Maslov index of a pullback bundle - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T21:49:45Z http://mathoverflow.net/feeds/question/23171 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23171/maslov-index-of-a-pullback-bundle Maslov index of a pullback bundle macbeth 2010-05-01T11:28:02Z 2010-05-03T17:46:34Z <p>This question has bugged me as I read McDuff-Salamon's book on pseudoholomorphic curves. I'll use their terminology.</p> <p>Let $\Sigma$ be a compact surface possibly with boundary, $M$ an almost-complex manifold, and $L$ a totally real submanifold of $M$. A map</p> <p>$u:(\Sigma, \partial\Sigma)\to(M, L)$</p> <p>gives rise to a <em>bundle pair</em> over $\Sigma$: a complex vector bundle $u^*TM$ over $\Sigma$, together with a totally real sub-bundle $u^*TL$ over $\partial\Sigma$.</p> <p><strong>Question:</strong> 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 $u$?</p> <p><strong>Motivating special case:</strong> if $\partial\Sigma=\emptyset$, then the Maslov index of the bundle pair $(u^*TM, \emptyset)$ is $2\langle c_1(TM), u_*[\Sigma]\rangle$.</p> http://mathoverflow.net/questions/23171/maslov-index-of-a-pullback-bundle/23176#23176 Answer by Tim Perutz for Maslov index of a pullback bundle Tim Perutz 2010-05-01T12:59:18Z 2010-05-03T17:46:34Z <p>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 $[\Sigma,\partial \Sigma]$.</p> <p>When $\Sigma$ is closed, the Chern number of $u^\ast TM$ is the signed count of zeroes of a transversely-vanishing section $s$ of $u^\ast\Lambda^{max}_{\mathbb{C}}TM$. </p> <p>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 $\Lambda^{max}_{\mathbb{R}} TL$. </p> <p>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.</p> <p>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!</p>