Difference between smooth structures being equivalent or diffeomorphic.

Question

I have just begun a course in Differential Topology and am being introduced to topological and smooth manifolds. Smooth manifolds involve having a smooth structure. And there were further concepts of two smooth structures being equivalent or diffeomorphic of distinct and I'm massively confused by these as I don't really understand the definition of when are two smooth structures equivalent and have been unable to find out definitions of them being distinct or diffeomorphic. Any help in understanding this better would be appreciated. I understand that a smooth structure is a maximal smooth atlas, where a smooth atlas means that any two charts in that atlas are smoothly equivalent.

Answer 1

Double check that my definition agrees with yours: a smooth structure $\mathcal{U}$ on a paracompact, second-countable, Hausdorff topological space $M$ is collection of chart neighborhoods $(U,\phi)$ such that:

• the $U$ cover $M$,
• the charts $\phi: U\to\Bbb{R}^n$ is a topological embedding, and
• for every pair $(U,\phi), (V,\psi)$ the transition map $\phi\psi^{-1}$ is a smooth map with everywhere-invertible differential from $\psi(U\cap V)$ to $\phi(U\cap V)$
• $\mathcal{U}$ is maximal with respect to these conditions, so that given another smooth structure $\mathcal{V}$, if $\mathcal{U}\cup\mathcal{V}$ is a smooth structure, then $\mathcal{V}\subset\mathcal{U}$.

I'm going to guess at the precise definitions of equivalent and diffeomorphic provided by your text.

We might say two smooth structures $\mathcal{U}, \mathcal{V}$ are diffeomorphic provided there exists a homeomorphism $h:M\to M$ such that the pullback of $\mathcal{U}$, defined as the structure $$h^*\mathcal{U} = \{(h^{-1}U, h^*\phi)\ |\ (U,\phi)\in\mathcal{U}\},$$ is equal to $\mathcal{V}$. Note that $h$ need not be smooth with respect to $\mathcal{U}$ or $\mathcal{V}$.

We might say two smooth structures are equivalent provided they are diffeomorphic by the identity map.

Answer 2

Two differentiable manifolds are diffeomorphic to each other if there is a map (called a diffeomorphism) between them that is "smooth" and has an inverse which is "smooth".

As for the definition of "smooth"ness above, assuming your manifolds are embedded in Euclidean spaces: a map $f: M \to \mathbb{R}^n$ is smooth if at every point $x$ of (an $m$-manifold) $M$, there exist a relative neighborhood $N_x \cap M$, where $N_x$ is an open set in $\mathbb{R}^m$, and a differentiable function, say $g$, from $N_x$ to an open set in $\mathbb{R}^n$ such that $g\big|_{N_x \cap M} = f\big|_{N_x \cap M}$.