Small Heegaard genus and SU(2)

2023/09/18 by Baldwin, John A., Sivek, Steven · 2 citations
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2309.09780

Abstract

Let Y be a closed, orientable 3-manifold with Heegaard genus 2. We prove that if H₁(Y;ℤ) has order 1, 3, or 5, then there is a representation π₁(Y) → SU(2) with non-abelian image. Similarly, if H₁(Y;ℤ) has order 2 then we find a non-abelian representation π₁(Y) → SO(3). We also prove that a knot K in S³ is a trefoil if and only if there is a unique conjugacy class of irreducible representations π₁(S³∖ K) → SU(2) sending a fixed meridian to (\beginsmallmatrixi&0 0&-i\endsmallmatrix).

Cited by

• Representation varieties and genus-three Torelli maps
• Toroidal graph manifolds with small homology are not SU(2)-abelian

Related

• Heegaard genus of the connected sum of m-small knots
• Comparing Heegaard and JSJ structures of orientable 3-manifolds
• Small 3-manifolds of large genus
• Toroidal graph manifolds with small homology are not SU(2)-abelian
• SU(2)-abelian graph manifolds with a single JSJ torus