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

Gay, David T

Winter Braids VIII (Marseille, 2018), Winter Braids Lecture Notes (2018), Exposé no. 4, 19 p.

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.

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