  Most properties of acoustic structures derive from their material and
geometry. Few are explicitly related to the real-space topology of the
structures. Here, we discover a fundamental connection between the real-space
topology of acoustic structures and the topological properties of acoustic
fields. We find that the genus of acoustic cavities can protect the birth of
acoustic polarization singularities and give rise to chiral velocity fields
with nonzero spin density. These polarization singularities are characterized
by an integer or half-integer topological index. For a cavity with smooth and
hard boundaries, the total index of the surface singularities must be equal to
the Euler characteristic of the cavity and is decided solely by the number of
"holes" in the cavity surface, guaranteed by the Poincare-Hopf theorem. This
intriguing property is irrelevant to the specific material, geometric details,
or excitation properties. Besides, the polarization singularities can give rise
to nontrivial polarization Mobius strips and skyrmion textures of velocity
field with rich spatial and temporal properties. The study uncovers the
intrinsic effects of real-space topology on acoustic field properties and
enables sound manipulation via a new degree of freedom, i.e., structural
topology. The results can find applications in chiral sound-matter interactions
and acoustic manipulation of small particles.

Applied Physics