In $\mathrm{Sp}(2)$, a loop of degree zero moves $e_1$ along a null-homotopic loop in $\mathbb R^2\setminus\{0\}$

Let $\varphi:[0,1]\to\mathrm{Sp}(2,\mathbb R)=\mathrm{SL}(2,\mathbb R)$ be a continuous loop with $\varphi(0)=\varphi(1)=\mathrm{Id}$, let $\hat\rho:\mathrm{Sp}(2n,\mathbb R)\to S^1$ is the map $\hat\rho(A)=\det_{\mathbb C}C_A/|\det_{\mathbb C}C_A|$, where $C_A=\tfrac12(A-J_0AJ_0)$ is the complex linear part of $A$ (Gutt, eq. (9)), and suppose that a continuous argument $\theta$ of $\hat\rho\circ\varphi$, i.e. $\hat\rho(\varphi(t))=e^{i\theta(t)}$, satisfies $\theta(1)=\theta(0)$. Let $e_1=(1,0)\in\mathbb R^2$. Then there is a continuous map $H:[0,1]\times[0,1]\to\mathbb R^2\setminus\{0\}$ with

So for $n=1$ the vanishing of the Maslov index $\deg(\hat\rho\circ\varphi)$ forces the orbit loop $t\mapsto\varphi(t)e_1$ to be null-homotopic in $\mathbb R^2\setminus\{0\}$ relative to $e_1$. It is the two-dimensional input for the statement that the Maslov index is injective on $\pi_1(\mathrm{Sp}(2n,\mathbb R))$ (Wendl, Lecture 3, §3.4).

Formalization Note. $\mathbb R^{2n\times2n}$ is Mat n (real matrices indexed by Fin n ⊕ Fin n, first block $q$, second block $p$) and $J_0=\begin{pmatrix}0&-\mathrm{Id}\\ \mathrm{Id}&0\end{pmatrix}$ is Matrix.J; IsSymplectic A is membership in Matrix.symplecticGroup. Vectors of $\mathbb R^{2n}$ are functions Fin n ⊕ Fin n → ℝ and $Av$ is A *ᵥ v. Here $n=1$, $e_1$ is Pi.single (Sum.inl 0) 1, $\hat\rho$ is rhoHat, and the argument condition is IsArgLift (fun t => rhoHat (φ t)) θ.