I have a few fairly generic questions, with a specific application to symplectic geometry in mind. Let me pose the specific problem first:
Let a symplectic manifold $(M,\omega)$ be given. One is naturally led to consider the "space" $\mathcal{J}$ of almost complex structures compatible with $\omega.$ One then shows that this space is non-empty and contractible. Of course, this is really just a set until one equips it with a topology. My basic question is: what is this topology? I can think of one possibility, which rests on the apparent fact (stated without proof in Audin-Lafontaine) that one has a bundle $\mathcal{J}(\omega) \to M$ of compatible almost complex structures, with fiber $\mathcal{J}_p = \{ \text{complex structures of } T_pM \text{ compatible with } \omega_p \}.$ Our space $\mathcal{J}$ is then the space of sections of said bundle.
Working fiberwise, we can see that we have a bijection $\mathcal{J}_p \cong Sp_{2n}/U(n),$ so we could let $\mathcal{J}_p$ inherit the topology of that homogeneous space. Is there a natural way, then, to construct a topology on $\mathcal{J}$ using the topologies on the fibers $\mathcal{J}_p?$
More generally, is there a natural way to topologize the space of sections of a vector bundle that applies to this scenario?
Finally, how would one construct the bundle $\mathcal{J}(\omega)$ from the fibers and the base space? I suppose one could use the fiber bundle construction theorem, but then one would need specific transition maps.
Best Answer
The space of smooth sections of a bundle is a subspace of the space of smooth maps from the base to the total space. The natural topology is then the subspace topology once we know how topologize the space of smooth maps between two smooth manifolds. There are many natural topologies on such a space, depending on how many derivatives you want to keep track of. The coarsest topology is the compact-open topology, on the space of continuous maps. The next trick is to embed the target manifold in some Euclidean space. Then you also have $C^k$-topology for each $k$ where you require uniform convergence on compacts of (all) the first $k$ partial derivatives. Lastly, you can also use the $C^\infty$-topology.