| The purpose of this dissertation is to carry out a foundational study of C0-Hamiltonian geometry and C0-symplectic topology. After a brief review of some important results in symplectic topology that motivate the study of C0-symplectic topology, we outline our general approach to a topological complement to Hamiltonian geometry and symplectic topology. We show that Hamiltonian geometry and symplectic topology 'extend nicely' to homeomorphisms.;We define the group Sympeo(M, o) of symplectic homeomorphisms, and study its relation to measure-preserving homeomorphisms. We define the Hamiltonian topology on the space of Hamiltonian paths and on the group Ham(M, o) of Hamiltonian diffeomorphisms, and study the completion of the space of Hamiltonian paths with respect to the Hamiltonian metric. We also define and study the spaces of topological Hamiltonian paths and topological Hamiltonian functions, and the group of Hamiltonian homeomorphisms, denoted by Hameo(M, o). We provide many evidences for our thesis that Hameo(M, o) ⊂ Sympeo(M, o) is a 'good' generalization of Hamiltonian and symplectic diffeomorphisms.;Among many other results, we prove that Hameo(M, o) is a normal subgroup of Sympeo(M, o), and that Hameo( M, o) is path connected and so contained in the identity component Sympeo0(M, o) of Sympeo(M, o). We show that the spaces of topological Hamiltonian paths and functions contain non smooth elements, and that Ham(M, o) ⊊ Hameo(M, o) ⊂ Sympeo0( M, o). We consider many cases in which Hameo(M, o) is a proper subgroup of Sympeo0(M, o). For this purpose, we review the mass flow homemorphism for measure-preserving homemorphisms and the flux homemorphisms for symplectic and volume-preserving diffeomorphisms. This leads to a conjecture on the simpleness of the group of area-preserving homeomorphisms of the two sphere. We show that the two groups of Hamiltonian homeomorphisms arising from two different norms used in its construction coincide, and that the corresponding 'Hofer norms' for Hamiltonian homeomorphisms coincide as well. |
