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 $Hame{o}^w(M,\omega )$ of {\it Hamiltonian homeomorphisms} such that $$Ham(M,\omega )⊊Hameo(M,\omega )\subset Hame{o}^w(M,\omega )\subset Sympeo(M,\omega )$$ 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 $Sympe{o}_0(M,\omega )$ 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 $M\mathrm{\ne }{S}^2$. For the case of $S^2$, we conjecture that $Hameo(S^2,\omega )$ is still a proper subgroup of $Homeo^\omega_0(S^2) = Sympeo_0(S^2,\omega)$.