Lifting Hamiltonian loops to isotopies in fibrations (Q2865245)

This paper studies the homotopy of Lie subgroups \(\mathcal G\) of \(\mathrm{Diff}(M)\), the diffeomorphism group of a homogeneous space or a symplectic toric manifold \(M\).NEWLINENEWLINEIn particular, given a Lie group \(G\) and a closed subgroup \(H\), let \(M=G/H\). Each representation \(\Psi\) of \(H\) determines a \(G\)-equivariant principal bundle \({\mathcal P}\) on \(M\) endowed with a \(G\)-invariant connection. For any subgroup \({\mathcal G}\) of \(\text{Diff}(M)\) such that each vector field \(Z\in\mathrm{Lie}({\mathcal G})\) admits a lift to a vector field on \(\mathcal P\) preserving the connection the author proves that \(\#\,\pi_1({\mathcal G})\geq \#\,\Psi(Z(G))\).