Work

Essential loops in completions of Hamiltonian groups

Year: 2025

Type: article

Abstract: We initiate the study of the fundamental group of natural completions of the group of Hamiltonian diffeomorphisms, namely its C 0 -closure Ham ¯ ( M , ω ) in the set of all homeomorphisms, and its completion with respect to the spectral norm Ham ^ ( M , ω ) . We prove that in some situations, namely complex projective spaces and rational Hirzebruch surfaces, certain Hamiltonian loops that were known to be non-trivial in π 1 ( Ham ( M , ω ) ) remain non-trivial in π 1 ( Ham ^ ( M , ω ) ) . This yields in particular cases, including ℂ P 2 and the monotone S 2 × S 2 , the injectivity of the map π 1 ( Ham ( M , ω ) ) → π 1 ( Ham ^ ( M , ω ) ) induced by the inclusion. The same results hold for the Hofer completion of Ham ( M , ω ) . Moreover, whenever the spectral norm is known to be C 0 -continuous, they also hold for Ham ¯ ( M , ω ) . Our method relies on computations of the valuation of Seidel elements and hence of the spectral norm on π 1 ( Ham ( M , ω ) ) . Some of these computations were known before, but we also present new ones which might be of independent interest. For example, we show that the spectral pseudo-norm is degenerate when ( M ,

Source: Annales de la faculté des sciences de Toulouse Mathématiques

DOI: 10.5802/afst.1833

Authors: Vincent Humilière, Alexandre Jannaud, Rémi Leclercq