  A solenoidal manifold is the inverse limit space of a tower of proper
coverings of a compact manifold. In this work, we introduce new invariants for
solenoidal manifolds, their asymptotic Steinitz orders and their prime spectra,
and show they are invariants of the homeomorphism type. These invariants are
formulated in terms of the monodromy Cantor action associated to a solenoidal
manifold. To this end, we continue our study of invariants for minimal
equicontinuous Cantor actions. We introduce the three types of prime spectra
associated to such actions, and study their invariance properties under return
equivalence. As an application, we show that a nilpotent Cantor action with
finite prime spectrum must be stable. Examples of stable actions of the integer
Heisenberg group are given with arbitrary prime spectrum. We also give the
first examples of nilpotent Cantor actions which are wild.
