Research Projects

In 2012, Gay and Kirby~\cite{GAY} introduced the notion of a trisection of a closed oriented smooth $4-$manifold and the notion of a relative trisection of a compact $4-$manifold with connected boundary as an analog of Heegaard splittings of $3-$manifold. They also showed the existence of a trisection on a closed oriented smooth $4-$manifold as well as the existence of a relative trisection on a compact oriented smooth $4-$manifold with connected boundary. Recent developments in the area of trisections demonstrate rich connections and applications to other aspects of four-manifold topology, including a new approach to studying symplectic manifolds and their embedded submanifolds \cite{LMS20} \cite{Lam19} \cite{LM18}, and to surface knots (embedded in $S^4$ and other more general 4-manifolds) \cite{MZ17} \cite{MZ18} along with associated surgery operations \cite{GM18} \cite{KM20}. A particular interest to the trisection community is the construction of new \cite{KT18} and the adaptation of established invariants in the trisection framework. A $(g,k)$ \textbf{\emph{trisection}} $\mathcal{T}$ of a closed oriented $4$--manifold $X$ is a decomposition of $X$ into three submanifolds $X_1,X_2$ and $X_3$ each diffeomorphic to $\natural_k S^1 \times D^3$ such that each $X_i\cap X_j$ is diffeomorphic to $\natural_g S^1 \times D^2$ and $X_1\cap X_2\cap X_3$ is diffeomorphic to closed oriented surface $\Sigma_g$ of genus $g$. Similarly, a $(g,k; p,b)$ \textbf{\emph{relative trisection}} $\mathcal{T}$ of a compact connected oriented $4$--manifold $Y$ is a decomposition of $Y$ into three submanifolds $Y_1,Y_2$ and $Y_3$ each diffeomorphic to $\natural_k S^1 \times D^3$ with $Y_1\cap Y_2\cap Y_3=\Sigma_{g,b}$ is a compact oriented surface of genus $g$ with $b$ boundary components such that each $X_i\cap X_j$ is diffeomorphic the cobordism between $\Sigma_{g,b}$ and a genus $p$ surface $\Sigma_{p,b}$ obtained by compressing certain non-separating $g-p$ curves on $\Sigma_{p,b}$. A relative trisection on a compact oriented $4$--manifold $Y$ induces an open book on $\partial Y$ with pages $\Sigma_{g,b}$. By an open book decomposition on a manifold is a way to express the manifold as a locally trivial fiber bundle over the circle $S^1$ in the complement of co-dimension $2$ submanifold. In ~\cite{LINV} Castro, et al., introduced the notion of Murasugi sum --a special plumbing operation-- of trisections which is analogue of the Murasugi sum for open books of manifolds \cite{GAB}. We have also discussed this notion with a schematic approach in the preprint \cite{GPS_TMUR}. \textbf{In this project our main focus is to study relative trisections of compact $4$--manifolds, their Murasugi sums and related properties.} \textbf{We also aim to investigate the connection of relative trisections with contact and symplectic topology}.