Quantitative Heegaard Floer cohomology and the Calabi invariant

On two remarkable groups of area-preserving homeomorphisms

“…If (ϕ t ) is a path of homeomorphisms of S 2 given by Theorem 1.4, then by the item (1) from the theorem, the time-1 map ϕ = ϕ 1 is an element of FHomeo(S 2 ). Both ( 1) and (2) imply that on the one hand we have ϕ H = E, but on the other hand the d H -distance of ϕ to any element of Ham(S 2 ) is greater than or equal to E. The latter property yields ϕ / ∈ Hameo(S 2 ). Indeed, assuming the contrary, that is ϕ ∈ Hameo(S 2 ), we obtain a sequence…”

“…Recently, a number of breakthrough works addressed the Fathi question, where it was first solved in the case of a two-disc [1], then in the two-sphere case [3,9], and finally for general surfaces of finite type and finite area [2]. Moreover, the works [3,9] have largely contributed to Hofer geometry, in particular solving the Polterovich-Kapovich question.…”

“…We denote by d H the Hofer distance on Ham(M, ω), that is, d H (φ, ψ) = φ −1 ψ H . Now we recall the definitions of the groups Ham(M, ω) (Hamiltonian homeomorphisms), Hameo(M, ω) (strong Hamiltonian homeomorphisms [8]) and FHomeo(M, ω) (finite energy Hamiltonian homeomorphisms [1,3,2]). Definition 1.1.…”

“…In that case the time-1 map φ 1 is called a strong Hamiltonian homeomorphism of M, and the set of all such homeomorphisms is denoted by 1 Hameo(M, ω). It is well known [8,2] that Hameo(M, ω) ⊂ FHomeo(M, ω) are normal subgroups of Ham(M, ω). For symplectic surfaces (M, ω) of finite type and area it was shown [1,3,9,2] that FHomeo(M, ω) is in fact a proper normal subgroup of Ham(M, ω).…”

“…It is well known [8,2] that Hameo(M, ω) ⊂ FHomeo(M, ω) are normal subgroups of Ham(M, ω). For symplectic surfaces (M, ω) of finite type and area it was shown [1,3,9,2] that FHomeo(M, ω) is in fact a proper normal subgroup of Ham(M, ω). This answers the Fathi question to the negative.…”