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Watanabe's theta graph diffeomorphism, constructed using Watanabe's clasper surgery construction which turns trivalent graphs in 4-manifolds into parameterized families of diffeomorphisms of 4-manifolds, is a diffeomorphism of $S^4$ representing a potentially nontrivial smooth mapping class of $S^4$. The "(1,2)-subgroup" of the smooth mapping class group of $S^4$ is the subgroup represented by diffeomorphisms which are pseudoisotopic to the identity via a Cerf family with only index 1 and 2 critical points. This author and Hartman showed that this subgroup is either trivial or has order 2 and explicitly identified a diffeomorphism that would represent the nontrivial element if this subgroup is nontrivial. Here we show that the theta graph diffeomorphism is isotopic to this one possibly nontrivial element of the (1,2)-subgroup. To prove this relation we develop a diagrammatic calculus for working in the smooth mapping class group of $S^4$., Comment: 16 pages, 11 figures

Perron and Quinn gave independent proofs in 1986 that every topological pseudo-isotopy of a simply-connected, compact topological 4-manifold is isotopic to the identity. Another result of Quinn is that every smooth pseudo-isotopy of a simply-connected, compact, smooth 4-manifold is smoothly stably isotopic to the identity. From this he deduced that $\pi_4(\operatorname{TOP}(4)/\operatorname{O}(4)) =0$. A replacement criterion is used at a key juncture in Quinn's proofs, but the justification given for it is incorrect. We provide different arguments that bypass the replacement criterion, thus completing Quinn's proofs of both the topological and the stable smooth pseudo-isotopy theorems. We discuss the replacement criterion and state it as an open problem., Comment: 22 pages, 18 figures

Links in $S^3$ can be encoded by grid diagrams; a grid diagram is a collection of points on a toroidal grid such that each row and column of the grid contains exactly two points. Grid diagrams can be reinterpreted as front projections of Legendrian links in the standard contact 3-sphere. In this paper, we define and investigate triple grid diagrams, a generalization to toroidal diagrams consisting of horizontal, vertical, and diagonal grid lines. In certain cases, a triple grid diagram determines a closed Lagrangian surface in $\mathbb{CP}^2$. Specifically, each triple grid diagram determines three grid diagrams (row-column, column-diagonal and diagonal-row) and thus three Legendrian links, which we think of collectively as a Legendrian link in a disjoint union of three standard contact 3-spheres. We show that a triple grid diagram naturally determines a Lagrangian cap in the complement of three Darboux balls in $\mathbb{CP}^2$, whose negative boundary is precisely this Legendrian link. When these Legendrians are maximal Legendrian unlinks, the Lagrangian cap can be filled by Lagrangian slice disks to obtain a closed Lagrangian surface in $\mathbb{CP}^2$. We construct families of examples of triple grid diagrams and discuss potential applications to obstructing Lagrangian fillings., Comment: 27 pages, 23 figures, comments very welcome! v2: Final version, accepted for publication in the Journal of Symplectic Geometry

This is an expository account of the author's collaboration with Rob Kirby leading up to the theory of trisections of smooth 4-manifolds. This article was written for inclusion in an upcoming issue of Celebratio Mathematica dedicated to Rob Kirby's mathematical work., Comment: 21 pages, 9 figures; to be published in Celebratio Mathematica

Isotopy classes of diffeomorphisms of the 4-sphere can be described either from a Cerf theoretic perspective in terms of loops of 5-dimensional handle attaching data, starting and ending with handles in cancelling position, or via certain twists along submanifolds analogous to Dehn twists in dimension two. The subgroup of the smooth mapping class group of the 4-sphere coming from loops of 5-dimensional handles of index 1 and 2 coincides with the subgroup generated by twists along Montesinos twins (pairs of 2-spheres intersecting transversely twice) in which one of the two 2-spheres in the twin is unknotted. In this paper we show that this subgroup is in fact trivial or cyclic of order two., Comment: 15 pages, 7 figures

One way to better understand the smooth mapping class group of the 4-sphere would be to give a list of generators in the form of explicit diffeomorphisms supported in neighborhoods of submanifolds, in analogy with Dehn twists on surfaces. As a step in this direction, we describe a surjective homomorphism from a group associated to loops of 2-spheres in $S^2 \times S^2$'s onto this smooth mapping class group, discuss two natural and in some sense complementary subgroups of the domain of this homomomorphism, show that one is in the kernel, and give generators as above for the image of the other. These generators are described as twists along Montesinos twins, i.e. pairs of embedded 2-spheres in the 4-sphere intersecting transversely at two points., Comment: 25 pages, 8 figures; revised to include acknowledgement of support from the Max Planck Institute of Mathematics and the National Science Foundation

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., Comment: 28 pages, 9 figures, notes from mini-course at Winter Braids VIII, School on Braids and Low-dimensional Topology

This paper deals with the history of the following problem: "Can an arbitrary rectangle be dissected into 3 non-rectangular congruent regions?" We present a new elementary proof that the answer is indeed no.

We study embedded spheres in 4-manifolds (2-knots) via doubly pointed trisection diagrams, showing that such descriptions are unique up to stabilization and handleslides, and we describe how to obtain trisection diagrams for certain cut-and-paste operations along 2-knots directly from doubly pointed trisection diagrams. The operations described are classical surgery, Gluck surgery, blowdown, and $(\pm4)$-rational blowdown, and we illustrate our techniques and results with many examples., Comment: 33 pages, 22 figures

Given a handle decomposition of a 4-manifold with boundary, and an open book decomposition of the boundary, we show how to produce a trisection diagram of a trisection of the 4-manifold inducing the given open book. We do this by making the original proof of the existence of relative trisections more explicit, in terms of handles. Furthermore, we extend this existence result to the case of 4-manifolds with multiple boundary components, and show how trisected 4-manifolds with multiple boundary components glue together., Comment: 17 pages, 12 figures

We offer a new proof that two closed oriented 4-manifolds are cobordant if their signatures agree, in the spirit of Lickorish's proof that all closed oriented 3-manifolds bound 4-manifolds. Where Lickorish uses Heegaard splittings we use trisections. In fact we begin with a subtle recasting of Lickorish's argument: Instead of factoring the gluing map for a Heegaard splitting as a product of Dehn twists, we encode each handlebody in a Heegaard splitting in terms of a Morse function on the surface and build the 4-manifold from a generic homotopy between the two functions. This extends up a dimension by encoding a trisection of a closed 4-manifold as a triangle (circle) of functions and constructing an associated 5-manifold from an extension to a 2-simplex (disk) of functions. This borrows ideas from Hatcher and Thurston's proof that the mapping class group of a surface is finitely presented., Comment: 19 pages, 5 figures; references added, minor changes to exposition in introduction

We establish a correspondence between trisections of smooth, compact, oriented $4$--manifolds with connected boundary and diagrams describing these trisected $4$--manifolds. Such a diagram comes in the form of a compact, oriented surface with boundary together with three tuples of simple closed curves, with possibly fewer curves than the genus of the surface, satisfying a pairwise standardness condition. This should be thought of as the $4$--dimensional analog of a sutured Heegaard diagram for a sutured $3$--manifold. We also give many foundational examples., Comment: The previous version had mistakes in the proof of the algorithm to find the monodromy of the induced open book decomposition using the relative trisection diagram. This version improves the statement of the algorithm and corrects the proof

A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies, and the closed 4-manifold, with homomorphisms between them induced by inclusion, form a commutative diagram of epimorphisms, which we call a trisection of the 4-manifold group. A trisected 4-manifold thus gives a trisected group; here we show that every trisected group uniquely determines a trisected 4-manifold. Together with Gay and Kirby's existence and uniqueness theorem for 4-manifold trisections, this gives a bijection from group trisections modulo isomorphism and a certain stabilization operation to smooth, closed, connected, oriented 4-manifolds modulo diffeomorphism. As a consequence, smooth 4-manifold topology is, in principle, entirely group theoretic. For example, the smooth 4-dimensional Poincar\'e conjecture can be reformulated as a purely group theoretic statement., Comment: 7 pages, 2 figures

Donaldson showed that every closed symplectic 4-manifold can be given the structure of a topological Lefschetz pencil. Gay and Kirby showed that every closed 4-manifold has a trisection. In this paper we relate these two structure theorems, showing how to construct a trisection directly from a topological Lefschetz pencil. This trisection is such that each of the three sectors is a regular neighborhood of a regular fiber of the pencil. This is a 4-dimensional analog of the following trivial 3-dimensional result: For every open book decomposition of a 3-manifold M, there is a decomposition of M into three handlebodies, each of which is a regular neighborhood of a page., Comment: 9 pages, 9 figures

We use parameterized Morse theory on the pages of an open book decomposition to efficiently encode the contact topology in terms of a labelled graph on a disjoint union of tori (one per binding component). This construction allows us to generalize the notion of the front projection of a Legendrian knot from the standard contact $\mathbb{R}^3$ to arbitrary closed contact $3$-manifolds. We describe a complete set of moves on such front diagrams, extending the standard Legendrian Reidemeister moves, and we give a combinatorial formula to compute the Thurston-Bennequin number of a nullhomologous Legendrian knot from its front projection., Comment: 34 pages, 14 figures

We show that any smooth, closed, oriented, connected 4--manifold can be trisected into three copies of $\natural^k (S^1 \times B^3)$, intersecting pairwise in 3--dimensional handlebodies, with triple intersection a closed 2--dimensional surface. Such a trisection is unique up to a natural stabilization operation. This is analogous to the existence, and uniqueness up to stabilization, of Heegaard splittings of 3--manifolds. A trisection of a 4--manifold $X$ arises from a Morse 2--function $G:X \to B^2$ and the obvious trisection of $B^2$, in much the same way that a Heegaard splitting of a 3--manifold $Y$ arises from a Morse function $g : Y \to B^1$ and the obvious bisection of $B^1$., Comment: 38 pages, 29 figures; minor improvements to exposition, more examples, and more discussion

Given a Morse 2-function $f: X^4 \to S^2$, we give minimal conditions on the fold curves and fibers so that $X^4$ and $f$ can be reconstructed from a certain combinatorial diagram attached to $S^2$. Additional remarks are made in other dimensions., Comment: 13 pages, 10 figures. Replaced because the main theorem in the original is false. The theorem has been corrected and counterexamples to the original statement are given

We show that under appropriate hypotheses, a plumbing of symplectic surfaces in a symplectic 4-manifold admits strongly convex neighborhoods. Moreover the neighborhoods are Lefschetz fibered with an easily-described open book on the boundary supporting the induced contact structure. We point out some applications to cut-and-paste constructions of symplectic 4-manifolds., Comment: 12 pages, 2 figures