The gg-areas and the commutator length

The commutator length of a Hamiltonian diffeomorphism $f\in \mathrm{Ham}(M, \omega)$ of a closed symplectic manifold $(M,\omega)$ is by definition the minimal $k$ such that $f$ can be written as a product of $k$ commutators in $\mathrm{Ham}(M, \omega)$. We introduce a new invariant for Hamiltonian diffeomorphisms, called the $k_+$-area, which measures the "distance", in a certain sense, to the subspace $\mathcal{C}_k$ of all products of $k$ commutators. Therefore this invariant can be seen as the obstruction to writing a given Hamiltonian diffeomorphism as a product of $k$ commutators. We also consider an infinitesimal version of the commutator problem: what is the obstruction to writing a Hamiltonian vector field as a linear combination of $k$ Lie brackets of Hamiltonian vector fields? A natural problem related to this question is to describe explicitly, for every fixed $k$, the set of linear combinations of $k$ such Lie brackets. The problem can be obviously reformulated in terms of Hamiltonians and Poisson brackets. For a given Morse function $f$ on a symplectic Riemann surface $M$ (verifying a weak genericity condition) we describe the linear space of commutators of the form $\{f,g\}$, with $g\in\mathcal{C}^\infty(M,\mathbb{R})$.Comment: 13 pages, 2 figures. To appear in International Journal of Mathematics, Vol. 24, No. 7 (2013). Revised version: misprint correcte

The homotopy type of the space of symplectic balls in rational ruled 4-manifolds

Let M:=(M^{4},\om) be a 4-dimensional rational ruled symplectic manifold and denote by w_{M} its Gromov width. Let Emb_{\omega}(B^{4}(c),M) be the space of symplectic embeddings of the standard ball B^4(c) \subset \R^4 of radius r and of capacity c:= \pi r^2 into (M,\om). By the work of Lalonde and Pinsonnault, we know that there exists a critical capacity \ccrit \in (0,w_{M}] such that, for all c\in(0,\ccrit), the embedding space Emb_{\omega}(B^{4}(c),M) is homotopy equivalent to the space of symplectic frames \SFr(M). We also know that the homotopy type of Emb_{\omega}(B^{4}(c),M) changes when c reaches \ccrit and that it remains constant for all c \in [\ccrit,w_{M}). In this paper, we compute the rational homotopy type, the minimal model, and the cohomology with rational coefficients of \Emb_{\omega}(B^{4}(c),M) in the remaining case c \in [\ccrit,w_{M}). In particular, we show that it does not have the homotopy type of a finite CW-complex.Comment: 38 pages; revised versio

The topology of the space of symplectic balls in rational 4-manifolds

We study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball $B^4(c) \subset \R^4$ into 4-dimensional rational symplectic manifolds. We compute the rational homotopy groups of that space when the 4-manifold has the form $M_{\lambda}= (S^2 \times S^2, \mu \omega_0 \oplus \omega_0)$ where $\omega_0$ is the area form on the sphere with total area 1 and $\mu$ belongs to the interval $[1,2]$. We show that, when $\mu$ is 1, this space retracts to the space of symplectic frames, for any value of $c$. However, for any given $1 < \mu < 2$, the rational homotopy type of that space changes as $c$ crosses the critical parameter $c_{crit} = \mu - 1$, which is the difference of areas between the two $S^2$ factors. We prove moreover that the full homotopy type of that space changes only at that value, i.e the restriction map between these spaces is a homotopy equivalence as long as these values of $c$ remain either below or above that critical value.Comment: Typos corrected, 2 minor corrections in the text. Numbering consistant with the published versio