$\mu_{Mas}(\gamma_1,\gamma_2)=\mu_{Mas}(\gamma_3,\gamma_4)$ if $\gamma_1\simeq\gamma_3$ and $\gamma_2\simeq\gamma_4$ are homotopic by a homotopy through admissible pairs.

Also, it is not difficult to see from the construction of the Maslov index that

-   $\mu_{Mas}(\gamma_1,\gamma_2)=\mu_{Mas}(\gamma_1,L_0)$ in case that $\gamma_2(\lambda)=L_0$ for some $L_0\in\Lambda(n)$ and all $\lambda\in I$,

-   $\mu_{Mas}(\gamma_1,\gamma_2)=-\mu_{Mas}(\gamma_2,\gamma_1)$ for any admissible pair $(\gamma_1,\gamma_2)$.

Finally, let us define the Maslov index for a non-admissible pair of paths. It is important to note that in this case there are different definitions in the literature. Here we follow [@Cappell], and note that given $L_1, L_2\in\Lambda(n)$ such that $L_1\cap L_2\neq\{0\}$, there is $\varepsilon>0$ such that $e^{\Theta J}L_2\in\Lambda(n)$ and $L_1\cap e^{\Theta J}L_2=\{0\}$ for all $0<|\Theta|\leq\varepsilon$. We define the Maslov index as

$$\mu_{Mas}(\gamma_1,\gamma_2)=\mu_{Mas}(\gamma_1,e^{-\Theta J}\gamma_2),$$ where $\Theta$ is such that $\gamma_1(0)\cap e^{-\Theta' J}\gamma_2(0)=\gamma_1(1)\cap e^{-\Theta' J}\gamma_2(1)=\{0\}$ for all $0<|\Theta'|\leq\Theta$. By the homotopy invariance, it is clear that this definition does not depend on the choice of $\Theta$. Also, it coincides with the previous definition in case that the pair of paths is admissible.

### The Paths $\gamma_{nor}$ and $\gamma'_{nor}$ {#section-gammanor}

The aim of this section is to compute the Maslov index for two elementary paths that will also become important in our proof of Theorem \[main\] below. The examples also show that is 