A topological rotary engine includes a first transmission mechanism, a second transmission mechanism, a valve mechanism, a rotor, and a cylinder. The rotor is arranged in an inner chamber of the cylinder. A cross section of the rotor is a curved-side topological polygon having n sides. A cross section of the inner chamber of cylinder is a curved-side topological polygon having n+1 sides, and n is an even number greater than or equal to 4. An outer topological curved surface of the rotor is meshed with an inner topological curved surface of the cylinder. The rotor reversely revolves around an axis of the cylinder with an eccentricity as a radius while rotating, and divides the cylinder into n+1 independent chambers. The cylinder is provided with n+1 fuel injection nozzles and n+1 spark plugs, which cooperate with the rotor and the valve mechanism.