The group of Hamiltonian homeomorphisms and $C^0$ symplectic topology

The main purpose of this paper is to carry out some of the foundational study of $C^0$-Hamiltonian geometry and $C^0$-symplectic topology. We introduce the notions of the strong and the weak {\it Hamiltonian topology} on the space of Hamiltonian paths, and on the group of Hamiltonian diffeomorphisms. We then define the {\it group} $Hameo(M,\omega)$ and the space $Hameo^w(M,\omega)$ of {\it Hamiltonian homeomorphisms} such that $$Ham(M,\omega) \subsetneq Hameo(M,\omega) \subset Hameo^w(M,\omega) \subset Sympeo(M,\omega)$$ where $Sympeo(M,\omega)$ is the group of symplectic homeomorphisms. We prove that $Hameo(M,\omega)$ is a {\it normal subgroup} of $Sympeo(M,\omega)$ and contains all the time-one maps of Hamiltonian vector fields of $C^{1,1}$-functions. We prove that $Hameo(M,\omega)$ is path connected and so contained in the identity component $Sympeo_0(M,\omega)$ of $Sympeo(M,\omega)$. In the case of an orientable surface, we prove that the {\it mass flow} of any element from $Hameo(M,\omega)$ vanishes, which in turn implies that $Hameo(M,\omega)$ is strictly smaller than the identity component of the group of area preserving homeomorphisms when $M \neq S^2$. For the case of $S^2$, we conjecture that $Hameo(S^2,\omega)$ is still a proper subgroup of $Homeo^\omega_0(S^2) = Sympeo_0(S^2,\omega)$.