6 problems

Let $Y$ be a closed oriented 3-manifold satisfying … If $Y$ is not $S^3$, then Kirby's $SU(2)$ representation conjecture. there exists a homomorphism … with non-abelian image. This…

Let $K\subset S^3$ be a knot. The $1$-surgery on $K$ is the closed $3$-manifold $S^3_1(K)$, and a closed $3$-manifold is $SU(2)$-abelian if its fundamental group admits no irreduci…

Let $Y$ be a toroidal rational homology $3$-sphere. The refined toroidal small-homology conjecture. If $|H_1(Y;\mathbb{Z})| \le 6$, then $Y$ is not $SU(2)$-abelian. This strengthen…

Let $Y$ be a rational homology $3$-sphere. The SU(2)-abelian implies Heegaard Floer L-space conjecture. If $Y$ is $SU(2)$-abelian, then it is a Heegaard Floer L-space. Here, a Heeg…

Let $Y$ be a toroidal rational homology $3$-sphere. The toroidal small-homology conjecture. If $|H_1(Y;\mathbb{Z})| \le 4$, then $Y$ is not $SU(2)$-abelian. This conjecture predict…

SU(2) representation conjecture. If $Y\not\cong S^3$, then there is a nontrivial homomorphism