# Classification of smooth atlases

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.

• The question is unclear. As I understand the terms, a differentiable manifold has a unique smooth maximal atlas. The same topological manifold can have more than one differentiable structure, that is, is can be more than differentiable manifold, that is, it can be given more than one maximal smooth atlas (where "smooth atlas" means that any two charts in the atlas are compatible with each other and "maximal" means that every chart compatible with the atlas is in the atlas). – Tom Goodwillie Feb 17 '12 at 16:08
• Your $\mathcal A'$ should not be called a new atlas on the same differentiable manifold, but rather a new atlas on the same topological manifold; or you can say that it makes a new differentiable manifold $M'$ with the same underlying topological manifold as $M$. Are you wondering when $M$ and $M'$ are diffeomorphic? – Tom Goodwillie Feb 17 '12 at 16:11
• Of course, they are in your class of examples ($f$ gives a diffeomorphism between $M$ and $M'$). – Tom Goodwillie Feb 17 '12 at 16:14
• I still do not understand the question. What can you mean by equivalence between two atlases? It appears that you mean compatibility, in the sense that comparing the charts of one atlas and the charts of the other atlas leads to diffeomorphisms between open subsets of $\mathbb R^n$. But then if your atlases are maximal (meaning maximal w.r.t. this compatibility) then they are never equivalent; and if they are not required to be maximal then two atlases are equivalent if and only if they both determine the same maximal atlas. Am I missing the point? – Tom Goodwillie Feb 17 '12 at 16:52
• @Tom Goodwillie: You are right, any equivalence class of atlases contains a maximum atlas, which is the union of all other atlases in the equivalence class. And two maximal atlases are either identical, or not equivalent. So we can restate my questions in terms of maximal atlases instead of the equivalence classes of atlases. The questions remain. – Cristi Stoica Feb 17 '12 at 17:14

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.
• Thank you for your answer. It is correct, so I up-vote it, but it is not what I asked. I did not ask for the equivalence relation on the set of atlases on $M$, but for the equivalence relation on the set of atlases on $M$, under very specific constraints, which I described in the question, and which are in fact the research-related part of the question. One condition is that the atlases coincide on a subset of $M$, and the other is that a given function or tensor field is smooth in these atlases. Thanks. – Cristi Stoica Mar 7 '12 at 6:04