Integral symplectic-module structure of the Torelli and homology-cylinder Lie algebras

Determine, for every degree kβ‰₯1, the structures of the integral symplectic-group modules Ξ“_kπ“˜/Ξ“_{k+1}π“˜ and Y_kπ“˜π“’/Y_{k+1}π“˜π“’.

Background

The paper explains that rational descriptions of the Torelli Lie algebra and the Lie algebra of homology cylinders are comparatively well developed, whereas integral coefficients introduce torsion phenomena, particularly 2-torsion.

The unresolved task is to give a complete Sp(H)-module description in every degree, including the torsion submodules. The degree-one and degree-two cases are cited as understood, but the general-degree problem remains.

References

With integral coefficients, the principal unresolved issues regarding the Torelli Lie algebra and the Lie algebra of homology cylinders can be formulated as follows, in each degree $k \geq 1$:-0.3cm]

— 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, paragraph beginning β€œWith integral coefficients”