Question

In http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf the Arnold conjecture from Symplectic Geometry is shown for the case of montone symplectic manifolds $(M, \omega)$ (i.e. we have $\int_{S^2} v^*\omega= \tau \int_{S^2} v^*c_1$ for every smooth map $v: S^2 \rightarrow M$ for the Chern class $c_1$ and some $\tau>0$), which furthermore fulfill the condition $\int_{S^2} v^*\omega \in \mathbb{Z}$ for every smooth map $v: S^2 \rightarrow M$. My question now is: Can we immediately deduce the Arnold conjecture without the condition $\int_{S^2} v^*\omega \in \mathbb{Z}$, if we have first shown it under that assumption? Is there any simple argument that the Arnold conjecture for general monotone symplectic manifolds is true, if its is true under the further condition $\int_{S^2} v^*\omega \in \mathbb{Z}$? For example on page 5 of http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf the author says that one can normalize the symplectic form such that this condition holds. Can you maybe tell me, why such a normalization is possible and why the Arnold conjecture for general monotone manifolds follows after considering the normalized case? Every help will be appreciated.

Answer 1

The normalization $$\int_{S^2} v^\ast \omega \in \Bbb Z \text{ for all smooth } v: S^2 \longrightarrow M$$ is achieved by setting $\omega \mapsto \frac{1}{\tau} \omega$ and looking at the monotonicity condition $$\int_{S^2} v^\ast c_1(M) = \tau \int_{S^2} v^\ast \omega \text{ for all smooth } v: S^2 \longrightarrow M.$$ $c_1(M)$ is an integral class, so after making the substitution $\omega \mapsto \frac{1}{\tau}\omega$ we see that the required integrality condition holds.

Salamon uses this normalization in order to make sure the symplectic action "functional" $$\mathcal{A}_H: \mathcal{L}_0 M \longrightarrow \Bbb R /\Bbb Z,$$ $$\mathcal{A}_H(\gamma) = - \int_{D^2} u^\ast \omega - \int_0^1 H_t(\gamma(t)) ~dt,$$ where $u: D^2 \longrightarrow M$ is such that $u|_{\partial D^2} = \gamma$, is well-defined up to addition of an integer. We can see that choosing another $u': D^2 \longrightarrow M$ such that $u'|_{\partial D^2} = \gamma$ changes $\mathcal{A}_H(\gamma)$ by an integer amount by "gluing" $u$ and $u'$ along their common boundary to get a map $u \# u': S^2 \longrightarrow M$ and noting that we have $$\int_{S^2} (u \# u')^\ast \omega \in \Bbb Z$$ by our assumption.

The integrality assumption is really just Salamon's way of avoiding the more technical general approach. To get an actual symplectic action functional, you really want to work in the Novikov covering space $\widetilde{\mathcal{L}_0 M}$ of the space of contractible loops in $M$. If you work in this more complicated setting, you get a bona fide symplectic action functional $$\mathcal{A}_H: \widetilde{\mathcal{L}_0 M} \longrightarrow \Bbb R,$$ $$\mathcal{A}_H([\gamma, u]) = - \int_{D^2} u^\ast \omega - \int_0^1 H_t(\gamma(t)) ~dt.$$ You don't need the normalization once you're working with the symplectic action functional on the Novikov covering space.

Salamon briefly mentions the Novikov covering space on page 41 of his notes. A much more detailed reference for this is chapter 18 of Oh's notes.