I read that every topological manifold in dimension 1,2 or 3 has a unique differential structure (up to diffeomorphism).
However, I can give $\mathbb{R}$ two different atlases as in Is the maximal atlas for a topological manifold unique? like $(\mathbb{R},x)$ and $(\mathbb{R},x^3)$, which both cover $\mathbb{R}$ but are not compatible at $0$ (i.e the transition function from the second to the first is not $C^\infty$ in $0$).
What am I missing here? I believe this question has already been asked in the past, for instance here Number of Differentiable Structures on a Smooth Manifold but I fail to understand what "unique up to diffeomorphism" means in this context.
Thanks
Best Answer
One says that a smooth structure on a topological manifold $X$ is unique up to a diffeomorphism if the following holds:
For any two differentiable structures $S_1, S_2$ on $X$, there exists a diffeomorphism $(X,S_1) \to (X,S_2)$.
Equivalently, one can say that for any two smooth manifolds $M_1, M_2$ homeomorphic to $X$, the manifolds $M_1, M_2$ are diffeomorphic to each other.
I prefer the second formulation since it does not tempt anybody to assume that the diffeomorphism is the identity map. (Since $M_1, M_2$ are not assumed to be the same as sets.)
In the specific example of $X={\mathbb R}$ and $S_1=({\mathbb R}, x)$, $S_2=({\mathbb R}, x^3)$, the "unique up to a diffeomorphism" means "there exists a diffeomorphism $M_1\to M_2$," where $M_i=(X, S_i)$, $i=1,2$. The diffeomorphism in this particular example is given by the map $X\to X$, $$ x\mapsto x^{1/3}. $$