What's meant by the number of "distinct $C^k$ differential structures" other than the amount of distinct maximal atlases?

Question

When reading the Wiki page on differential structures, I'm struck by the exceptional case of $R = 4$.

However, the definition of differential structure leaves me nonplussed, as it seems to just be another name for "maximal atlas" in the context of https://i.stack.imgur.com/MBce6.png, which says: "There is an "essentially unique" smooth structure for any topological manifold of dimension smaller than 4."

When they say "smooth structure" do they simply mean a unique maximal atlas (C-infinity) or is there more to it?

Thanks so much for helping with the clarification.

Answer 1

You are on the right track, but not quite there yet.

First, a smooth structure is indeed nothing more than a maximal $C^\infty$ atlas.

However, to say that the smooth structure is "essentially unique" does not mean that there exists a unique maximal $C^\infty$ atlas. Such a strong uniqueness property is always false: for any smooth manifold $M$ there exists a non-smooth homeomorphism $h : M \to M$, and using $h$ you can then define a different maximal $C^\infty$ atlas by transport of structure, which means replacing your given atlas $\{\phi_i : U_i \to \mathbb R^m\}$ with the new atlas $\{\phi_i \circ h^{-1} : h(U_i) \to \mathbb R^m \}$. For example, even the standard structure on $\mathbb R$ can be changed by transport of structure using the non-smooth homeomorphism $h(x) = \sqrt[3]{x}$.

Instead, for the smooth structure to be "essentially unique" usually means that it is unique up to transport of structure: for any two maximal $C^\infty$ atlases $\{\phi_i : U_i \to \mathbb R^m\}$ and $\{\psi_j : V_j \to \mathbb R^m\}$ there exists a homeomorphism $h : M \to M$, and a bijection between the index sets, such that if $i \leftrightarrow j$ correspond under that bijection then $V_j = h(U_i)$ and $\psi_j = \phi_i \circ h^{-1}$. This, of course, is equivalent to the statement that $h$ is a diffeomorphism from the structure $\{\phi_i : U_ \to \mathbb R^m\}$ to the structure $\{\psi_j : V_j \to \mathbb R^m\}$.

Answer 2

Yes, a $C^k$ (/differential/smooth) structure on a manifold $M$ is exactly a maximal atlas $\mathscr{A}$ with all transition functions of class $C^k$.

A caveat: Exotic $C^k$ structures on a manifold $M$ are not merely maximal atlases which are not the same. One typically says that two $C^k$ structures (i.e. atlases) $\mathscr{A}$ and $\mathscr{A}'$ on $M$ are equivalent (or diffeomorphic) of there is a function $f:M\to M$ such that, with respect to the two structures, $f:(M,\mathscr{A})\to(M,\mathscr{A}')$ is a $C^k$ diffeomorphism.

For instance, $\mathbb{R}^n$ has infinitely many smooth structures, but all of them are equivalent for $n\neq 4$. This is what is meant by "essentially unique" in the linked snippet.