On the Birman exact sequence of the subgroups of the mapping class group of genus three

We prove that for any finite index subgroup of the mapping class group containing the Johnson subgroup, the profinite Birman exact sequence does not split in genus $g\ge 3$, extending prior results of Hain and the second author for $g\ge 4$. For the Torelli group, we prove that the graded Lie algebra version of the Birman exact sequence admits no section with symplectic equivariance, extending Hain’s result from $g\ge 4$ to $g=3$. These results are deduced by our main tool, relative completion, with the help of Hodge theory and representation theory of symplectic groups, along with explicit structural obstructions coming from hyperelliptic mapping class groups.