Chillingworth-homomorphism analogue for the Torelli subgroup of an outer automorphism group

Determine whether there is a meaningful analogue of the Chillingworth homomorphism for the Torelli subgroup \(IO_r\leq \operatorname{Out}(F_r)\), as suggested by the color-permutation homomorphism arising from the uniquely \((r-1)\)-colorable graph of spheres representing a fixed primitive homology class.

Background

The paper proves that the graph of spheres in a fixed primitive homology class of $H_2(M_r;\mathbb Z)$ is uniquely $(r-1)$-colorable. This induces a nontrivial homomorphism from $IO_r$, the kernel of the action of $\operatorname{Out}(F_r)$ on homology, to the symmetric group $S_{r-1}$. In the surface setting, an analogous coloring action of the Torelli group is identified with the Chillingworth homomorphism.

The authors explicitly leave unresolved whether the homomorphism obtained in the free-group setting has a comparably meaningful interpretation. The problem is motivated by extending the surface/free-group analogy, and the paper states that pursuing this direction will form the subject of future work.