Exotic Smoothness II: What is so Exotic?

In the previous post I tried to give a brief overview of what a smooth manifold is, and why we need to use them for physics. During the construction of a manifold, some choices were made, and in this post I will discuss how these choices might not be unique, and why we should care. By way of review, a graphical representation of the process of constructing manifolds is given by the following illustration:

Hierarchy_v2

Equivalent Atlases

Recall that an atlas on a manifold is a collection of sets U_i and charts \phi_i:U_i\to U with some restrictions on the kind of functions we can have as overlaps \phi_{ij}=\phi_i\circ \phi_j^{-1}\to U_i\cup U_j. If the overlap functions are at least continuous, we have a topological manifold, and if they are smooth we have a smooth manifold. We can actually have anything in between (or other possibilities, like piecewise linear manifolds), but these two categories are sufficient for us to understand. The space U is a subset in some vector space upon which the manifold is modeled; in physics this is often \mathbb{R}^n or \mathbb{C}^n. Banach spaces can also be considered, as long as one keeps in mind that infinite-dimensional spaces have a few quirks that you must keep track of. A nice text which uses Banach spaces wherever possible is  Foundations of Differential Geometry by Lang.

So, how do we know that our choice of atlas \{(U_i,\phi_i)\} is unique?

Definition: Two atlases \{(U_i,\phi_i)\}, \{(V_i,\psi_i)\} on a smooth manifold are compatible if their union is again an atlas. This compatibility is an equivalence relation, and an equivalence class of an atlas is called a differentiable structure. 

Ok, so let’s say we have all the possible sets \mathcal{U}_i on the manifold, and all the possible charts on these sets \Phi_i. We start to group them into atlases

(\mathcal{U}_1,...,\mathcal{U}_j,\Phi_1,...,\Phi_j),~(\mathcal{U}_{j+1},...,\mathcal{U}_k,\Phi_{j+1},...,\Phi_{k}),...

with smooth overlap functions. Now we check to see if they are unique, which means taking a union of all pairs to see if they form another valid atlas. This gives us a bunch of equivalence classes of smooth structures, which one could denote them with a representative from each class

[\{U_i,\phi_i\}],[\{V_i,\psi_i\}],...,\quad U_i,V_i,...\in\mathcal{U}_i~\phi_i,\psi_i,...\in \Phi_i.

(If you know what I am talking about, don’t fret with the notation, I am just trying to be illustrative).

Definition: A topological manifold with inequivalent smooth structures is said to posses exotic smooth structures.

At this stage it’s not obvious why these things might be called “exotic” – from this constructionist point of view, they appear to be totally reasonable. To see why this “exotic” label might be appropriate, we need to know a few more things about manifolds.

Up to this point we have been studying manifolds intrinsicly, but looking at their points, sets, charts, and atlases. One can also compare manifolds by studying the maps between them. By considering the properties of maps

f:M\to N

between manifolds M and N we can learn something about the relative properties of the manifolds. For instance, if f is a continuous map with continuous inverse f^{-1}, we refer to M and N as homeomorphic. If f is smooth with a smooth inverse, M and N are diffeomorphic. A simple example of this can be found by looking at maps between different sized circles. Consider the explicit embedding of a 1-sphere of radius r in \mathbb{C}, that is

\mathbb{S}_r^1=\{(z\in\mathbb{C}||z|=r\}.

Now let’s define the map f:\mathbb{S}^1_a\to \mathbb{S}^1_b by

f(r,\theta)=(\frac{b}{a}r,\theta).

As long as a,b\neq 0 this map is continuous with an continuous inverse, so \mathbb{S}_a^1 and \mathbb{S}^1_b are homeomorphic. In fact, this map and it’s inverse are smooth as well so they are also diffeomorphic. In fact, since a and b are arbitrary we have just proved that all 1-spheres are diffeomorphic. Notice that there was no essential difference between determining if these manifolds are homeomorphic and determining if they are diffeomorphic. This might be what we expect; functions that are continuous but not smooth are not too far on the fringe of mathematical curiousity, but most functions we write down are smooth.

Ok, technically we only proved that “all 1-spheres represented as subsets of \mathbb{C} are diffeomorphic”; we didn’t use the chart construction outlined in the previous post. If we used the charts (which we generally want to do, since explicitly embeddings of manifolds are rather hard to come by), we would have a diagram like the following (for any i and j with U_i\cup V_j\neq 0:

Exotic_Overlap

Notice that in order for the diagram to be commutative the function f:M\to N must satisfy the same conditions as the overlap functions. Since the charts are only local, this would be a local condition like

f_{ij}:=\psi_i\circ f\circ \phi_j^{-1}:\phi_j(U_j)\to \psi_i(V_i)

for any atlas \{(U_i,\phi_i)\} for M and atlas \{(V_i,\psi_i)\} for N. However, since we already know that the two atlases we have chosen are not compatible, there is at least one overlap

\phi_i\circ \psi_j^{-1}:\psi_j(U_i\cup V_j)\to \phi_i (U_i\cup V_j)

that will not be smooth (if it was, this would be a valid overlap function in the smooth category and the atlases would be equivalent). Thus we have another definition for exotic smoothness:

Definition: Two manifolds which are homeomorphic but not diffeomorphic are exotic smooth structures (with respect to each other).

I hope it is now more clear why we called them “exotic” in the first place; for something to be such a structure, it must be impossible  to find a smooth function f:M\to N, while it is  possible to find such a continuous function. All the functions in all the world are checked, and not a single one works.

I will stop here; next time I will give some more details about the existance of exotic smooth structures and then why they are of particular importance in physics.

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