nLab smooth structure

Idea

A differential structure on a topological space $X$ is the extra structure of a differential manifold on $X$. A smooth structure on $X$ is the extra structure of a smooth manifold.

Definition

Definition

(smooth structure)

Let $X$ be a topological manifold and let

$$\left( \mathbb{R}^n \underoverset{\simeq}{\phi_i}{\longrightarrow} U_i \subset X \right)_{i \in I} \phantom{AAA} \text{and} \phantom{AAA} \left( \mathbb{R}^{n} \underoverset{\simeq}{\psi_j}{\longrightarrow} V_j \subset X \right)_{j \in J}$$

be two atlases, both making $X$ into a smooth manifold (this def.).

Then there is a diffeomorphism of the form

$$f \;\colon\; \left( X \;,\; \left( \mathbb{R}^n \underoverset{\simeq}{\phi_i}{\longrightarrow} U_i \subset X \right)_{i \in I} \right) \overset{\simeq}{\longrightarrow} \left( X\;,\; \left( \mathbb{R}^{n} \underoverset{\simeq}{\psi_j}{\longrightarrow} V_j \subset X \right)_{j \in J} \right)$$

precisely if the identity function on the underlying set of $X$ constitutes such a diffeomorphism. (Because if $f$ is a diffeomorphism, then also $f^{-1}\circ f = id_X$ is a diffeomorphism.)

That the identity function is a diffeomorphism between $X$ equipped with these two atlases means (by definition) that

$$\underset{{i \in I} \atop {j \in J}}{\forall} \left( \phi_i^{-1}(V_j) \overset{\phi_i}{\longrightarrow} V_j \overset{\psi_j^{-1}}{\longrightarrow} \mathbb{R}^n \phantom{AA} \text{is smooth} \right) \,.$$

Hence diffeomorphsm induces an equivalence relation on the set of smooth atlases that exist on a given topological manifold $X$. An equivalence class with respect to this equivalence relation is called a smooth structure on $X$.

Properties

Theorem

(uniqueness of smooth structure on Euclidean space in $d \neq 4$)

For $n \in \mathbb{N}$ a natural number with $n \neq 4$, there is a unique (up to isomorphism) smooth structure on the Cartesian space $\mathbb{R}^n$.

This was shown in (Stallings 62).

Theorem

In $d = 4$ the analog of this statement is false. One says that on $\mathbb{R}^4$ there exist exotic smooth structures.

Exotic smooth structures

Many topological spaces have canonical or “obvious” smooth structures. For instance a Cartesian space $\mathbb{R}^n$ has the evident smooth structure induced from the fact that it can be covered by a single chart – itself.

From this example, various topological spaces inherit a canonical smooth structure by embedding. For instance the $n$-sphere may naturally be thought of as the collection of points

$$S^n \hookrightarrow \mathbb{R}^n$$

given by $S^n = \{\vec x \in \mathbb{R}^n | \sum_i (x^i)^2 = 1\}$ and this induces a smooth structure of $\mathbb{S}^n$.

But there may be other, non-equivalent smooth structures than these canonical ones. These are called exotic smooth structures. See there for more details.

Related concepts

• smooth structure on a topos

References

• John Stallings, The piecewise linear structure of Euclidean space , Proc. Cambridge Philos. Soc. 58 (1962), 481-488. (pdf)

See also

• Wikipedia, Smooth structure