of $E$. Then for any $\theta \in {{\mathbb{R}}}$, consider the symplectomorphism $g_\theta$ which preserves $D$ and $E$ and such that for any $x \in E$, $$g_{\theta}  x = \begin{cases} \exp ( i
\theta) x \qquad  \text{ if } x \in D \\  x   \qquad \text{ if } x \perp D
\end{cases}$$ Then $ \theta \rightarrow (g_\theta, \exp ( -i \theta /2 ))$ is a path of automorphisms connecting $( {\operatorname{id}}, 1)$ with $( {\operatorname{id}},  -1)$.

Definition of the index {#sec:definition-index}
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Consider the complex structure $j$ of $S$ such that $\ker (j - i {\operatorname{id}}) = E$. Then $(X, Y) \rightarrow {\omega}(X, jY)$ is a scalar product of $S$. Let ${\operatorname{Sym}}(S, j)$ be the space of linear endomorphisms of $S$ symmetric with respect to this scalar product. For any $A \in {\operatorname{Sym}}(S, j)$, define the square root $$\operatorname{det}^{1/2} \bigl(\tfrac{1}{2} {\operatorname{id}}+  i A \bigr)$$ in such a way that it depends continuously on $A$ and is positive for $A=0$. It is well-defined because ${\operatorname{Sym}}(S, j )$ is contractible, as a vector space. Observe also that any symmetric endomorphism $A$ is diagonalisable with a real spectrum so that $\tfrac{1}{2} {\operatorname{id}}+ i A$ is invertible.

Let ${\operatorname{Sp}}_* (S)$ be the set of symplectic 