at.algebraic topology - Maslov index of a pullback bundle

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 $u^astLambda^{max}_{mathbb{C}}TM$.

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 $zmapsto z$ of the trivial line bundle over the disc satisfies the standard Maslov-index 2 boundary condition. Done!