nLab smooth structure

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

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.

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