smooth structure


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Manifolds and cobordisms



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.



(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.



(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).


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.


  • 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 (