  We give the definition and explore the algebraic structure of a class of
quantum symmetries, called topological symmetries, which are generalizations of
supersymmetry in the sense that they involve topological invariants similar to
the Witten index. A topological symmetry (TS) is specified by an integer n>1,
which determines its grading properties, and an n-tuple of positive integers
(m_1,m_2,...,m_n). We identify the algebras of supersymmetry, p=2
parasupersymmetry, and fractional supersymmetry of order n with those of the
Z_2-graded TS of type (1,1), Z_2-graded TS of type (2,1), and Z_n-graded TS of
type (1,1,...,1), respectively. We also comment on the mathematical
interpretation of the topological invariants associated with the Z_n-graded TS
of type (1,1,...,1). For n=2, the invariant is the Witten index which can be
identified with the analytic index of a Fredholm operator. For n>2, there are n
independent integer-valued invariants. These can be related to differences of
the dimension of the kernels of various products of n operators satisfying
certain conditions.
