Arnold's conjecture and symplectic reduction (Q679524)

In the seventies V. I. Arnold conjectured that any exact symplectomorphism of a closed symplectic manifold has at least as many fixed points as the minimal number of critical points for a smooth real-valued function on the manifold. This was proved for several classes of manifolds. In particular, \textit{B. Fortune} [Invent. Math. 81, 29-46 (1985; Zbl 0566.58007)] proved it for \(\mathbb{C} P^n\). Here the authors show that Fortune's approach is universal in the sense that it is possible to translate Arnold's conjecture on any closed symplectic manifold to a critical point problem with symmetry on loops in \(\mathbb{R}^{2n}\) with its standard symplectic structure.