Constructing a Closed 4-manifold

Given a trisection diagram \boldsymbol{(}\bf{F_g},\boldsymbol{\alpha, \beta, \gamma)}, we can uniquely construct a smooth, connected, closed 4-manifold as follows. We first begin by constructing 3-dimensional handlebodies, \bf{H}_{\boldsymbol{\alpha}}, \bf{H}_{\boldsymbol{\beta}}, \bf{H}_{\boldsymbol{\gamma}} by attaching 3-dimensional 2-handles to \bf{F_g} along the \bf{g}-tuples \boldsymbol{\alpha}, \boldsymbol{\beta}, and \boldsymbol{\gamma}, respectively. We then attach \bf{H}_{\boldsymbol{\alpha}} \boldsymbol{\times} \bf{I} to \bf{F_g}\boldsymbol{\times} \bf{D^2} along \bf{F_g }\boldsymbol{\times [-\varepsilon, \varepsilon]} (where we view \boldsymbol{[-\varepsilon, \varepsilon]} as a subset of \boldsymbol{\partial}\bf{D^2}), and likewise for \bf{H}_{\boldsymbol{\beta}} \boldsymbol{\times} \bf{I}, \bf{H}_{\boldsymbol{\gamma}}\boldsymbol{\times} \bf{I}. After smoothing corners, this gives us a smooth 4-manifold with boundary. A schematic of this intermediate step is given below.

closedconstruction (2)

By definition, we require that any two of \boldsymbol{\alpha, \beta, \gamma} give a Heegaard diagram for\bf{\boldsymbol{\#}^g\boldsymbol{(}S^1\boldsymbol{\times} S^2\boldsymbol{)}}. Thus we have constructed a 4-manifold with three disjoint boundary components, each of which is diffeomorphic to \bf{\boldsymbol{\#}^g\boldsymbol{(}S^1\boldsymbol{\times} S^2\boldsymbol{)}}. Finally, we make use of a theorem of Laudenbach and Poénaru which tells us that there is a unique way to fill in this boundary components. This associates to the given diagram \boldsymbol{(}\bf{F_g}, \boldsymbol{\alpha, \beta, \gamma)} a unique, closed 4-manifold \bf{X}, as desired.