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The Lagrangian Grassmannian is the space of all Lagrangian subspaces of a symplectic vector space. The unitary group acts transitively on $\mathrm{LGr}({\mathbb{R}}^{2n})$, and the stabilizer of ${\mathbb{R}}^n$ is the orthogonal group $O(n)$, giving

The Maslov index assigns an integer to a loop of Lagrangian subspaces. Under $\mathrm{LGr}({\mathbb{R}}^{2n})\cong U(n)/O(n)$, it is the degree of the map induced by $A\mapsto \det (A{)}^2$.

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 $\ell$.