Rational Homology 3-Spheres and SL(2,ℂ) Representations

We use instanton gauge theory to prove that if Y Y is a closed, orientable 3 3 -manifold such that H 1 ( Y ; Z ) H₁(Y;Z) is nontrivial and either 2 2 -torsion or 3 3 -torsion, and if Y Y is neither # r R P 3 #ʳ RP³ for some r ≥ 1 r≥ 1 nor ± L ( 3 , 1 ) ± L(3,1) , then there is an irreducible representation π 1 ( Y ) → SL ⁡ ( 2 , C ) π ₁(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 R P 3