Generalization of Weinstein's Morphism

TL;DR

This paper generalizes Weinstein's morphism to higher homotopy groups of Hamiltonian diffeomorphisms on symplectic manifolds, demonstrating nontriviality of certain homotopy groups for complex projective spaces and their blow-ups.

Contribution

It introduces a new generalized Weinstein's morphism applicable to higher homotopy groups, revealing nontrivial homotopy groups in specific symplectic manifolds.

Findings

Nontrivial homotopy groups rac{2k-1}{ ext{th}} of Ham(CP^n) and its blow-up.

Generalization of Weinstein's morphism to higher homotopy groups.

Application to symplectic manifolds with explicit nontrivial homotopy groups.

Abstract

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