In this paper, acoustic wave propagation in a two- or three-dimensional phononic crystal consisting of Helmholtz resonators embedded in a fluid matrix is studied. The band structures are calculated to discuss the influence of the geometry topology of Helmholtz resonators on the bandgap characteristics. It is shown that a narrow bandgap will appear in the lower frequency range due to the resonance of the Helmholtz resonators. The width and position of this resonance bandgap can be tuned by adjusting the geometrical parameters of the Helmholtz resonator. The position of the resonance bandgap can be evaluated by the resonance frequency of the Helmholtz resonator. A decrease in the size of the opening generally results in a lower position and a smaller width of the bandgap. The system with one opening exhibits a wider bandgap in a lower position than the system with two openings.