A homotopical viewpoint at the Poisson bracket invariants for tuples of sets (Q2216138)

The paper discusses a construction in the intersection of Poisson and symplectic geometry. In a first result, the author proves a theorem which states a relation between two invariants for a symplectic manifold \((M,\omega)\). The two invariants are called \(\mathrm{pb}_3\) and \(\mathrm{pb}_4\). The first invariant is defined as a lower bound for the supremum norm of the Poisson bracket of two smooth maps on \(M\) with compact support endowed with additional signature properties that are associated to three compact subsets \(X,Y,Z\) in \(M\). Denote this invariant associated to \(X,Y,Z\) by \(\mathrm{pb}_3(X,Y,Z)\). The second invariant is similarly defined, with the difference that now one considers four compact subsets \(X_0,X_1,Y_0,Y_1\) of \(M\) with the properties \(X_0\cap X_1=Y_0\cap Y_1=\emptyset\). Denote this invariant associated to \(X_0,X_1,Y_0,Y_1\) by \(\mathrm{pb}_4(X_0,X_1,Y_0,Y_1)\). The relation proven in this paper is \[ \mathrm{pb}_4(X_0,X_1,Y_0,Y_1)=2\cdot \mathrm{pb}_3(X_0,Y_0,X_1\cup Y_1). \] Further, the author describes a generalization of the invariants mentioned before by considering compact convex subsets \(\Delta\) of \(\mathbb{R}^2\) which measure a lower bound for the supremum norm of the Poisson bracket of the components of a tuple \(\Phi=(\Phi_1,\Phi_2)\colon M\to \Delta\). Similarly as before, the new invariants depend on some conditions of these tuples on some subsets \(X_1,\ldots,X_N\) of \(M\). Denote these new invariants by \(\mathrm{Pb}_N(X_1,\ldots,X_N)\). The generalization of the above result states that for cyclically intersecting subsets \(X_1,\ldots,X_N\) of \(M\) one has \[ \mathrm{pb}_N(X_1,\ldots,X_N)=\mathrm{pb}_{N-1}(X_1,\ldots,X_{N-1}\cup X_N) \] when \(N\geq 4\). Another result considers the existence of a limit for these invariants. In particular, it is proven that \[ \lim_{K_n\searrow X_1\cap X_N}\mathrm{pb}_{N+1}(\overline{X_1\setminus K_n},X_2,\ldots,X_{N-1},\overline{X_N\setminus K_n},K_n)=\mathrm{pb}_N(X_1,\ldots,X_N), \] where \(X_1,\ldots,X_N\) are compact subsets of \(M\) intersecting cyclically and \(K_n\) is a decreasing sequence of compact neighborhoods of \(X_1\cap X_N\), converging to \(X_1\cap X_N\) in the Hausdorff distance and such that \(K_1\cap\left(\bigcup_{j=2}^{N-1}X_j\right)=\emptyset\). Then, for a compact subset \(X\) of \(M\), the author introduces another invariant \(Pb_{X}(\alpha)\) which is defined through a modification of the invariant \(\mathrm{pb}_N\). In particular, in the new invariant the target of the tuple of functions on \(M\) is given by the unit ball in \(\mathbb{R}^2\) and moreover the homotopy class on \(X\) is given by \(\alpha\in H^1(X,\mathbb{Z})\). Then it is proven that \[ Pb_3(X_1,X_2,X_3)=Pb_X(\alpha), \] where \(X_1,X_2,X_3\) are compact subsets of \(M\) such that \(X_1\cap X_2\cap X_3=\emptyset\), \(X=X_1\cup X_2\cup X_3\) and \(\alpha\) is the class determined by \(X\) in the sense that \(\alpha=[f]\) with \(f\colon X\to S^1\) such that \(f\vert_{X_i}\subseteq \gamma_i\) where \(S^1=\gamma_1\cup\gamma_2\cup\gamma_3\) is a decomposition of the circle into three consecutive arcs ordered counterclockwise.