Let $M_n$ be the connect sum of $n$ copies of $S^2 times S^1$. A classical theorem of Laudenbach says that the mapping class group $text{Mod}(M_n)$ is an extension of $text{Out}(F_n)$ by a group $(mathbb{Z}/2)^n$ generated by sphere twists. We prove
that this extension splits, so $text{Mod}(M_n)$ is the semidirect product of $text{Out}(F_n)$ by $(mathbb{Z}/2)^n$, which $text{Out}(F_n)$ acts on via the dual of the natural surjection $text{Out}(F_n) rightarrow text{GL}_n(mathbb{Z}/2)$. Our splitting takes $text{Out}(F_n)$ to the subgroup of $text{Mod}(M_n)$ consisting of mapping classes that fix the homotopy class of a trivialization of the tangent bundle of $M_n$. Our techniques also simplify various aspects of Laudenbachs original proof, including the identification of the twist subgroup with $(mathbb{Z}/2)^n$.
We give two proofs that appropriately defined congruence subgroups of the mapping class group of a surface with punctures/boundary have enormous amounts of rational cohomology in their virtual cohomological dimension. In particular we give bounds tha
t are super-exponential in each of three variables: number of punctures, number of boundary components, and genus, generalizing work of Fullarton-Putman. Along the way, we give a simplified account of a theorem of Harer explaining how to relate the homotopy type of the curve complex of a multiply-punctured surface to the curve complex of a once-punctured surface through a process that can be viewed as an analogue of a Birman exact sequence for curve complexes. As an application, we prove upper and lower bounds on the coherent cohomological dimension of the moduli space of curves with marked points. For $g leq 5$, we compute this coherent cohomological dimension for any number of marked points. In contrast to our bounds on cohomology, when the surface has $n geq1$ marked points, these bounds turn out to be independent of $n$, and depend only on the genus.
