  We construct a realization of the algebra of the Z_3-graded topological
symmetry of type (1,1,1) in terms of a pair of operators D_1: H_1 -> H_2, and
D_2: H_2 -> H_3 satisfying [D_1D_1^\dagger,D_2^\dagger D_2]=0. We show that the
sequence of the restriction of these operators to the zero-energy subspace
forms a complex and establish the equality of the corresponding topological
invariants with the analytic indices of these operators.
