  Gravitational theta-sectors are investigated in spatially locally homogeneous
cosmological models with flat closed spatial surfaces in 2+1 and 3+1 spacetime
dimensions. The metric ansatz is kept in its most general form compatible with
Hamiltonian minisuperspace dynamics. Nontrivial theta-sectors admitting a
semiclassical no-boundary wave function are shown to exist only in 3+1
dimensions, and there only for two spatial topologies. In both cases the
spatial surface is nonorientable and the nontrivial no-boundary theta-sector
unique. In 2+1 dimensions the nonexistence of nontrivial no-boundary
theta-sectors is shown to be of topological origin and thus to transcend both
the semiclassical approximation and the minisuperspace ansatz. Relation to the
necessary condition given by Hartle and Witt for the existence of no-boundary
theta-states is discussed.
