Differentiable structure on the real line

Question

The usual differentiable structure on real line was obtained by taking ${F}$ to be the maximal collection containing the identity map, Let ${F_1}$ be the maximal collection containing $t\mapsto t^3$. I need to show $F_1\neq F$, but $(\mathbb{R},F)$ and $(\mathbb{R},F_1)$ are diffeomorphic.

Well, first of all, what does it mean by maximal collection? I am not clearly getting and how to define a diffeomorphism between these ordered pairs, that also not clear to me. Please help.

Answer 1

The maximality of $F$ means that if $(U, \varphi)$ [here $U$ is an open subset of our manifold; in this case, of $\mathbb R$] is another chart which is compatible with each $(V, \psi) \in F$ — in the sense that each transition map \[ \varphi^{-1} \circ \psi\colon \psi(U \cap V) \to \varphi(U \cap V) \] is a diffeomorphism in the sense of calculus on $\mathbb R$ — then $\varphi$ is actually in $F$.

So take $U = \mathbb R$ and $\varphi(t) = t^3$. Is this compatible with $F$? For the second part, note that it follows from all of these definitions that if a function $f\colon (\mathbb R, F) \to (\mathbb R, F_1)$ is a diffeomorphism with respect a choice of global coordinates on both sides, then it is a diffeomorphism of manifolds.

Answer 2

What is a "maximal collection"?

A differentiable structure is a collection of open neighborhoods and chart maps which satisfies a series of conditions:

• Open neighborhoods cover the manifold
• Open neighborhoods are homeomorphic to the model space $\mathbb{R}^n$
• On intersections of open neighborhoods, transition maps are smooth

To require maximality of a differentiable structure $\mathcal{A}$ is to require that if $\mathcal{A}$ is contained in any other differentiable structure $\mathcal{A}'$ which also satisfies these properties, then $\mathcal{A}' = \mathcal{A}$.

How to define a diffeomorphism between smooth manifolds?

A diffeomorphism between smooth manifolds is a homeomorphism that is smooth and smoothly invertible in every coordinate chart. Just as a homeomorphism defines an equivalence of topology between topological spaces, a diffeomorphism defines an equivalence of smooth structures.

So it looks like your job is to write down the maximal smooth structure $\mathcal{A}_1$ containing the identity map as a chart map, the maximal smooth structure $\mathcal{A}_2$ containing $t\mapsto t^3$, and then a self-homeomorphism of $\mathbb{R}$ (the underlying topological space) which is smooth and smoothly invertible in each coordinate chart in $\mathcal{A}_1$ and $\mathcal{A}_2$.