Good morning everybody, I am currently reading through the book of Banyaga "Structure of classical diffeomorphism groups" link, and I am particularly interested in the question of factorizing volume-preserving diffeomorphisms on a compact smooth and connected $n$-dimensional manifold $M$ which are isotopic to the identity.

Now, chapter 5 of the book in particular deals with this infinite dimensional Lie group, and at page 129 they take into consideration exactly this question, and they provide the factorization specifically for maps $h$ isotopic to the identity, under the constraint that they belong to the kernel of the so-called flux homomorphism $S_\omega$.

Its definition is as follows:

assume $\omega$ to be a volume form on $M$ and let $\varphi_t\in\textrm{Diff}^\infty_\omega(M)$ be a smooth isotopy to the identity. Let $i(\cdot)$ denote the interior product and define $$I_{\varphi_t}(\omega)=\int_0^1(\varphi_t^*i(\dot\varphi_t)\omega)dt.$$

The cohomology class $[I_{\varphi_t}(\omega)]\in H^{n-1}(M,\mathbb R)$ depends only on the homotopy class relatively to fixed ends of the isotopy $\varphi_t\in\textrm{Diff}^\infty_\omega(M)$.

Let $G_\omega(M)$ be the group consisting of homotpy classes $[\varphi_t]$ of isotopies $\varphi_t\in\textrm{Diff}^\infty_\omega(M)$, relatively to fixed ends. The mapping $$S_\omega:G_\omega(M)\ni[\varphi_t]\mapsto I_{\varphi_t}(\omega)\in H^{n-1}(M,\mathbb R)$$ is the so-called flux homomorphism. Therefore $[\varphi_t]\in\ker S_\omega\Leftrightarrow I_{\varphi_t}(\omega)$ is an exact $n-1$ form. You can read more on $S_\omega$ in chapter 3 of the book I mentioned above.

My question now is as follows: let $\textrm{Diff}^\infty_\omega(M)_0$ be the subgroup of $\textrm{Diff}^\infty_\omega(M)$ of isotopic to the identity volume-preserving diffeomorphisms on $M$. Is it possible to find a sufficiently small neighborhood of the identity map $O\subset \textrm{Diff}^\infty_\omega(M)_0$ such that $f\in\ker S_\omega$ for any $f\in O$? I am pretty confident this is possible if $M$ is also simply connected, but in the general setting I explained at the beginning? I was wondering this could also be the case if all the maps in $O$ had common support within a contractible set $U\subset M$ but is this a full neighborhood of the identity map in $\textrm{Diff}^\infty_\omega(M)_0$?

Thanks in advance to all those who will reply and help me clarify this point.

-Guido-