Kernel of flux homomorphism (Calabi invariant) for volume-preserving maps on a compact manifold

Question

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-

Answer 1

If M is simply connected and closed (and oriented, but this is the case if it has a volume form), then $H^{1}(M)$ is zero and hence by Poincaré duality $H^{n-1}(M)$ is also zero. So the flux homomorphism is trivial and this answers your question.

Otherwise I think that the answer to your question is no. If you have a volume preserving vector field X on your manifold, with flow $f^t$ then the flux of $f^t$ is equal to a constant time t. Indeed, fix a real number $a$, as an isotopy from the identity to $f^a$ you take the path $\varphi_{t}=f^{ta}$ with $t$ between $0$ and $1$. The vector field you call $\dot \varphi_{t}$ is just $aX$. The form $\iota_{aX}\omega$ is invariant by $\varphi_{t}$ so the integral you wrote to define the flux of $f^a$ is just the integral from $0$ to $1$ of the constant $(n-1)$-form $\iota_{aX}\omega$.

So the flux of $f^a$ is $a$ times the class of the form $\iota_{X}\omega$.

If this class is nonzero, then for small $a$ you get volume preserving diffeomorphisms arbitrarily close to the identity (in the $C^{\infty}$ topology) and with nonzero flux.

And you can find examples of pairs M, X like this. For instance you can take M to be any closed oriented surface of positive genus. If you pick any closed $1$-form $\alpha$, there always exists a volume preserving vector field $X_{\alpha}$ such that $\iota_{X_{\alpha}}\omega = \alpha$. So the cohomology class of $\iota_{X}\omega$ can be any class in $H^1$.

The condition to have zero flux is necessary to be able to write a diffeomorphism as a composition of diffeomorphisms compactly supported in open sets diffeomorphic to balls. (and isotopic to the identity inside these balls)

There is a recent book by Bounemoura (in french), about the same topics as Banyaga's book, but focusing mainly on surfaces, and with connections to more recent developments. You might find it interesting (it is called Simplicité des groupes de transformations de surfaces. [The simplicity of surface transformation groups] Ensaios Matemáticos [Mathematical Surveys], 14).