MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This question has bugged me as I read McDuff-Salamon's book on pseudoholomorphic curves. I'll use their terminology.

Let $\Sigma$ be a compact surface possibly with boundary, $M$ an almost-complex manifold, and $L$ a totally real submanifold of $M$. A map

$u:(\Sigma, \partial\Sigma)\to(M, L)$

gives rise to a bundle pair over $\Sigma$: a complex vector bundle $u^*TM$ over $\Sigma$, together with a totally real sub-bundle $u^*TL$ over $\partial\Sigma$.

Question: Is there a nice description for the Maslov index of this bundle pair, in terms of a topological invariant of $u$? For instance, in terms of the homology class $u_*[\Sigma]\in H_2(M, L)$, or in terms of the homotopy equivalence class of $u$?

Motivating special case: if $\partial\Sigma=\emptyset$, then the Maslov index of the bundle pair $(u^*TM, \emptyset)$ is $2\langle c_1(TM), u_*[\Sigma]\rangle$.

share|cite|improve this question
up vote 8 down vote accepted

When you have a vector bundle on a manifold $X$ with boundary, trivialised over $\partial X$, there are characteristic classes valued in $H^\ast (X,\partial X)$. Here, when $L$ is orientable, the Maslov index is twice the first Chern class of $u^\ast TM$ relative to the trivialisation on the boundary induced by $L$, evaluated on $[\Sigma,\partial \Sigma]$.

When $\Sigma$ is closed, the Chern number of $u^\ast TM$ is the signed count of zeroes of a transversely-vanishing section $s$ of $u^\ast\Lambda^{max}_{\mathbb{C}}TM$.

When there is an orientable boundary condition, the relative Chern number is the same thing, but you choose $s$ non-vanishing along the boundary and tangent to the real line sub-bundle $\Lambda^{max}_{\mathbb{R}} TL$.

This doesn't make sense when $u^*|_{\partial \Sigma} TL \to \partial \Sigma$ is not orientable: its top exterior power then has no non-vanishing section. Besides, the Maslov index is odd in this case.

ADDED: Here's a proof using the method of Robbin's appendix to McDuff-Salamon ("$J$-holomorphic curves and symplectic topology"). Robbin characterises the boundary Maslov index as an invariant of bundle pairs (complex vector $E$ bundle over a surface, totally real sub-bundle $F$ over the boundary) which is additive under direct sum and under sewing boundaries and is suitably normalised for line bundles over the disc. The uniqueness proof, by "pair-of pants induction", still applies when $F$ is assumed orientable. The invariant "twice the relative Chern number" evidently satisfies the direct sum and sewing properties, and the section $z\mapsto z$ of the trivial line bundle over the disc satisfies the standard Maslov-index 2 boundary condition. Done!

share|cite|improve this answer
Thanks! Can you suggest a good reference for this material? – macbeth May 1 '10 at 14:08
Well, I would have suggested McDuff-Salamon... You could try the "Intro to symp. topology" if the "J-hol. curves" book doesn't have what you want. When I'm in the office I'll try to find a precise reference. – Tim Perutz May 2 '10 at 13:42
Cheers. McD-S as a rule seem uninterested in the with-boundary case; I couldn't find this treated there. – macbeth May 2 '10 at 15:58
It follows from what they do say - see added paragraph. [Hm. Macbeth bamboozled by McDuff. It sounds strangely familiar... I feel I should warn you that McDuff is not a "man of woman born"!] – Tim Perutz May 3 '10 at 17:51
The holomorphic bubbles have caused me much toil and trouble. Thanks for the extra details! – macbeth May 3 '10 at 20:04

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.