Relative Trisections of Fiber Bundles over the Circle

Dissler, Rudy

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 5, pp. 1373-1412

For an oriented $4$-dimensional fiber bundle over $S^1$, we build a relative trisection from a sutured Heegaard splitting of the fiber. We provide an algorithm to explicitly construct the associated relative trisection diagram, from a sutured Heegaard diagram of the fiber. As an application, we glue our relative trisection diagrams with existing diagrams to recover trisected closed fiber bundles over $S^1$ and trisected spun manifolds, and to provide trisections for $4$-dimensional open-books.

Nous construisons une trisection pour un fibré sur le cercle, orienté et compact, à partir d’un scindement de Heegaard suturé de la fibre. Nous donnons un algorithme pour construire les diagrammes de trisection relatifs associés, à partir d’un diagramme de Heegaard suturé de la fibre. Enfin, nous recollons nos diagrammes à des diagrammes de trisections relatifs déjà existants, retrouvant ainsi les trisections de fibrés sur le cercle fermés, les trisections de variétés spun, et produisant des trisections pour les livres ouverts de dimension $4$.

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