Genuine back-of-the-envelope calculation!

Mathematical Loose Ends

A year of indulgence

Last Spring I decided to take about a year off work. One of things I wanted to do was mathematics research; I wanted to see if I could solve any nice problems and get my work published. I ended up spending about seven months working essentially full time on mathematics and in the end I had two papers published, including this one:
K-theory, LQEL manifolds and Severi varieties,
which has been accepted by Geometry & Topology.

Eventually I turned my attention toward other non-mathematical endeavours but I did have a few mathematical ideas that I would have liked to work on if I had had even more time. As time has now run out and I will shortly return to employment, I decided to type up a few of these ideas in case I ever wish to think about them again. Mathematical culture is such that revealing partially-formed (and possibly incorrect!) ideas tends to be rare but I have decided to be brave.

Here is a letter containing some ideas I think are worth pursuing. I would be delighted to hear from anybody who takes an interest in these problems and happy to clarify or to expand upon the comments I have made.

Bonus remarks on K-theory

In the spirit of revealing mathematical scaffolding, I'm going to add a few remarks about a simple result that helped me believe K-theory was the right tool when I was working on the paper I mentioned above.

The result is due to Karoubi and can be found in the following old paper (in French, as it happens):
M. Karoubi, Les isomorphismes de Chern et de Thom-Gysin en K-Théorie, Séminaire Cartan-Schwartz, 16e année, 1963/64, no. 16 (1964).

Proposition [Karoubi 1964, proposition 2.1]

If: $$ ch : K(S^n) \to H^*(S^n, \mathbb{Q}) $$ is the Chern character map for the n-dimensional sphere, then: $$ Im(ch) = H^*(S^n, \mathbb{Z}) $$

A priori for a (torsion-free) space the image of the Chern character is just a maximal-rank lattice in the rational cohomology and need not coincide with the lattice that is the integral cohomology. That it does for the sphere should have consequences. Here is one:

Corollary

Let \( S^{2n}\) be a sphere carrying an almost complex structure, then \( n = 1 \mbox{ or } 3\).

Proof If \( E \to S^{2n}\) is a complex rank n vector bundle, let \( c_n(E) \in H^{2n}(S^{2n}, \mathbb{Z}) \simeq \mathbb{Z}\) be the top Chern class, then: $$ (n-1)! \mid c_n(E) $$ This follows from the proposition since \( ch(E) = (-1)^n \frac{c_n(E)}{(n-1)!}\) (because the sphere only has cohomology is degrees 0 and 2n). Now take \( E = TS^{2n}\) and use \( c_n(TS^{2n}) = \chi(S^{2n}) = 2\) to get the condition: $$ (n-1)! \mid 2 $$ This leaves only the \(n=2\) case which is most easily dealt with by means of an Euler-class calculation (though it can also be done easily enough using only K-theory).

I found this result strongly motivating because I knew that it was easy to use characteristic classes to show that n must be odd if \( S^{2n}\) carries an almost complex structure but that it was substantially harder to get the full result that n could in fact only be 1 or 3 (this is really due to the existence of the octonions). So when I saw how easily the result could be obtained using K-theory, I was impressed. Furthermore I knew that the Severi varieties could be shown to have dimension either \( 2^n \mbox{ or } 3\cdot 2^n\) using characteristic classes but that this was about as far as one could get with this approach (Severi varieties have dimension 2, 4, 8 or 16). With Karoubi's result (and others) in mind I decided to see what would happen if I expressed the ideas of those who had been using characteristic classes for the Severi variety problem in K-theory. Immediately I saw it was the more natural language and the result essentially fell out in one piece.

Elliptic operators

I have one last comment. In the proof of the corollary to Karoubi's result mentioned above we noted that we had the following divisibility condition for vector bundles on spheres: $$ \frac{c_n(E)}{(n-1)!} \in \mathbb{Z} $$ This raises the question: what is this integer that is intrinsically associated to the vector bundle E? In fact, by the index theorem, it is the index of Dirac operator coupled to E.

With this in mind and bearing in mind the divisibility property for LQEL manifolds which comes up in my paper, I can't help wondering, what is the integer: $$ \frac{n-\delta}{2^{[(\delta - 1)/2]}} $$ that is intrinsically associated to an LQEL manifold? I don't know the answer but I could believe it is also the index of an appropriate elliptic operator.