  The closed, k=1, FRW model coupled to a massless scalar field is investigated
in the framework of loop quantum cosmology using analytical and numerical
methods. As in the k=0 case, the scalar field can be again used as emergent
time to construct the physical Hilbert space and introduce Dirac observables.
The resulting framework is then used to address a major challenge of quantum
cosmology: resolving the big-bang singularity while retaining agreement with
general relativity at large scales. It is shown that the framework fulfills
this task. In particular, for states which are semi-classical at some late
time, the big-bang is replaced by a quantum bounce and a recollapse occurs at
the value of the scale factor predicted by classical general relativity. Thus,
the `difficulties' pointed out by Green and Unruh in the k=1 case do not arise
in a more systematic treatment. As in k=0 models, quantum dynamics is
deterministic across the deep Planck regime. However, because it also retains
the classical recollapse, in contrast to the k=0 case one is now led to a
cyclic model. Finally, we clarify some issues raised by Laguna's recent work
addressed to computational physicists.
