Date of Graduation

7-2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor/Mentor

Clay, Matthew

Committee Member

Morris, Jeremy Van-Horn

Second Committee Member

Day, Matthew

Third Committee Member

Rieck,Yoav

Keywords

Differential Geometry; Geometric Group Theory; Manifold and Cell Complexes; Topology

Abstract

Laudenbach proved that the mapping class group of the connected sum of $n$ copies of $S^2 \times S^1$ is an extension of $\text{Out}(F_n)$ by a finite group. Brendle, Broaddus and Putman proved that this exact sequence splits. We provide an explicit section $s$ of this split exact sequence. Given a locally finite graph $\Gamma$, Udall proved that the mapping class group of the doubled handlebody associated to $\Gamma$ is an extension of $\map(\Gamma)$ by a possibly infinite direct product of $\Z_2$, and proved that this exact sequence splits. We provide an explicit formula for this section $s$ of $\Psi$ restricted to $\pmap(\Gamma)$, and in the case that $E(\Gamma) < \infty$, we provide a formula for a section $s : \map(\Gamma)\xrightarrow{}\map(M_\Gamma)$ of $\Psi$.

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