3.8 (Agol) — Twisted homology products after cutting along a surface

Definitions and mathematical statement

Let $M$ be a $3$-manifold, and let $\phi:\pi_1(M)\to{\mathbb{Z}}$ be dual to a properly embedded cooriented surface $(\Sigma,\partial\Sigma)\subset(M,\partial M)$. In the customary compact oriented setting this means that the cohomology class of $\phi$ is Poincaré--Lefschetz dual to $[\Sigma]\in H_2(M,\partial M;{\mathbb{Z}})$. Let $N$ be the manifold obtained by cutting along $\Sigma$, with the two resulting boundary copies $\Sigma_+$ and $\Sigma_-$.

For $\alpha:\pi_1(M)\to\operatorname{SL}_2({\mathbb{C}})$, let $E_\alpha$ be the rank-two complex local system associated with its standard representation, restricted to $N$ and its boundary copies. Say that $N$ is an $\alpha$-twisted homology product if both inclusions induce isomorphisms \[ H_i(\Sigma_\pm;E_\alpha)\longrightarrow H_i(N;E_\alpha) \quad\text{for every }i, \] equivalently $H_i(N,\Sigma_\pm;E_\alpha)=0$ for all $i$ and both signs.

Agol's question, item 3.8, asks to characterize exactly those $\alpha\in\operatorname{Hom}(\pi_1(M),\operatorname{SL}_2({\mathbb{C}}))$ for which this holds. The excerpt does not impose connectedness, primitivity or nonvanishing of $\phi$, or norm-minimizing/taut conditions on $\Sigma$; such conditions cannot be silently added.

The source says that the Dunfield--Friedl--Jackson conjecture would imply the property when $M$ is hyperbolic and $\alpha$ is discrete faithful. This is a supplied conditional expectation whose precise hypotheses, especially on the dual surface, are absent here; it is not an unconditional assertion for every surface dual to $\phi$.

Short English statement

For a manifold cut along a surface dual to an integral cohomology class, determine which two-dimensional complex representations make both boundary copies of the surface twisted-homology equivalent to the cut manifold.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10900016$, exact source code catalogue entry, attributed to Agol.
• Delp, Hoffoss and Manning, Problems In Groups, Geometry, and Three-Manifolds (2015), subsection 3.8, PDF page 3: https://arxiv.org/pdf/1512.04620#page=3; abstract record: https://arxiv.org/abs/1512.04620. Supplied access date: 2026-07-29; source-tex extraction checked against rendered PDF text by the source record.
• The source invokes the Dunfield--Friedl--Jackson conjecture and the registry mentions twisted-Alexander/homology follow-ups, without supplying exact papers or the full conditional statement.