Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 146 / 1 / c / Solution

On the quotient from part (b), the Maslov mapis well defined because $\det (Q{)}^2=1$ for $Q\in O(n)$. Consider the loop of Lagrangian subspacesIts endpoints agree as unoriented subspaces, and $\mu \circ \ell (t)=e^{2\pi it}$ has degree one. If $\eta$ is the generator of $H^1(S^1;\mathbb{Z})$, thenConsequently ${\mu }^{\ast }\eta \ne 0$, provingThe same integer is the Maslov index of $\ell$.