Rational homology 3-spheres and SL(2,$\mathbb{C}$) representations

Abstract

We use instanton gauge theory to prove that if $Y$ is a closed, orientable $3$-manifold such that $H_1(Y;\mathbb{Z})$ is nontrivial and either $2$-torsion or $3$-torsion, and if $Y$ is neither ${\#}^r{\mathbb{RP}}^3$ for some $r\ge 1$ nor $\pm L(3,1)$, then there is an irreducible representation ${\pi }_1(Y)\to \mathrm{SL}(2,\mathbb{C})$. 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 ${\mathbb{RP}}^3$, answering a conjecture of Przytycki (Kirby problem 1.92(F)) unless every summand but one is ${\mathbb{RP}}^3$. 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 $H_1(M)$, then $M$ admits a degree-1 map onto the twisted $I$-bundle over the Klein bottle.

70 pages, 11 figures

References in corpus (2)

• Instanton L-spaces and splicing
• 3-manifolds without any embedding in symplectic 4-manifolds