# Research

My current research is in trisections of smooth 4-manifolds. Trisections are a 4-dimensional analog of Heegaard splittings of 3-manifolds. A trisection of a smooth, closed 4-manifold $\bf{X}$ is a decomposition $\bf{X\boldsymbol{=} X_1 \boldsymbol{\cup} X_2\boldsymbol{\cup} X_3}$ three 4-dimensional 1-handlebodies $\bf{X_i \cong \natural^k S^1 \times B^3}$ whose pairwise intersections are 3-dimensional handlebodies $\bf{X_i\boldsymbol{\cap} X_j\boldsymbol{\cong}\boldsymbol{\#}^g S^1\boldsymbol{\times} S^2}$ and triple intersection $\bf{X_1 \cap X_2 \cap X_3}$ is a surface.

There are many analogies between the theories of trisections and Heegaard splittings. For example, every smooth closed 4-manifold admits a trisection which is unique up to the appropriate notion of stabilization. There are also trisection diagrams $\bf{\boldsymbol{(}F,\boldsymbol{\alpha, \beta, \gamma)}},$ where $\bf{F}$ is a closed, genus $\bf{g}$ surface and each of $\boldsymbol{\alpha, \beta, \gamma}$ are $\bf{g}$-tuples of non-separating, simple, closed curves in $\bf{F}$ such that each triple $\bf{\boldsymbol{(}F, \boldsymbol{\alpha, \beta)}},\bf{\boldsymbol{(}F, \boldsymbol{\beta, \gamma)}},\bf{\boldsymbol{(}F, \boldsymbol{\alpha, \gamma)}}$ is a Heegaard diagram for $\bf{\#^g S^1 \times S^2.}$