  A unique classification of the topological effects associated to quantum
mechanics on manifolds is obtained on the basis of the invariance under
diffeomorphisms and the realization of the Lie-Rinehart relations between the
generators of the diffeomorphism group and the algebra of infinitely
differentiable functions on the manifold. This leads to a unique
("Lie-Rinehart") C* algebra as observable algebra; its regular representations
are shown to be locally Schroedinger and in one to one correspondence with the
unitary representations of the fundamental group of the manifold. Therefore, in
the absence of spin degrees of freedom and external fields, the first homotopy
group of the manifold appears as the only source of topological effects.
