From Heegaard splittings to trisections; porting 3 -dimensional ideas to dimension 4

Abstract

These notes summarize and expand on a mini-course given at CIRM in February 2018 as part of Winter Braids VIII. We somewhat obsessively develop the slogan “Trisections are to 4 –manifolds as Heegaard splittings are to 3 –manifolds”, focusing on and clarifying the distinction between three ways of thinking of things: the basic definitions as decompositions of manifolds, the Morse theoretic perspective and descriptions in terms of diagrams. We also lay out these themes in two important relative settings: 4 –manifolds with boundary and 4 –manifolds with embedded 2 –dimensional submanifolds.