  This paper provides a topological interpretation for number theoretic
properties of quantum invariants of 3-manifolds. In particular, it is shown
that the p-adic valuation of the quantum SO(3)-invariant of a 3-manifold M, for
odd primes p, is bounded below by a linear function of the mod p first betti
number of M. Sharper bounds using more delicate topological invariants related
to Massey products are given as well.
