smooth structure in nLab
Definition
(smooth structure)
Let XX be a topological manifold and let
(ℝ n⟶≃ϕ iU i⊂X) i∈IAAAandAAA(ℝ n⟶≃ψ jV j⊂X) j∈J \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,(ℝ n⟶≃ϕ iU i⊂X) i∈I)⟶≃(X,(ℝ n⟶≃ψ jV j⊂X) j∈J) 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 −1∘f=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
∀i∈Ij∈J(ϕ i −1(V j)⟶ϕ iV j⟶ψ j −1ℝ 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.
Many topological spaces have canonical or “obvious” smooth structures. For instance a Cartesian space ℝ 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
given by S n={x→∈ℝ 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.