On Gromov's Theorem on Symplectic Homotopy

Question

I want to understand the proof of the following theorem due to Gromov which I'll state in the context of Euclidean spaces. While I tried to read the proof from Macduff-Salamon, it turned out that my geometry/topology knowledge is too poor to understand the proof. Is there any reference where the proof is explained in detail, preferably in the context of Euclidean spaces?

Theorem:

Let $n>2$ be even and $\Omega\subset \mathbb{R}^{n}$ be a smooth, bounded and connected domain. Let $f,g\in C^{\infty}\left(\Omega ;\Lambda^2\left(\mathbb{R}^{n}\right)\right) $ be closed, non-degenerate two-forms in $\Omega$ such that $g-f$ is exact in $\Omega$. If there is a smooth homotopy of non-degenerate two-forms in $\Omega$ connecting $f$ to $g$, there is a homotopy $H_t$ of closed, non-degenerate two-forms connecting $f$ and $g$ with $H_t-f$ is exact in $\Omega$, for all $t\in [0,1]$.

Thank you.

Answer 1

I didn't know that the proof of that fact was in McDuff-Salamon (in which one? Introduction to symplectic topology?).

You can also find it in Eliashberg-Mishachev, Introduction to the h-principle. There it is given as an application of the h-principle for Diff(V)-invariant open differential relations on open manifolds. The methods are not very hard, but they require a lot of time to get used to.

Here is the idea (the general proof is quite technical):

Suppose $\Omega$ is a ball. By your assumptions (I'm using some arguments that are trivial in this simple situation, but which are necessary in more general cases), you have close to the origin a family of non-degenerate $2$-forms $\omega_t$ (which are not necessarily closed).

Restricting to a small neighborhood of the origin, you can replace the family by a closed family ... choose for example the family of $2$-forms $\tilde \omega_t(x) := \omega_t(0)$ that are constant on $\Omega$. (Again, because our situation is very simple, \omega_t(x) is defined for all of $\Omega$ but usually, this definition would only work on a small chart containing the origin).

We can smooth out the family $\tilde \omega_t(x)$ so that it agrees close to $t=0$ and $t=1$ with the initial family $\omega_t$. This gives us now a new $2$-form $\tilde \omega_t$ that is defined for $t<\epsilon$ and $t>1-\epsilon$ on all of $\Omega$, and inbetween it is only defined in a small neighborhood of the origin. The forms $\tilde \omega$ are all closed and non-degenerate, where they are defined.

Now you can use an isotopy on $\Omega$ that is the identity for $t=0$ and $t=1$, and compresses in between the whole of $\Omega$ into the small neighborhood of the origin on which $\tilde \omega_t$ is defined.

Pulling back the family $\tilde \omega_t$ with this isotopy gives you a globally defined family of symplectic forms that are everywhere defined and that agree with $\omega_0$ and $\omega_1$ respectively.

Hope this helped.