We use instanton gauge theory to prove that if Y is a closed,
orientable 3-manifold such that H1(Y ; Z) is nontrivial and either 2-
torsion or 3-torsion, and if Y is neither #rRP3
for some r ≥ 1 nor
±L(3, 1), then there is an irreducible representation π1(Y ) → SL(2, C).
We apply this to show that the Kauffman bracket skein module of a
non-prime 3-manifold has nontrivial torsion whenever two of the prime
summands are different from RP3
, answering a conjecture of Przytycki
(Kirby problem 1.92(F)) unless every summand but one is RP3
. As part
of the proof in the 2-torsion case, we also show that if M is a compact,
orientable 3-manifold with torus boundary whose rational longitude has
order 2 in H1(M), then M admits a degree-1 map onto the twisted
I-bundle over the Klein bottle.