JOURNAL OF THE KOREAN MATHEMATICAL SOCIETY, v.53, no.4, pp.195 - 834
Publisher
KOREAN MATHEMATICAL SOC
Abstract
The main purpose of this paper is to propose a scheme of a proof of the nonsimpleness of
the group HomeoΩ(D2 , ∂D2 ) of area pre- serving homeomorphisms of the 2-disc D2. We first
establish the exis- tence of Alexander isotopy in the category of Hamiltonian homeomor- phisms.
This reduces the question of extendability of the well-known Calabi homomorphism Cal : DiffΩ(D1 ,
∂D2) → R to a homomorphism Cal : Hameo(D2 , ∂D2 ) → R to that of the vanishing of the basic phase
function fF, a Floer theoretic graph selector constructed in [9], that is associated to the graph
of the topological Hamiltonian loop and its nor- malized Hamiltonian F on S2 that is obtained via
the natural embedding D2 ֒→ S2. Here Hameo(D2, ∂D2 ) is the group of Hamiltonian homeo- morphisms
introduced by Mu¨ller and the author [18]. We then provide an evidence of this vanishing conjecture
by proving the conjecture for the special class of weakly graphical topological Hamiltonian loops on D2 via
the associated Hamiton-Jacobi equation.
c 2016 Korean Mathematical Society