I'm following a course on differential geometry and from some background I've gotten used to the fact that a topological manifold $M$ is called a smooth/differentiable manifold if we can equip $M$ with some smooth atlas $\mathcal{A}$, then the pair $(M, \mathcal{A})$ is called a differentiable manifold.
Now reading Lee's book I've found out that we actually require $\mathcal{A}$ to be something called maximal. He first states that
Our plan is to define a “smooth structure” on $M$ by giving a smooth atlas, and to
define a function $f : M \to \Bbb R$ to be smooth if and only if $f \circ \varphi^{-1}$ is smooth in the sense of ordinary calculus for each coordinate chart $(U, \varphi)$ in the atlas. There is one minor technical problem with this approach: in general, there will be many possible atlases that give the “same” smooth structure, in that they all determine the same collection of smooth functions on $M$.
Then he goes on to state that
However, it is more straightforward to make the following definition: a smooth atlas $\mathcal{A}$ on $M$ is maximal if it is not properly contained in any larger smooth atlas. This just means that any chart that is smoothly compatible with every chart in $\mathcal{A}$ is already in $\mathcal{A}$.
The lecturer in the course I'm taking says that in practice we only need to consider some atlas $\mathcal{A}$ for $M$ instead of the maximal one which is causing my confusion. Why is this true?
Lee also gives the proposition $1.17$ which states that
Let $M$ be a topological manifold, then every smooth atlas $\mathcal{A}$ for $M$ is contained in a unique maximal smooth atlas, called the smooth structure determined by $\mathcal{A}.$
and presumably this is the reason why we can consider some arbitary atlas $\mathcal{A}$ instead of the maximal one?
Why does this imply that we don't need to consider the maximal one? I don't think it's obvious.
Best Answer
A smooth manifold structure on a topological manifold $M$ is a choice of maximal atlas for $M$.
Each atlas $\mathcal{A}$ is contained in a unique maximal atlas consisting of the charts on $M$ that are compatible with $\mathcal{A}$. So it is clear that specifying any atlas automatically determines a unique choice of maximal atlas, hence a smooth manifold structure.
Two atlases $\mathcal{A}_1$ and $\mathcal{A}_2$ are said to be equivalent if $\mathcal{A}_1\cup\mathcal{A}_2$ is itself an atlas. This defines an equivalence relation. Each equivalence class contains a unique maximal atlas, and therefore defines a smooth manifold structure.
So to define a smooth manifold structure on $M$, you can either specify a maximal atlas directly, or specify an equivalence class of atlases, or just any atlas.
Reference: Lee, J.M., Manifolds and Differential Geometry (Indian Edition), Amer. Math. Soc., 2009