  We discuss some consequences of the fact that symmetry groups appearing in
compactified (super-)gravity may be non-simply connected. The possibility to
add fermions to a theory results in a simple criterion to decide whether a
3-dimensional coset sigma model can be interpreted as a dimensional reduction
of a higher dimensional theory. Similar criteria exist for higher dimensional
sigma models, though less decisive. Careful examination of the topology of
symmetry groups rules out certain proposals for M-theory symmetries, which are
not ruled out at the level of the algebra's. We conclude with an observation on
the relation between the ``generalized holonomy'' proposal, and the actual
symmetry groups resulting from E_10 and E_11 conjectures.
