}$ or (real-)analytic structure on the manifold rather than a smooth one. Similarly, a complex structure can be defined by requiring the transition maps to be holomorphic.

References

↑ Callahan, James J. (1974). "Singularities and plane maps". Amer. Math. Monthly. 81 (3): 211–240. doi:10.2307/2319521. JSTOR 2319521.

Hirsch, Morris (1976). Differential Topology. Springer-Verlag. ISBN 3-540-90148-5.
Lee, John M. (2006). Introduction to Smooth Manifolds. Springer-Verlag. ISBN 978-0-387-95448-6.
Sepanski, Mark R. (2007). Compact Lie Groups. Springer-Verlag. ISBN 978-0-387-30263-8.

[1] Callahan, James J. (1974). "Singularities and plane maps". Amer. Math. Monthly. 81 (3): 211–240. doi:10.2307/2319521. JSTOR 2319521.

[1]