Problem 1-7 (d) in John M. Lee's Introduction To Smooth Manifolds asks to
… verify that the atlas consisting of two charts $(\mathbb{S}^n \setminus \{N\}, \sigma)$ and $(\mathbb{S}^n \setminus \{S\}, \tilde{\sigma})$ defines a smooth structure on $\mathbb{S}^n$.
, where $N = (0,\ldots,0,1) \in \mathbb{R}^{n+1}$ and $S = (0,\ldots,0,-1) \in \mathbb{R}^{n+1}$, $\sigma$ is the stereographic projection from $N$ and $\tilde{\sigma}$ is the stereographic projection from $S$. Lee defines a smooth structure to be a maximal smooth atlas on a topological manifold. I have managed to prove that $\sigma$ and $\tilde{\sigma}$ are bijective, that the transition map $\tilde{\sigma} \circ \sigma^{-1}$ is a diffeomorphism, and hence the charts $(\mathbb{S}^n \setminus \{N\}, \sigma)$ and $(\mathbb{S}^n \setminus \{S\}, \tilde{\sigma})$ are smoothly compatible. Thus, the atlas consisting of these two charts is smooth.
I have struggled to prove that these two charts form a maximal smooth atlas on $\mathbb{S}^n$. After some frustration, I checked this site and found that this smooth atlas is not actually maximal, according to the following post:
How to show this atlas is maximal on the sphere $S^n$?
So is the term smooth structure in the question a typo? Should it have instead asked just to prove that the two charts form a smooth atlas on $\mathbb{S}^n$?
Thanks.
Best Answer
As you might recall, every smooth atlas $\mathcal{A}$ for a topological manifold $M$ is contained in a $\textbf{unique maximal smooth atlas}$ ($\textit{Proposition 1.17}$ ). So by showing that $\mathcal{A} = \{(\Bbb{S}^n \smallsetminus N,\sigma), (\Bbb{S}^n\smallsetminus S,\tilde{\sigma})\}$ is a smooth atlas for $\Bbb{S}^n$, you automatically have a unique smooth structure determined by $\mathcal{A}$. Put it another way, $\mathcal{A}$ defines a smooth structure on $\Bbb{S}^n$. So it's not a typo.