Y-equivalence classification of homology cylinders

Prove that two homology cylinders over a compact connected oriented surface are diffeomorphic if and only if they are Y_k-equivalent for every k≥1, equivalently, establish that finite-type invariants classify homology cylinders up to diffeomorphism.

Background

The paper studies the Y-filtration on the monoid of homology cylinders and its associated graded Lie algebra. The filtration is intended to play a role analogous to the lower central series of the Torelli group.

The proposed classification would assert that the successive Y-equivalence relations detect all distinctions between homology cylinders, paralleling the residual nilpotence of the Torelli group and linking finite-type invariants with diffeomorphism classification.

References

Conjecturally, it is expected that two homology cylinders are diffeomorphic if and only if they are $Y_k$-equivalent for every $k \ge 1$, which would parallel the residual nilpotency of~$\mathcal{I}$. This conjecture is equivalent to asserting that finite-type invariants classify homology cylinders up to diffeomorphism.

— On the Sp-structure of the torsion of the Lie algebra of homology cylinders  (2609.26409 - Faes et al., 22 Sep 2026) in Section 1, Introduction