Kirby Problem 3.52

Definitions and mathematical statement

For a connected closed three-manifold $M$, a unitary rank-two representation is a homomorphism \[ \rho:\pi_1(M)\longrightarrow\operatorname{SU}(2),\qquad \operatorname{SU}(2)=\{A\in M_2(\mathbb C):A^*A=I,\ \det A=1\}. \] It is nontrivial if some group element has image different from $I$. It has abelian image if $\rho(g)\rho(h)=\rho(h)\rho(g)$ for all $g,h\in\pi_1(M)$. Nontriviality does not require irreducibility or a nonabelian image.

(a) Does every closed connected three-manifold other than $S^3$ admit a nontrivial such representation? The homomorphism may have a kernel and need not distinguish all elements of the fundamental group. Basepoint changes conjugate the input group and do not affect its existence.

(b) Classify the closed three-manifolds with nonabelian fundamental group for which every homomorphism to $\operatorname{SU}(2)$ has abelian image. Equivalently, for every representation all image matrices can be simultaneously conjugated into a maximal torus of $\operatorname{SU}(2)$, the subgroup of diagonal matrices $\operatorname{diag}(z,\bar z)$ with $|z|=1$.

These questions are compatible: a manifold counted in (b) can still have nontrivial abelian representations answering (a). Conversely, replacing the target by $\operatorname{SL}_2(\mathbb C)$ changes the question, since its noncompact representations need not be conjugate into the unitary subgroup. No geometric decomposition or homology restriction is imposed on the main domain.

Short English statement

Do all non-spherical closed three-manifolds have nontrivial SU(2) representations, and which nonabelian groups only have abelian SU(2) images?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2850 (KP-3.52), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 3.52. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.