  Symmetry plays a key role in modern physics, as manifested in the
revolutionary topological classification of matter in the past decade. So far,
we seem to have a complete theory of topological phases from internal
symmetries as well as crystallographic symmetry groups. However, an intrinsic
element, i.e., the gauge symmetry in physical systems, has been overlooked in
the current framework. Here, we show that the algebraic structure of crystal
symmetries can be projectively enriched due to the gauge symmetry, which
subsequently gives rise to new topological physics never witnessed under
ordinary symmetries. We demonstrate the idea by theoretical analysis, numerical
simulation, and experimental realization of a topological acoustic lattice with
projective translation symmetries under a $Z_2$ gauge field, which exhibits
unique features of rich topologies, including a single Dirac point, M\"{o}bius
topological insulator and graphene-like semimetal phases on a rectangular
lattice. Our work reveals the impact when gauge and crystal symmetries meet
together with topology, and opens the door to a vast unexplored land of
topological states by projective symmetries.
