element in$H_1(\Lambda(n),\Lambda_0(L_0))$. The *Maslov index* of the path is the integer obtained from the sequence of isomorphisms .\
Let us note from the very definition the following three properties of the Maslov index:

-   If $\gamma_1,\gamma_2$ are homotopic by a homotopy having endpoints in $\Lambda_0(L_0)$, then

    $$\mu_{Mas}(\gamma_1,L_0)=\mu_{Mas}(\gamma_2,L_0).$$

-   If $\gamma_1, \gamma_2$ are such that $\gamma_1(1)=\gamma_2(0)$, then

    $$\mu_{Mas}(\gamma_1\ast\gamma_2,L_0)=\mu_{Mas}(\gamma_1,L_0)+\mu_{Mas}(\gamma_2,L_0).$$

-   If $\gamma(\lambda)\in\Lambda_0(L_0)$ for all $\lambda\in I$, then $\mu_{Mas}(\gamma,L_0)=0$.

Let us point out that (iii) also follows from (i) and (ii) independently of the construction.\
The Maslov index can easily be generalised to a pair of paths in $\Lambda(n)$. To this aim let us call a pair of paths $(\gamma_1,\gamma_2)$ *admissible* if

$$\gamma_1(0)\cap\gamma_2(0)=\gamma_1(1)\cap\gamma_2(1)=\{0\}.$$ In what follows we consider $\mathbb{R}^{2n}\times\mathbb{R}^{2n}$ as a symplectic space with respect to the symplectic form $(-\omega_0)\times\omega_0$. Note that the diagonal $\Delta$ is in $\Lambda(2n)$, as well as $L_1\times L_2$ for any $L_1,L_2\in\Lambda(n)$. Moreover, $L_1\cap L_2\neq\{0\}$ if and only if $(L_1\times L_2)\cap\Delta\neq\{0\}$. Hence it is natural to define the *Maslov index* for a pair $(\gamma_1,\gamma_2)$ of admissible paths in $\Lambda(n)$ as

$$\mu_{Mas}(\gamma_1,\gamma_2)=\mu_{Mas}(\gamma_1\times\gamma_2,\Delta).$$ Note that the basic properties which we previously mentioned carry over immediately, i.e.,

-   $\mu_{Mas}(\gamma_1,\gamma_2)=0$ if $\gamma_1(\lambda)\cap\gamma_2(\lambda)=\{0\}$ for all $\lambda\in I$.

-   $\mu_{Mas}(\gamma_1\ast\gamma_3,\gamma_2\ast\gamma_4)=\mu_{Mas}(\gamma_1,\gamma_2)+\mu_{Mas}(\gamma_3,\gamma_4)$ if $\gamma_{1}(1)=\gamma_{3}(0)$ and $\gamma_{2}(1)=\gamma_{4}(0)$.

-   