  We examine the topology of the subset of controls taking a given initial
state to a given final state in quantum control, where "state" may mean a pure
state |\psi>, an ensemble density matrix \rho, or a unitary propagator U(0,T).
The analysis consists in showing that the endpoint map acting on control space
is a Hurewicz fibration for a large class of affine control systems with vector
controls. Exploiting the resulting fibration sequence and the long exact
sequence of basepoint-preserving homotopy classes of maps, we show that the
indicated subset of controls is homotopy equivalent to the loopspace of the
state manifold. This not only allows us to understand the connectedness of
"dynamical sets" realized as preimages of subsets of the state space through
this endpoint map, but also provides a wealth of additional topological
information about such subsets of control space.
