  The relation between manifold topology, observables and gauge group is
clarified on the basis of the classification of the representations of the
algebra of observables associated to positions and displacements on the
manifold. The guiding, physically motivated, principles are i) locality, i.e.
the generating role of the algebras localized in small, topological trivial,
regions, ii) diffeomorphism covariance, which guarantees the intrinsic
character of the analysis, iii) the exclusion of additional local degrees of
freedom with respect to the Schroedinger representation. The locally normal
representations of the resulting observable algebra are classified by unitary
representations of the fundamental group of the manifold, which actually
generate an observable, "topological", subalgebra. The result is confronted
with the standard approach based on the introduction of the universal covering
${\tilde{\cal M}}$ of $\cal{M}$ and on the decomposition of $L^2({\tilde{\cal
M}})$ according to the spectrum of the fundamental group, which plays the role
of a gauge group. It is shown that in this way one obtains all the
representations of the observables iff the fundamental group is amenable. The
implications on the observability of the Permutation Group in Particle
Statistics are discussed.
