Generalization of Weinstein's Morphism

TL;DR

This work generalizes Weinstein's morphism to higher homotopy groups of the Hamiltonian diffeomorphism group by defining $\mathcal{A}: \pi_{2k-1}(\mathrm{Ham}(M,\omega)) \to \mathbb{R}/\mathcal{P}_{2k}(M,\omega)$ for $2 \le k \le n$, interpreting $\mathcal{A}$ as an averaged $\omega^k$-volume associated to traces of $(2k-1)$-spheres. It proves nontriviality of $\pi_{2k-1}(\mathrm{Ham}(\mathbb{C}P^n,\omega_{FS}))$ for all $1 \le k \le n$ by constructing explicit generators from unitary actions and a dense symplectic ball in $\mathbb{C}P^n$, with a concrete formula for $\mathcal{A}$ on these generators. The paper also extends these results to Cartesian products and to symplectic blow-ups: products yield additive behavior of $\mathcal{A}$, and lifts to the blow-up $(\widetilde{\mathbb{C}P}^n, \widetilde{\omega}_{\rho})$ produce elements of infinite order in $\pi_{2k-1}$ for transcendental weights $\rho$, evidenced by a polynomial obstruction in $\rho$ that cannot vanish. These results deepen our understanding of higher-degree obstructions to triviality in Hamiltonian diffeomorphism groups and offer a pathway to detect nontrivial higher homotopy via Weinstein-type invariants.

Abstract

We introduce a generalization of Weinstein's morphism, defined on π_{2k-1}(Ham(M,ω)) for 1 < k \leq n, where (M,ω) is a 2n-dimensional symplectic manifold. Using this morphism, we show that for n > 1 and 1 < k \leq n, the homotopy groups π_{2k-1}(Ham(CP^n,ω_{FS})) and π_{2k-1}(Ham(\tilde CP^n,\tildeω_ρ)) are nontrivial. Here, (\tilde CP^n,\tildeω_ρ) denotes the symplectic one-point blow-up of (CP^n,ω_FS) of weigh ρ.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.4
• proof
• Proposition 2.5
• proof
• Lemma 3.6
• Lemma 3.7
• proof