  Artificial structures have been widely used to manipulate sound fields. Most
properties of these structures derive from the material and geometry. Few are
explicitly related to the structural topology in the real space. Here, we
discover a fundamental connection between the real-space topology of acoustic
structures and the topological properties of sound fields. We find that the
genus of acoustic cavities can protect the birth of singularities in velocity
polarization and isopressure line fields. The total topological index of the
surface singularities is equal to the cavities' Euler characteristic. This
intriguing relationship is rooted in the Poincare-Hopf theorem and is
irrelevant to the specific material, geometric details, or excitation
properties. The isopressure line singularities lead to acoustic hotspots and
quiet zones. The velocity polarization singularities give rise to nontrivial
polarization Mobius strips and skyrmion textures. The results enable sound
sculpturing with structural topology and can find applications in acoustic
communications, acoustic sensing, and noise control.
