The group of strictly contact homeomorphisms

Let $M$ be a smooth, closed and connected manifold carrying a cooriented contact structure. Based on the approach of S. M\"uller and Y.-G. Oh we construct a metric topology on the space of strictly contact isotopies of $M$ and use it to define a certain completion that we call the collection of topological strictly contact isotopies. This collection of isotopies is a topological group. The set of time-one maps of these isotopies also forms a topological group satisfying a transformation law that restricts to the usual transformation law for strictly contact diffeomorphisms. Suppose the contact form on $M$ is regular. We prove that the topological strictly contact isotopy associated to a topological contact Hamiltonian function is unique and consequently we extend the group laws for smooth basic contact Hamiltonian functions to the collection of topological contact Hamiltonian functions. We next prove the Buhovsky-Seyfaddini-Viterbo uniqueness theorem for contact Hamiltonians, completing the one-to-one correspondence between topological flows and Hamiltonians. Furthermore the group of strictly contact homeomorphisms is a central extension of the group of Hamiltonian homeomorphisms of the quotient of $M$ by the flow of the Reeb field of the regular contact form. Finally, using this $S^1$-extension, we prove the analogue of a theorem of M\"uller for Hamiltonian diffeomorphisms that says the group of strictly contact homeomorphisms is independent of a certain choice of norm used in the construction.