Transversality and $C^l$, $C^{\infty}$ spaces of almost complex structures

Question

Recently I have been trying to get a grip on transversality results in Floer homology. That is suppose we the section $\partial_{J,H}: W^{1,p}(u^*(TM))\rightarrow L^{p}(u^*(TM))$ and we want to prove that there exists a generic set of almost complex structures $\mathcal{J}_\text{reg}$ for which its derivative is surjective.

To prove this the idea is to consider an universal moduli space, prove this is a banach manifold, using the fact that set of regular points $R(u)$ is dense. Then one considers the projection $\pi: \mathcal{M}_\text{univ}\rightarrow \mathcal{J}$ and proves this is a Fredholm map, and then the desired result will follow from the Sard–Smale theorem.

Now there a few technicalities here. First of all this direct approach only works in the $C^l$-topology of almost complex structures and to get a result on the $C^{\infty}$-topology one needs to use a method due to Taubes. This is one thing I would like to understand. Why is the completion of $\mathcal{J}$ in the $C^l$-topology a banach manifold and not in the $C^{\infty}$-topology? I am not looking for all the analytical details regarding the proof of this, but I would like to have some kind of intuition on why this happens.

Any insight is appreciated.

Answer 1

There are several different issues happening here. The space of $C^\infty$ is not a Banach space because the topology is generated by a countable family of semi-norms not by one norm. You might try to think about why you can’t generate the topology by some magical norm. OK so why do we care about $C^\infty$ anyway. The point is applying Sard Smale. This theorem (like Sard’s Thm) requires more regularity as the formal dimension ( index) goes up so of you want one perturbation to rule all modulo spaces no matter the index you need it to be smooth. How do you marry these two issues. The idea is to find Banach spaces contained in $C^\infty$. The trick appears first to my knowledge in Floer’s paper “The unregularized gradient flow of the symplectic action”. Perhaps he mentions that he learned the idea from Taubes. What you do is pick a rapidly decreasing sequence $\epsilon_k$ and take the norm $$\sum_k \epsilon_k \|\cdot\|_{C^k}. $$ Floer shows in that paper that for suitable sequences there are plenty (enough to get transversality) of perturbations with this norm finite but this is not all of $C^\infty$ . Thus there are Banach spaces of perturbations contained the smooth perturbations. We can now apply Sard Smale to a countable family of moduli space problems to get still a dense set of perturbations that are regular for all of these.