Classification of smooth atlases

Question

Let $\mathcal{A}$ be a smooth maximal atlas on a manifold $M$. Let $f:M\to M$ be a smooth invertible function, whose inverse is not smooth (for example $f:\mathbb R\to \mathbb R$, $f(x)=x^3$). Then $f$ induces an atlas $\mathcal{A}'$ which is not compatible with $\mathcal{A}$.

Since the relation of compatibility between atlases is an equivalence, the problem of classifying them appears. To simplify, we can try to classify the smooth atlases which are compatible on $M-N$, where $N\subset M$, but are incompatible on $N$. The simplest case seems to be when $N$ contains only a point.

Another problem is to find all maximal atlases for which a given function on $M$, or other object, for example a tensor, is smooth.

Are there any studies of these kinds of classifications of atlases?

---

Update: I don't ask about exotic smooth structures. The compatibility of atlases is "finer", but both are named "smooth structures", which leads to confusions (see http://en.wikipedia.org/wiki/Smooth_structure#Confusion_about_terminology).

---

Update 2 (example): Let's consider the manifold $\mathbb R$ with the maximal atlas $\mathcal A$ generated by $id:\mathbb R\to \mathbb R$. Let the atlas $\mathcal A'$ be generated by $f(x)=x^3$. Then not all functions on $\mathbb R$ which are smooth in one atlas are smooth in the other too.

Now let's consider the set $N$ containing only the origin. When we restrict the charts from $\mathcal A$ and $\mathcal A'$ to $M-N$, we obtain two compatible atlases, although $\mathcal A$ and $\mathcal A'$ are not compatible. This justifies the first problem:

Q1: Find all the maximal atlases on $\mathbb R$ whose restriction on $M-N$ is compatible with $\mathcal A|_{M-N}$.

Since not all functions on $\mathbb R$ which are smooth in one atlas are smooth in the other, the following problem arises:

Q2: Find all the maximal atlases in which a given function is smooth.

Answer 1

To me it looks like you're asking for the equivalence relation on the set of atlases on $M$, where one atlas is considered equivalent to the other if the identity map $Id : M \to M$ (where $Id(x)=x$ always) is a diffeomorphism between $M$ with one atlas and $M$ with the other.

In the case of manifolds that admit only one smooth structure, like $\mathbb R$, the group of homeomorphisms of $M$, $Homeo(M)$ acts transitively on the maximal atlases of $M$. This is basically just a restatement that there is only one smooth structure up to diffeomorphism. The point stabilizers of the action of $Homeo(M)$ on the maximal atlases on $M$ is $Diff(M)$ ($M$ having a particular smooth structure). So the object that classifies your maximal atlases up to the identity map being a diffeomorphism is just

$$Homeo(M) / Diff(M)$$

the above is in the case that $M$ admits precisely one smooth structure up to diffeomorphism. In general, you'd have a disjoint union of these, one for every possible smooth structure on $M$.

$Homeo(M)/Diff(M)$ is always a big, infinite object provided $dim(M)\geq 1$. For example, the points of non-differentiability of a homeomorphism of $M$ is a well-defined invariant of $Homeo(M)/Diff(M)$ which is equipped with a smooth structure. I'm not aware of people that study this object specifically since it's not clear to me what one might want to do with it, but of course there's lots of related things. For example, germs of homeomorphisms are a related topic that has to do with smoothing theory. The classical reference of Kirby and Siebenmann covers some of this topic. The difference in the homotopy-types of $Homeo(M)$ and $Diff(M)$ is also a classical topic of study in manifold theory. For example, there's relatively little known about the homotopy-type of $Diff(S^n)$ when $n \geq 4$. When $n \geq 5$, $\pi_0 Diff(S^n)$ is isomorphic to the group of homotopy $(n+1)$-spheres. But there's not much known about higher homotopy.