How to choose a standard smooth structure for a manifold?

Question

Given any smooth manifold $(M,\mathscr{A})$ with a specified smooth structure $\mathscr{A}$, we can identify uncountably many distinct smooth structures $(\mathscr{B}_s)_{s \ge 0}$ such that $(M,\mathscr{B}_s)$ is also a smooth manifold.

So how do we go about choosing a standard smooth structure to work with and do calculations with? How can we justify any of the uncountably many choices as the "best" one?

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Context: My confusion is a result of solving the following problem (i.e. my solution raises more questions than it answers) from Lee's Introduction to Smooth Manifolds, 1-6 on p.30:

Let $M$ be a nonempty topological manifold of dimension $n \ge 1$. If $M$ has a smooth structure, show that it has uncountably many distinct ones. [Hint: first show that for any $s>0$, $F_s(x)=|x|^{s-1}x$ defines a homeomorphism from $\mathbb{B}^n$ to itself, which is a diffeomorphism if and only if $s=1$.]

This answer and this question both seem to imply that all of these smooth structures should be diffeomorphic, but if they were diffeomorphic, then wouldn't the smoothness of transition maps between coordinate charts imply that they are equal (since any smoothness structure is a maximal atlas)? Anyway, even if they were all diffeomorphic, that still doesn't resolve the issue of which one should be the "standard" one and which one to choose for calculations, etc.

For example, on p. 40 of this same book, it says that:

...$\mathbb{R}^4$ has uncountably many distinct smooth structures, no two of which are diffeomorphic to each other! The existence of nonstandard smooth structures on $\mathbb{R}^4$ (called fake $\mathbb{R}^4$'s) was first proved by Simon Donaldson and Michael Freedman in 1984 as a consequence of their work on the geometry and topology of compact 4-manifolds...

So when working with $\mathbb{R}^4$, how does one decide which smooth structure is the "standard" smooth structure, and how can someone verify that they are working with the correct smooth structure?

My attempt: Does the answer have to do with the fact that any topological manifold already comes pre-equipped with a collection of charts which are appropriate homeomorphisms but not necessarily smoothly compatible with each other, and thus any smooth structure is a strict subset of the family of charts inherited from the topological manifold structure? Such that any two distinct smooth structures are both strict subsets of the family of charts from the underlying topological manifold? So they are necessarily equivalent up to homeomorphism? I still don't see how to show that they are equivalent up to diffeomorphism. Also the fact that any smooth structure on a set induces a topological manifold structure on the same set would seem to suggest that any smooth atlas is not strictly contained in the family of charts resulting only from the topology of the underlying space, although I am not sure either way.

Answer 1

The point of this problem is to help the reader understand the difference between two distinct concepts. Suppose $M$ is a topological manifold, and $\mathscr A_1$ and $\mathscr A_2$ are two different smooth structures on $M$ (i.e., maximal smoothly compatible atlases). Then we can ask two questions about $\scr A_1$ and $\scr A_2$:

1. What does it mean for $\scr A_1$ and $\scr A_2$ to be the same smooth structure on $M$?
2. What does it mean for the smooth manifolds $(M,\scr A_1)$ and $(M,\scr A_2)$ to be diffeomorphic to each other?

In question 1, we are given two different atlases on $M$, and the question is whether each chart of $\scr A_1$ is smoothly compatible with each chart in $\scr A_2$ and vice versa. The problem you quoted (Problem 1-6) asks you to construct uncountably many smooth structures on a given manifold that are distinct, in the sense that the charts of one are not smoothly compatible with the charts of another. This is possible even on $\mathbb R$; indeed, as the problem states, it is possible on any positive-dimensional topological manifold as long as it admits at least one smooth structure.

The second question (whether two given smooth structures on $M$ result in smooth manifolds that are diffeomorphic to each other) is a completely different question. The result that @levap refers to about $\mathbb R$ (Problem 15-13 in my book) says that if $\scr A_1$ and $\scr A_2$ are two smooth structures on $\mathbb R$, then there is a map $F\colon \mathbb R\to\mathbb R$ that is a diffeomorphism from $(\mathbb R,\scr A_1)$ to $(\mathbb R,\scr A_2)$. Another way of saying this is that given a smooth chart $(U,\phi)\in\scr A_2$, the chart $(F^{-1}(U),\phi\circ F)$ will be a chart for $\mathbb R$ that is compatible with all the charts in $\mathscr A_1$ (and thus, by maximality, is already in $\mathscr A_1$). It doesn't say that every chart in $\mathscr A_2$ is already smoothly compatible with those in $\mathscr A_1$. And it does not contradict the fact that there are many distinct smooth structures on $\mathbb R$; it just says that any two of them are related to each other by such a map.