Injectivity and image of the mapping-cylinder homomorphism

Determine, for every degree kโ‰ฅ1, whether the Sp(H)-homomorphism Gr_k๐–ผ: ฮ“_k๐“˜/ฮ“_{k+1}๐“˜ โ†’ Y_k๐“˜๐“’/Y_{k+1}๐“˜๐“’ induced by the mapping-cylinder construction is injective, and determine its image.

Background

The mapping-cylinder construction embeds the Torelli group into the monoid of homology cylinders and induces a graded homomorphism between the corresponding Lie algebras.

A complete understanding requires not only the module structures of the source and target graded pieces but also a determination of whether the graded mapping-cylinder map loses information and, if so, an explicit description of its image. The paper later notes that injectivity remains unaddressed in degree three.

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] \begin{enumerate} \item Describe both $\Gamma_k \mathcal{I} / \Gamma_{k+1} \mathcal{I}$ and $Y_k \mathcal{IC} / Y_{k+1}$ as $Sp(H)$-modules.-0.3cm] \item Decide whether the $Sp(H)$-homomorphism $Gr_k \mathbf{c}:\Gamma_k \mathcal{I} / \Gamma_{k+1} \mathcal{I} \to Y_k \mathcal{IC} / Y_{k+1}$ induced by~eq:mcc is injective, and determine its image.-0.3cm] \end{enumerate}

— 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 โ€œWith integral coefficientsโ€