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　nontrivial　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.　©　2022　authors.　Published　by　the　American　Physical　Society.