  We show that pairing in an ultracold Fermi gas under
spin-orbital-angular-momentum coupling (SOAMC) can acquire topological
characters encoded in the quantized angular degrees of freedom. The resulting
topological superfluid is the angular analog of its counterpart in a
one-dimensional Fermi gas with spin-orbit coupling, but characterized by a Zak
phase defined in the angular-momentum space. Upon tuning the SOAMC parameters,
a topological phase transition occurs, which is accompanied by the closing of
the quasiparticle excitation gap. Remarkably, a topological vortex state can
also be stabilized by deforming the Fermi surface, which is topologically
non-trivial in both the coordinate and angular-momentum space, offering
interesting potentials for applications in quantum information and quantum
control. We discuss how the topological phase transition and the exotic vortex
state can be detected experimentally.
