Some results of Hamiltonian homeomorphisms on closed aspherical surfaces (Q2196618)

A symplectic manifold \((M,\omega)\) is said to be symplectically aspherical if \(\omega\) and \(c_1\), the first Chern class of \(M\), both vanish on \(\pi_2(M)\). The famous Gromov-Eliashberg Theorem, claiming that the group of symplectic diffeomorphisms is \(C^0\)-closed in the full group of diffeomorphisms, makes the authors interested in defining a symplectic homeomorphism as a homeomorphism which is a \(C^0\)-limit of symplectic diffeomorphisms. This becomes a central theme of what is now called \(C^0\)-symplectic topology that has been studied extensively. On closed symplectically aspherical manifolds, \textit{M. Schwarz}, [Pac. J. Math. 193, No. 2, 419--461 (2000; Zbl 1023.57020)] proved a classical result: the action function of a nontrivial Hamiltonian diffeomorphism is not constant. In this article, the author generalises Schwarz's theorem to the \(C^0\)-case on closed aspherical surfaces. As an application, the author proves that the contractible fixed points set (and consequently the fixed points set) of a nontrivial Hamiltonian homeomorphism is not connected. The author also obtains a similar result for an orientation-preserving nonwandering point homeomorphism of the two-sphere. In the end, the author gives further applications based on the \(C^0\)-Schwarz Theorem.