I will write $(M,\xi)=\mathbf{OB}(P,\phi)$ to denote that $M$ admits an open book decomposition with page $P$ and monodromy $\phi$ supporting a contact structure $\xi$. I will focus on the case where $M=S^{2n+1}$ is a sphere.
Almost contact structures on $S^{2n+1}$ are classified by the group $\pi_{2n+1}(SO(2n+2)/U(n+1))$, which is given by $\mathbb{Z}_{n!},$ $\mathbb{Z},$ $\mathbb {Z}_{n!/2},$ or $ \mathbb{Z}\oplus \mathbb{Z}_2$ according to whether $n\equiv 0,1,2,3 \mod 4$ respectively. Now, assume that we have chosen a stable trivialization of $TS^{2n+1}$ in such a way that the class of the standard contact structure on $S^{2n+1}$ corresponds to zero.
- What is the element in this group corresponding to the overtwisted contact structure $(S^{2n+1},\xi_{OT})=\mathbf{OB}(D^*S^n,\tau^{-1})$, where $\tau$ is the Dehn-Seidel twist?
In dimension three, the corresponding group is $\mathbb Z$, and the corresponding element is $\pm$ a generator depending on your sign convention (this information is given by the Hopf invariant). I wonder if this still holds in higher dimensions. Moreover, the case $n=4k+3$ is also slightly exceptional, as there is a $\mathbb Z_2$ summand in the group.
- If $n=4k+3$, is there a geometric way to flip the sign of the second summand $\mathbb Z_2$ (e.g. inverting the orientation of the hyperplane distribution)?
Thank you for any inputs.