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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.

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Maximal = cannot add anything to it while preserving the consistency relations. Re-read the definition of a differentiable structure carefully. –  user31373 Jun 28 '12 at 21:14
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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.

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There are a lot of little things to say here, but I don't want to write a book in this space. Most texts (Lee's Introduction to Smooth Manifolds and Spivak's A Comprehensive Introduction to Differential Geometry, Vol. 1 come to mind) will talk about this in great detail. It is impossible to physically write down all of charts in the resulting atlas, but that's almost the point of the maximality condition. –  Dylan Moreland Jun 28 '12 at 21:34
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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 $\mathcal{A}_2$.

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