The professor of an introduction to general relativity made a remark that confused me. It is not merely that I find it unintuitive, I also find it hard to wrap my head around what it means:
In dimensions $1$ to $3$, a topological manifold can
be given only one differential structure (and finitely many in $5-7$) [I think what he means
here is that we can give it an infinite amount of
differentially-compatible atlasses, but they are all diffeomorphic],
but in dimension $4$, we can give it an uncountable number of
non-diffeomorphic differential structures.
First of all, I am not 100% sure if I understand what it means.
For example is it correct to say that to give a topological manifold a particular differential structure is to choose a smooth Atlas $A$ or any other one that has smooth transition maps to $A$'s charts?
To test my understanding, please tell me if the following re-formulation is correct:
Take any ${1,2,3}$-dimensional differentiable manifold $M$. Take the set $X$ of all manifolds that are homeomorphic to $M$. Then all elements of $X$ are also diffeomorphic to eachother.
Take any $4$-dimensional differentiable manifold $M$. Take the set $Y$ of all manifolds that are homeomorphic to $M$. Then there are an uncountable number of subsets $U_{\alpha \in R}$ of $Y$ such that for all $\alpha \in R$, all elements of $U_{\alpha}$ are diffeomorphic to eachother, but for every $\alpha, \beta \in R, ($with $\alpha \neq \beta)$, no element of $U_{\alpha}$ is diffeomorphic to an element of $U_{\beta}$.
Then the questions this post is all about:
- Why is there only one particular differential structure for ${1,2,3}$ dimensional manifolds (proof-wise perhaps, but mostly intuitively)? I feel like I can't understand the $4$ dimensional weirdness, until I intuitively understand the ${1,2,3}$ case.
- What should we even imagine when we say that a $4$-dimensional manifold has multiple differential structures? We can't easily picture $4$ dimensional space, but is there some intuitive way to show what this would mean?
Additional remarks
$(1)$. Additional analysis/question: If my above re-formulation is correct, then the theorem does not imply that for any particular differentiable $4$-manifold (i.e. any particular set of points in $R^{d>=4}$ forming a $4$-sub-manifold) there are infinite possible differential structures. For example, if we take any particular differentiable $4$-manifold embedded in $R^{d>=4}$, then that manifold will only have a unique differentiable structure, correct?
Best Answer
The $1$-dimensional case is trivial, and the $2$-dimensional case is classical (but harder than one might expect). The case of dimension $3$ was proved by Moise in the 50s. Higher dimensions are different: The first distinct smooth structures on the same manifold were presented by Milnor for $S^7$. Furthermore, any compact, PL manifold in dimension $n\not = 4$ has only finitely many distinct smooth structures.
The case $n = 4$ is very different from the lower- and higher-dimensional cases, and the intuition there probably doesn't apply. The glib one-line answer is that in low-dimensions, geometry dominates; in high-dimensions, the $h$-cobordism theorem and its extensions dominate, and the subject becomes surgery theory. The problem with dimension $4$ is that the Whitney trick fails spectacularly. There are nice $4$-manifolds that have no smooth structure (i.e., a manifold $X$ not homeomorphic to any smooth manifold $Y$), and there are nice $4$-manifolds that have multiple smooth structures. For example, there exist uncountably many manifolds that are homeomorphic to $\mathbb{R}^4$, but no two of which are diffeomorphic. The $E_8$-manifold is compact and simply connected, but it can't be given a smooth structure. The details of which manifolds have a smooth structure are complicated, but the existence of a PL structure for a compact manifold $X$ is detected by the Kirby-Siebenmann class $\kappa\in H^4(X, \mathbb{Z}_2)$. In particular, if $X$ has dimension $<4$, then this class vanishes. (Of course, that's obscures exactly where the class comes from; it's a bit like reading the punchline without the joke.)
You asked for the intuition behind that, and the best answer I can come up with is that the naive intuition that one can take a improve a reasonable homeomorphism $X \to Y$ to a "nearby" smooth map fails completely in higher dimensions. Along similar lines, there are characteristic classes attached to manifolds that are invariant under diffeomorphisms but not under arbitrary homeomorphisms, and so the two categories of structures are distinguishable. Dimension $4$ is just particularly weird. Low-dimensional topology has a very different flavor from high-dimensional topology, and dimension $4$ is very different even from dimension $3$.