nLab
smooth structure

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\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)

Manifolds and cobordisms

Contents

Idea

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

Definition

Definition

(smooth structure)

Let XX be a topological manifold and let

( nϕ iU iX) iIAAAandAAA( nψ jV jX) jJ \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 XX into a smooth manifold (this def.).

Then there is a diffeomorphism of the form

f:(X,( nϕ iU iX) iI)(X,( nψ jV jX) jJ) 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 XX constitutes such a diffeomorphism. (Because if ff is a diffeomorphism, then also f 1f=id Xf^{-1}\circ f = id_X is a diffeomorphism.)

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

iIjJ(ϕ i 1(V j)ϕ iV jψ j 1 nAAis smooth). \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 XX. An equivalence class with respect to this equivalence relation is called a smooth structure on XX.

Properties

Theorem

(uniqueness of smooth structure on Euclidean space in d4d \neq 4)

For nn \in \mathbb{N} a natural number with n4n \neq 4, there is a unique (up to isomorphism) smooth structure on the Cartesian space n\mathbb{R}^n.

This was shown in (Stallings 62).

Theorem

In d=4d = 4 the analog of this statement is false. One says that on 4\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 n\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 nn-sphere may naturally be thought of as the collection of points

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

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

References

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

See also

Revised on June 21, 2017 16:12:34 by Urs Schreiber (88.77.226.246)