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We discuss the topology of the symmetry groups appearing in compactified(super-)gravity, and discuss two applications. First, we demonstrate that for 3dimensional sigma models on a symmetric space G/H with G non-compact and H themaximal compact subgroup of G, the possibility of oxidation to a higherdimensional theory can immediately be deduced from the topology of H. Second,by comparing the actual symmetry groups appearing in maximal supergravitieswith the subgroups of SL(32,R) and Spin(32), we argue that these groups cannotserve as a local symmetry group for M-theory in a formulation of de Wit-Nicolaitype.Comment: 18 pages, LaTeX, 1 figure, 2 table

The topology of U-duality (sub)groups