Trisections of 4-manifolds with boundary

Significance

Trisections are decompositions of 4D spaces (4-manifolds) into three pieces; these decompositions become both richer and more subtle when the 4-manifolds have 3D boundaries and even more subtle when there are multiple boundary components. Here, we discuss this setup and show how to turn a certain standard decomposition of a 4-manifold with boundary, a handle decomposition, into a diagram drawn on a surface that describes that 4-manifold and its trisection given certain boundary conditions.

Abstract

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.

When developing the foundations of the study of trisections of 4-manifolds in ref. 1, Gay and Kirby (1) briefly discussed the case of 4-manifolds with connected boundary, in which the corresponding boundary data are open book decompositions on 3-manifolds. Castro (2) developed this case further, in particular showing the importance of this case by showing how to glue two trisected 4-manifolds along a common boundary to produce a trisection of a closed manifold. The authors of this work then worked out (3) the diagrammatic version of this theory, with the diagrammatic version of gluing appearing in the work of Castro and Ozbagci (4). This paper supplements these papers by first showing, through examples and a careful exposition of the existence proof in (1), how to explicitly turn a handle decomposition of a 4-manifold together with the data of a given open book on the boundary into a trisection of the 4-manifold described diagrammatically. Then, we extend the definitions to include trisections of 4-manifolds with multiple boundary components and discuss the corresponding theorems in that case. We end by implementing the method discussed earlier for turning handle decompositions into diagrams in this multiple boundary setting, showing how to get trisection diagrams for product cobordisms.

1. Transforming Kirby Diagrams into Relative Trisection Diagrams

An open book decomposition of a closed, oriented, connected 3-manifold $Y$ is an oriented link $\mathrm{\Lambda }\subset Y$ and a fibration $f:Y\backslash \mathrm{\Lambda }\to S^1$ such that, for each $t\in S^1$, we get a compact oriented surface $P_t=f^{-1}\left(t\right)$ with $\partial P_t=\mathrm{\Lambda }$. If $Y=\partial X$ for a compact, connected 4-manifold $X$, there is a notion defined in ref. 1 of a relative trisection of $X$ inducing the given open book on $Y$. In Section 2 below, we give a more general version of this definition in the case of multiple boundary components for $X$, but for the purposes of this section, we take the definition as understood, and the reader unfamiliar with the definition will gain an understanding by example as preparation for Section 2. (Such a reader may also choose to begin by reading Section 2.) In ref. 3, the authors show how to diagrammatically characterize trisections on 4-manifolds with connected boundary and how to understand the open book on the boundary.

Question.

Given a handle decomposition of a compact 4-manifold $X$ with boundary $Y$, involving a single 0-handle and some 1-, 2-, and 3-handles, described diagrammatically via a Kirby diagram (which is thus also a surgery diagram for $Y$) and given an open book decomposition of $Y$ described by explicitly drawing a single page in the surgery diagram for $Y$, how does one produce a trisection diagram for $X$ inducing the given open book decomposition on $Y$?

We answer this by giving a detailed proof paired with an extended example of the basic existence theorem.

Theorem 1 [Gay and Kirby (1)].

Given an open book decomposition of $Y=\partial X$, there exists a relative trisection of $X$ inducing the given open book. (Here, $Y$ and $X$ are connected.)

We show that this existence theorem can in fact be made explicit so as to produce a diagram for the trisection in the sense of Question.

Proof:

We begin with an example of the given data in Question. For $X$, we take the complement of the standard torus in $S^4$, and for the open book on $Y=T^3$, we take a standard open book with a twice-punctured torus as the page and monodromy given by a right-handed Dehn twist parallel to one boundary component and a left-handed twist parallel to the other boundary component. Fig. 1 shows a Kirby diagram for $X$: once drawn with dotted circle notation and once drawn with ball notation for the 1-handles. □

Fig. 1.

Kirby diagram for a handle decomposition of $X=S^4\backslash N\left(T^2\right)$ with one 0-handle, one 1-handle, and two 2-handles. The 2-handles are 0-framed.

We need to augment this diagram with an explicit embedding of a page of the open book decomposition that we have in mind. The diagram in Fig. 2, Top shows a generic fiber $F$ of the fibration of $T^3$ over $T^2$ and its location within the surgery description of $\partial X$. This fibration structure can be transformed into the open book under consideration via the three-component fibered link $C_0⨿C_1⨿C_2$ obtained by plumbing a positive Hopf band to a negative Hopf band along a boundary parallel arc. To be precise, we remove a disk $D$ with boundary from the fiber $F$ of the fibration and then realize $T^3$ as

$$\left(F\backslash D\right)\times S^1\underset{\partial D=C_0}{\cup }\left(S^3\backslash \nu \left(C_0\right)\right).$$

This decomposition gives the right open book for $T^3$. The page $P$ for this open book embedded into the surgery diagram is shown in the diagram in Fig. 2, Middle, and an ambient isotopy yields the final diagram. Note that the final diagram depicts a handle decomposition of $P$ involving one 0-handle and three 1-handles; this takes us closer to a “planar diagram” of the page, which will be convenient in the next stage.

Fig. 2.

Fibration and open book for $\partial X\cong T^3$.

With $X$ and $Y$ described by a Kirby diagram and the open book described via an embedded page $P$, the first step is to see $X$ as a relative cobordism between 3-manifolds with boundary and produce a new handle decomposition of $X$ adapted to this cobordism structure.

Definition 1:

Let $Y_{-}$ and $Y_{+}$ be 3D-oriented manifolds with boundary and $X$ be a 4D-oriented manifold with boundary. The triple $\left(X;Y_{-},Y_{+}\right)$ is a relative cobordism between $Y_{-}$ and $Y_{+}$ if

• i)the oriented boundary of $X$ can be decomposed as $-Y_{-}\cup N\cup Y_{+}$;
• ii)$-Y_{-}\cap Y_{+}=\varnothing$;
• iii)$\partial N=-\partial Y_{-}⨿\partial Y_{+}$;
• iv)$N\cong \left[-\mathrm{1,1}\right]\times \partial Y_{-}$ (and thus, $\partial Y_{-}\cong \partial Y_{+}$); and
• v)$N\cap Y_{\pm }=\partial Y_{\pm }$.

The manifold $X$ is then called a relative cobordism from $Y_{-}$ to $Y_{+}$. In addition, the manifold $N$ is sometimes referred to as the horizontal boundary component of $X$, and the manifolds $Y_{-}$ and $Y_{+}$ are referred to as the vertical boundary components. (We view our cobordisms as running from left to right, not from bottom to top.)

When $\partial X=Y$ has an open book decomposition, this gives us a relative cobordism structure in the following sense. Consider the open book $\left(\mathrm{\Lambda },f\right)$ on $Y$, and restrict the fibration $f$ to the complement $Y\backslash \nu \left(\mathrm{\Lambda }\right)$ of an open tubular neighborhood of the link $\mathrm{\Lambda }$. An identification of $S^1$ with a square $\partial \left(\left[-\mathrm{1,1}\right]\times \left[-\mathrm{1,1}\right]\right)$ and a decomposition of the square into its vertical and horizontal components $v_{\pm }=\left\{\pm 1\right\}\times \left[-\mathrm{1,1}\right],h_{\pm }=\left[-\mathrm{1,1}\right]\times \left\{\pm 1\right\}$ induces the following decomposition on $Y$:

$$Y=f^{-1}\left(v_{-}⨿v_{+}\right)\cup \left(\nu \left(\mathrm{\Lambda }\right)\cup f^{-1}\left(h_{-}⨿h_{+}\right)\right).$$

Then, if we set $Y_{\pm }=f^{-1}\left(v_{\pm }\right)$, we call $Y_{-}$ and $Y_{+}$ the vertical components of $Y$, and $N=\nu \left(\mathrm{\Lambda }\right)\cup f^{-1}\left(h_{-}⨿h_{+}\right)$ is the horizontal component of $Y$. Moreover, notice that, if $P$ is a page of the fibration $f$, then $Y_{\pm }\cong \left[-\mathrm{1,1}\right]\times P$ and $N\cong \nu \left(\partial P\right)\cup \left(\left[-\mathrm{1,1}\right]\times P⨿\left[-\mathrm{1,1}\right]\times P\right)\cong \left[-\mathrm{1,1}\right]\times D\left(P\right)$, where $D\left(P\right)$ denotes the double of $P$.

With respect to this decomposition of $Y=\partial X$, $X$ is now a relative cobordism from $Y_{-}\cong \left[-\mathrm{1,1}\right]\times P$ to $Y_{+}\cong \left[-\mathrm{1,1}\right]\times P$. Our first goal is to turn the given handle decomposition of $X$, which presents $X$ as a cobordism from $\varnothing$ to $Y$, into a handle decomposition presenting $X$ as a relative cobordism from $Y_{-}$ to $Y_{+}$. We will call a handle decomposition of the first type (starting with a 0-handle, to which 1-, 2-, and 3-handles are added) a “standard handle decomposition” and a handle decomposition of the second type (starting with $\left[-\mathrm{1,1}\right]\times Y_{-}\cong \left[-\mathrm{1,1}\right]\times \left[-\mathrm{1,1}\right]\times P$, to which 1-, 2-, and 3-handles are added along $\left\{1\right\}\times \left[-\mathrm{1,1}\right]\times P$) a “relative handle decomposition.” Intermediate between an arbitrary standard handle decomposition and a relative decomposition, we introduce Definition 2.

Definition 2:

A standard handle decomposition of a 4-manifold $X$, involving a single 0-handle and a positive number of 1-, 2-, and 3-handles, is compatible with a given open book $\left(\mathrm{\Lambda },f\right)$ on its boundary $\partial X$ if, for some page $P\subset \partial X$ of the open book, the 0-handle and some of the 1-handles form a neighborhood $\left[-\mathrm{1,1}\right]\times \left[-\mathrm{1,1}\right]\times P$ of $P$ in $X$, with $\left\{-1\right\}\times \left[-\mathrm{1,1}\right]\times P$ being a neighborhood of $P$ in $\partial X$; the remaining 1-, 2-, and 3-handles attached along $\left\{1\right\}\times \left[-\mathrm{1,1}\right]\times P$.

If a standard handle decomposition of $X$ is compatible with a given open book with page $P$, then $P$ can be given a handle decomposition involving a single 2D 0-handle and several 2D 1-handles with the following behavior.

• •The 2D 0-handle is completely contained in the 4D 0-handle, and it is disjointed from every attaching sphere of the 4D handle decomposition.
• •For each 2D 1-handle $H_1^2$, there is a unique 4D 1-handle $H_1^4$, such that the core of $H_1^2$ intersects the cocore of $H_1^4$ transversely exactly once (i.e., $H_1^2$ goes over $H_1^4$ geometrically once and no other 2D 1-handles go over $H_1^4$).

With a given standard handle decomposition of $X$ and a page $P$ visible in $Y=\partial X$, we will most likely need to add cancelling 1–2 pairs to achieve this compatibility. In our example, having drawn a picture in which the 2D 1-handles of $P$ are obvious, we see a natural place to put the 1–2 pairs: one of the 2D 1-handles already went over a 4D 1-handle, and now, we add two cancelling pairs to accommodate the other two 2D 1-handles. A planar diagram for $P$ can then be obtained via an ambient isotopy. In some sense, this is the hardest part of the process to implement in practice. The result of the isotopy is shown in Fig. 3, where the 0-handle for the page is drawn as a hexagon, and the handles are shown as identifications of some of the sides.

Fig. 3.

Kirby diagram for a handle decomposition of $X$ compatible with the given open book on $\partial X$. The grey 2-handles were added to cancel the 1-handles, and these, in turn, were added to accommodate the 2D handles of the page.

Returning to the general setting, we now assume that the handle decomposition of $X$ is compatible with the given open book. Let $X_1$ be the union of the 0-handle and the 1-handles. Since the handle decomposition of $X$ is compatible with the open book on $\partial X$, then $X_1$ can be realized as the union of ${\left[-\mathrm{1,1}\right]}^2\times P$ with some number $a_1$ of “purely 4D” 1-handles. Thus, $X_1$ is isomorphic to ${♮}^{k_1}{S}^1\times B^3$, and $\partial X_1$ is naturally decomposed into

$$\begin{array}{ccc}\hfill {\partial }_{\mathrm{b}\mathrm{d}\mathrm{r}\mathrm{y}}{X}_1& =\hfill & \partial X\cap X_1\cong Y_{-}\cong \left[-\mathrm{1,1}\right]\times P\text{and}.\hfill \\ \hfill {\partial }_{\mathrm{i}\mathrm{n}\mathrm{t}}{X}_1& =\hfill & \partial X_1\cap \hspace{0.17em}\mathrm{i}\mathrm{n}\mathrm{t}\left(X\right)\hfill \\ \hfill & =\hfill & \left(\left\{1\right\}\times Y_{-}\right)\backslash \left({⨿}^{a_1}\left(S^0\times D^3\right)\right)\cup \left({⨿}^{a_1}\left(D^1\times S^2\right)\right).\hfill \end{array}$$

That is, ${\partial }_{\mathrm{i}\mathrm{n}\mathrm{t}}{X}_1$ is the result of performing surgery on $\left[-\mathrm{1,1}\right]\times P$ along the attaching spheres of the purely 4D handles. To understand the effect of these handles, align the attaching regions of the 1-handles so that the equator of each $S^0\times \partial D^3$ is in $\left\{0\right\}\times P$. Then, removing a vertical half of $S^0\times D^3$ from each $\left[-\mathrm{1,0}\right]\times P$ and $\left[\mathrm{0,1}\right]\times P$ results in a space that is diffeomorphic to $I\times P$ and that has marked disks in its boundary. Additionally, the products $D^1\times S^2$ are attached along the northern hemisphere to $\left[\mathrm{0,1}\right]\times P\backslash a_1\left(S^0\times D^3\right)$ and along the southern hemisphere to $\left[-\mathrm{1,0}\right]\times P\backslash a_1\left(S^0\times D^3\right)$. Thus, ${\partial }_{\mathrm{i}\mathrm{n}\mathrm{t}}{X}_1$ has a sutured Heegaard splitting $H_{12}\underset{F}{\cup }{H}_{31}$, where $F$ is the surface (with boundary) that results from performing surgery on $P$ along the $a_1$ embedded 0-spheres, and each $H_{ij}$ is the result of attaching 3D 1-handles to $I\times P$.

Now let $L$ be the framed attaching link for the 2-handles; $L$ is in fact embedded in ${\partial }_{\mathrm{i}\mathrm{n}\mathrm{t}}{X}_1$ and can be projected onto the surface $\left\{0\right\}\times F$ as already illustrated in Fig. 3. If necessary (it is not in our example), use Reidemeister I moves to ensure that the handle framing agrees with the surface framing. Since at each crossing in the projection, there is a way of knowing which strand is to be thought of as the one passing under, the two points in $L$ that project into each one of the crossing points can be labeled as “over point” and “under point.” Then, if a diagram of the projection has $c$ crossing points, the link $L$ has a decomposition into $c$ vertices and $c$ edges by placing a vertex at each under point of $L$. Call an edge an “over edge” if it contains at least one over point. If necessary (again, it is not in our example), use Reidemeister II moves to ensure that each component has at least one over edge.

The next step involves the stabilizations of the sutured Heegaard splitting. These stabilizations can be performed at each crossing in the decomposition of $L$ induced by the projection, but it is worth noting that we can often be more efficient if the diagram for $L$ is not alternating: if one strand goes over several other understrands in succession, we can resolve this with one stabilization. Specifically, if $e_{ij}$ is the $j$th over edge in the component $L_i$ of $L$, then form the handlebody $H_{31}^{\prime }$ by removing from $H_{31}$ the 3D tubular neighborhoods of arcs in $H_{31}$ parallel relative (rel.) boundary to interior segments of the edges $e_{ij}$. In addition, form the space $H_{12}^{\prime }$ by adding these tubular neighborhoods to $H_{12}$, and set $F\prime =H_{12}^{\prime }\cap H_{31}^{\prime }$. Next, consider the disk $D_{ij}$ obtained as the isotopy rel. boundary that gives the arc parallel to $e_{ij}$, and notice that $D_{ij}$ is a compressing disk for $H_{31}^{\prime }$. Finally, resolve the crossings in $L$ by sliding each over strand $e_{ij}$ over $D_{ij}$ and into $F\prime$. This gives a higher genus sutured Heegaard splitting of ${\partial }_{\mathrm{i}\mathrm{n}\mathrm{t}}{X}_1$. Pushing $L$ into the interior of $H_{12}^{\prime }$, we are ready to produce the trisection (and simultaneously, recap the definition of a relative trisection) in much the same way as explained in ref. 1, lemma 14.

Let $N=\left[-ϵ,ϵ\right]\times H_{12}^{\prime }$ be a small tubular neighborhood of $H_{12}^{\prime }$ with $\left[-ϵ,0\right]\times H_{12}^{\prime }=N\cap X_1$, and set

• •$X_1$ as described,
• •$X_2$ as the union of $\left[0,ϵ\right]\times H_{12}^{\prime }$ and the 2-handles, and
• •$X_3=X\backslash \mathrm{i}\mathrm{n}\mathrm{t}\left(X_1\cup X_2\right)$.

To see that $X=X_1\cup X_2\cup X_3$ is a relative trisection, notice that

• •$F\prime =X_1\cap X_2\cap X_3$ is a compact surface with boundary,
• •both $H_{31}^{\prime }=X_3\cap X_1$ and $H_{12}^{\prime }=X_1\cap X_2$ are relative compression bodies compressing $F\prime$ down to the page $P$,
• •$X_1$ is diffeomorphic to ${♮}^{k_1}\left(S^1\times B^3\right)$ for some $k_1$, and
• •$X_1\cap \partial X\cong I\times P$.

It remains to verify that both $X_2$ and $X_3$ are diffeomorphic to ${♮}^{k_i}\left(S^1\times B^3\right)$ for some $k_2$ and $k_3$, with both $X_2\cap \partial X$ and $X_3\cap \partial X$ diffeomorphic to $I\times P$, and that $H_{23}^{\prime }=X_2\cap X_3$ is a relative compression body from $F\prime$ to $P$. This will be an unbalanced trisection if $k_1$, $k_2$, and $k_3$ are distinct, but standard trisection stabilizations can balance the trisection to get $k=k_1=k_2=k_3$.

To see that $X_2$ is isomorphic to ${♮}^{k_2}\left(S^1\times B^3\right)$, notice that the space $\left[0,ϵ\right]\times H_{12}^{\prime }$ is isomorphic to ${♮}^l\left(S^1\times B^3\right)$ (for some $l$), and it inherits a handle structure from that of $H_{12}^{\prime }$. Moreover, the belt sphere $\left\{0\right\}\times S^2$ of the 4D 1-handle $D^1\times D^3$ is the union of $\left[0,ϵ\right]\times {\beta }_j$ and $\left\{0,ϵ\right\}\times D_j$, where $D_j$ is a compression disk for $H_{12}^{\prime }$ and ${\beta }_j$ is its boundary. Notice also that the way that the stabilizations were performed guarantees that every component $L_i$ of $L$ intersects the $\beta$ belt spheres transversely. Therefore, fix one index $J_i$ for each $i$ to be the belt sphere that $L_i$ intersects transversely, and slide every other ${\beta }_{ij}$ over ${\beta }_{i{J}_i}$ so that it no longer intersects $L_i$. This shows that the 1-handle with core parallel to the over edge $e_{i{J}_i}$ and the 2-handle with attaching circle $L_i$ are a cancelling pair and therefore, that $X_2\cong {♮}^{k_2}\left(S^1\times B^3\right)$ (for $k_2$ equal to $l$ minus the number of components of $L$). This process will also help us produce enough $\gamma$ curves to get a trisection diagram.

Turning the relative handle decomposition “upside down” shows us that $X_3=X\backslash \mathrm{i}\mathrm{n}\mathrm{t}\left(X_1\cup X_2\right)$ is diffeomorphic to ${♮}^{k_3}\left(S^1\times B^3\right)$, $X_3\cap \partial X$ is diffeomorphic to $\left[-\mathrm{1,1}\right]\times P$, and the interior boundaries ${\partial }_{\mathrm{i}\mathrm{n}\mathrm{t}}{X}_1$ and ${\partial }_{\mathrm{i}\mathrm{n}\mathrm{t}}{X}_3$ are also diffeomorphic. Since the attaching link $L$ can be isotoped to be in the interior of $H_{12}^{\prime }$, the latter implies that $H_{23}^{\prime }={\partial }_{\mathrm{i}\mathrm{n}\mathrm{t}}{X}_2\backslash H_{12}^{\prime }$, the result of surgery on $H_{12}^{\prime }$ along the attaching circles of the 2-handles, is diffeomorphic to $H_{12}^{\prime }$ and thus, to the desired compression body.

We return to the example in Fig. 3. Notice that the attaching circles of the 2-handles have already been projected into $P$, with no Reidemeister I or II moves needed. Also, there are no purely 4D 1-handles and no 3-handles (i.e., the relative handle decomposition has only 2-handles). We need to stabilize the page five times to resolve the crossings in the 2-handle link $L$. (There are eight crossings, but some of them can be resolved in groups with a single stabilization, since the diagram is not alternating.) Thus, we get a trisection immediately with the following.

• •$X_1$ is $\left[-\mathrm{1,1}\right]\times \left[-\mathrm{1,1}\right]\times P$, where $P$ is the page, a genus 1 surface with two boundary components. Thus, $X_1\cong {♮}^3{S}^1\times B^3$.
• •${\partial }_{\mathrm{i}\mathrm{n}\mathrm{t}}{X}_1\cong \left[-\mathrm{1,1}\right]\times P$ is split along a genus 6 surface $\mathrm{\Sigma }$ with two boundary components obtained by stabilizing the splitting along $\left\{0\right\}\times P$ five times. This splits ${\partial }_{\mathrm{i}\mathrm{n}\mathrm{t}}{X}_1$ into two compression bodies $H_{31}$ and $H_{12}$, each of which compresses $\mathrm{\Sigma }$ down to $P$ by compressing along five simple closed curves. Thus, ignoring the sutures, each of $H_{12}$ and $H_{31}$ is a genus 8 handlebody.
• •$X_2$ is a thickening of $H_{12}$ together with the four 2-handles, each of which cancels one of eight 1-handles in $I\times H_{12}\cong {♮}^8{S}^1\times B^3$, so that $X_2\cong {♮}^4{S}^1\times B^3$.
• •Thus, $H_{23}$ is also a compression body from $\mathrm{\Sigma }$ to $P$, and since there are no 3-handles, $X_3$ is again $\left[-\mathrm{1,1}\right]\times \left[-\mathrm{1,1}\right]\times P\cong {♮}^3{S}^1\times B^3$.

This is an unbalanced trisection, and we can draw its diagram by understanding which curves on the genus 6 surface $\mathrm{\Sigma }$ bound disks in the three compression bodies. In the two compression bodies forming the boundary of $X_1$, namely $H_{31}$ and $H_{12}$, these are the five standard $\alpha$ (red in Fig. 4) (bounding in $H_{31}$) and $\beta$ (blue in Fig. 4) (bounding in $H_{12}$) curves corresponding to the five stabilizations indicated in Fig. 4. The four components of the attaching link $L$ for the 2-handles give four of five curves that bound disks in $H_{23}$ (green $\gamma$ curves in Fig. 4). To find the fifth $\gamma$ curve, we note that the four curves coming from $L$ cancel four of the $\beta$ curves; therefore, we can carry out this cancellation, and the missing $\gamma$ curve can be obtained as a parallel copy to the curve obtained as ${\beta }_C+{\beta }_D$. The result is shown in Fig. 4. Finally, we can stabilize to get a balanced trisection if desired (also shown in Fig. 4). □

Fig. 4.

A $\left(\mathrm{8,5};\mathrm{1,2}\right)$ relative trisection diagram for the exterior of the standard torus in $S^4$. The thick green curve is the $\gamma$ curve that is not an attaching circle of the 2-handles.

2. Trisections of 4-Manifolds with Multiple Boundary Components

We extend the definition of relative trisections to manifolds with $m>1$ boundary components by generalizing the construction in ref. 3. Given integers $\left(g,k;p,b\right)$ with $g\ge p$ and $2p+b-1\le k\le g+p+b-1$, we begin as in the cases of empty boundary or a unique boundary component with $Z_k=♮S^1\times B^3$ and $\partial Z_k={\#}^k{S}^1\times S^2$ but with a special decomposition of $\partial Z_k$ that will depend on the pages of the open book decompositions on the boundary components of $X$. To extend the definition in a precise way, let $g,k$ be two positive integers, and let $\left(p_j,b_j\right)$ for $j=1,\dots ,m$ be pairs of positive integers that parametrize the page of an open book decomposition $\left(P_j,{\mu }_j\right)=P_i{\times }_{{\mu }_j}{S}^1\hspace{0.17em}\cup \hspace{0.17em}\partial P_i\times D^2$ on the $j$th boundary component of $\partial X$. To make the exposition cleaner, let $p=\sum p_i$, $b=\sum b_i$, $l_i=2p_i+b_i-1$, and $l=\sum l_i=2p+b-1$.

One of the issues in trying to understand relative trisections is the presence of the binding of the open book decompositions. Our solution to work around this nuisance is to fix a 2D wedge $D=\left\{r{e}^{i\theta }\mid 0\le r\le 1;-\pi /3\le \theta \le \pi /3\right\}$ and consider the 4D product $U_j=P_j\times D\cong {♮}^{l_j}{S}^1\times B^3$ for $j=1,\dots ,m$. Although these are 4D 1-handlebodies, these are not quite yet the “correct” pieces for a trisection, both because they do not form a connected piece and because even connecting them might not give us the decomposition of $\partial Z_k$ required for the definition. To obtain the correct pieces, notice that the natural decomposition of $\partial D$ into its three pieces ${\partial }^{0}D=\left\{e^{i\theta }\mid -\pi /3\le \theta \le \pi /3\right\}$ and ${\partial }^{\pm }D=\left\{r{e}^{\pm i\pi /3}\mid 0\le r\le 1\right\}$ induces a decomposition on each $\partial U_i$ into

$${\partial }^0{U}_i=P_i\times {\partial }^{0}D\hspace{0.17em}\cup \hspace{0.17em}\partial P_i\times D,\hspace{0.17em}\text{and}\hspace{0.17em}{\partial }^{\pm }{U}_i=P_i\times {\partial }^{\pm }D.$$

Now, for $j=1,\dots ,m$, let $B_j$ be a 2D disk in the interior of $P_j$, and connect all of the $U_j$ along $B_j\times \left({\partial }^{-}{D}_{ϵ}\cup {\partial }^{+}{D}_{ϵ}\right)$, where $D_{ϵ}$ is a wedge with radius $0<ϵ<1$ and contained in $D$. This preserves the splitting, and we thus get $U=U_1♮\cdots ♮U_m\cong {♮}^l{S}^1\times B^3$, with a decomposition of its boundary into

$${\partial }^{0}U{=⨿}_{i=1}^m{\partial }^0{U}_i,\hspace{0.17em}\text{and}\hspace{0.17em}{\partial }^{\pm }U={\partial }^{\pm }{U}_1♮\cdots ♮{\partial }^{\pm }{U}_m.$$

To obtain the desired 4D pieces of the trisection, we proceed as in ref. 3 and connect $U$ to $V_n≔{♮}^n{S}^1\times B^3$, the boundary of which has the unique genus $n+s$ Heegaard splitting ${\partial }_s{V}_n={\partial }_s^{+}{V}_n\cup {\partial }_s^{-}{V}_n.$ Choosing $n=l-k$ and $s=g-n-p$ gives us $Z_k≔U♮V_n\cong {♮}^k{S}^1\times B^3$, with a boundary that inherits the decomposition ${\partial }_g{Z}_k={\partial }_g^{+}{Z}_k\cup {\partial }_m^0{Z}_k\cup {\partial }_g^{-}{Z}_k,$ with

$${\partial }_m^0{Z}_k={\partial }^{0}U\hspace{0.17em}\text{and}\hspace{0.17em}{\partial }_g^{\pm }{Z}_k=\left({\partial }^{\pm }{U}_1♮\cdots ♮{\partial }^{\pm }{U}_m\right)♮{\partial }_s^{\pm }{V}_n.$$

Here, we use the subscript $m$ on ${\partial }_m^0{Z}_k$ to indicate that the trisection is of a manifold with $m>1$ boundary components.

Definition 3:

A $\left(g,k;p_1,b_1,\dots ,p_m,b_m\right)$ relative trisection of a compact, connected, smooth, oriented 4-manifold $X$ with $m>0$ boundary components is a decomposition of $X$ into three codimension 0-submanifolds $X=X_1\cup X_2\cup X_3$ such that, for each $i=\mathrm{1,2,3}$,

• 1.there exists a diffeomorphism ${\phi }_i:X_i\to Z_k$; the diffeomorphism satisfies
• 2.${\phi }_i\left(X_i\cap X_{\mp 1}\right)={\partial }_g^{\pm }{Z}_k$ and ${\phi }_i\left(X_i\cap \partial X\right)={\partial }_m^0{Z}_k,$

where indices are taken mod 3.

Lemma.

A $\left(g,k;p_1,b_1;\dots ;p_m,b_m\right)$ relative trisection of a 4-manifold $X$ with $m>0$ boundary components induces an open book decomposition on each component ${\partial }_{i}X$ of $\partial X$ with page that is $P_i$, a genus $p_i$ surface with $b_i$ boundary components.

Proof:

Suppose that $X=X_1\cup X_2\cup X_3$ is a $\left(g,k;p_1,b_1;\dots ;p_m,b_m\right)$ relative trisection. Recall that $X_i\cap \partial X\cong P\times I\hspace{0.17em}\cup \hspace{0.17em}\partial P\times D$, where $P=\bigsqcup _{i=1}^m{P}_i$. We also denote $D_j=\left\{r{e}^{i\theta }\mid \left|r\right|\le 1,\hspace{0.17em}\frac{2\pi j}{3}\le \theta \le \frac{2\pi \left(j+1\right)}{3}\right\}$, for $j=\mathrm{1,2,3}$, to be a third of the unit disk associated with each $X_j$. Thus,

$$\begin{array}{ccc}\hfill \partial X& =\hfill & X_1\cap \partial X\hspace{0.17em}\cup \hspace{0.17em}{X}_2\cap \partial X\hspace{0.17em}\cup \hspace{0.17em}{X}_3\cap \partial X\hfill \\ \hfill & \cong \hfill & \left(P\times {\partial }^0{D}_1\hspace{0.17em}\cup \hspace{0.17em}P\times {\partial }^0{D}_2\hspace{0.17em}\cup \hspace{0.17em}P\times {\partial }^0{D}_3\right)\hfill \\ \hfill & \hfill & \bigcup \left(\partial P\times D_1\hspace{0.17em}\cup \hspace{0.17em}\partial P\times D_2\hspace{0.17em}\cup \hspace{0.17em}\partial P\times D_3\right)\hfill \\ \hfill & =\hfill & P{\times }_{\mu }{S}^1\hspace{0.17em}\cup \hspace{0.17em}\partial P\times D^2,\hfill \end{array}$$

where $P{\times }_{\mu }{S}^1$ is the $m$-component mapping cylinder of the monodromy map $\mu$ and $\partial P\times D^2$ is a regular neighborhood of the binding $\partial P$. Restricting $\mu$ to each $P_i$ gives the open book decomposition $\left(P_j,{\mu }_j\right)$ on each component of $\partial X$ as desired. □

Theorem 2.

Given an open book decomposition on each component of $\partial X$, there exists a relative trisection of $X$ inducing the given open book(s).

The proof in this case is essentially identical to the connected boundary component case proved earlier; rather than replicate that proof, we see how it works in this case in Section 5 by turning an interesting handle decomposition of $I\times M$ into a trisection diagram.

3. Relative Trisection Diagrams

Trisection diagrams allow us to describe any 4-manifold in terms of three collections of curves $\alpha ,\beta ,\gamma$ on a surface $\mathrm{\Sigma }$. The most notable characteristic of relative trisection diagrams for manifolds with multiple boundary components is that each collection $\alpha ,\beta ,\gamma$ contains a (at least one) separating curve. This is a requirement, since performing surgery on $\mathrm{\Sigma }$ along any one of $\alpha ,\beta ,\gamma$ results in $m>0$ surfaces with boundary, each one corresponding to a page of an open book induced on a component of $\partial X$.

Definition 4:

A $\left(g,k;p_1,b_1;\dots ;p_m,b_m\right)$ relative trisection diagram is a 4-tuple $\left(\mathrm{\Sigma },\alpha ,\beta ,\gamma \right)$, such that

• i)$\mathrm{\Sigma }$ is a genus $g$ surface with $b$ boundary components;
• ii)each of $\alpha =\left\{{\alpha }_1,\dots ,{\alpha }_{g-p},{\tilde{\alpha }}_1,\dots ,{\tilde{\alpha }}_{m-1}\right\}$ is a collection of $g-p+m-1$ disjoint, simple, closed curves such that ${\alpha }_1,\dots ,{\alpha }_{g-p}\subset \alpha$ are essential, and each ${\tilde{\alpha }}_1,\dots ,{\tilde{\alpha }}_{m-1}\subset \alpha$ is a separating curve ($\beta$ and $\gamma$ are similarly defined); and
• iii)each triple $\left(\mathrm{\Sigma },\alpha ,\beta \right),\left(\mathrm{\Sigma },\beta ,\gamma \right),\left(\mathrm{\Sigma },\alpha ,\gamma \right)$ is handleslide diffeomorphic to the standard diagram $\left(\mathrm{\Sigma },ϵ,\delta \right)$ in Fig. 5.

Fig. 5.

Standard position of $\left(\mathrm{\Sigma },ϵ,\delta \right)$. There are $m-1$ separating curves, ${\widetilde{ϵ}}_1,\dots ,{\widetilde{ϵ}}_{m-1}$, and $\widetilde{{\delta }_1},\dots ,{\tilde{\delta }}_{m-1}$. Each ${\widetilde{ϵ}}_i$ and ${\tilde{\delta }}_i$ separates $\mathrm{\Sigma }$ into two pieces: one with curves and one without. The latter corresponds to the page of the induced open book on a boundary component of the corresponding 4-manifold.

Theorem 3.

There is a one-to-one correspondence between relatively trisected 4-manifolds up to diffeomorphism and relative trisection diagrams up to handleslides and diffeomorphisms of the trisection surface.

Proof:

To obtain a relative trisection diagram from a relatively trisected 4-manifold $X=X_1\cup X_2\cup X_3$, let $\mathrm{\Sigma }=X_1\cap X_2\cap X_3$. Define $\alpha \subset \mathrm{\Sigma }$ to be the simple closed curves corresponding to the boundaries of compressing disks in the compression body $X_1\cap X_2$; similarly, let $\beta$ correspond to the boundaries of compression disks in $X_2\cap X_3$, and $\gamma$ correspond to those in $X_1\cap X_3$.

Let $\mathrm{\Delta }$ denote the 2-simplex with vertices $v_{\alpha }$, $v_{\beta }$, and $v_{\gamma }$ labeled clockwise, and let $e_{\alpha }$, $e_{\beta }$, and $e_{\gamma }$, respectively, denote the edges opposite to these vertices. Let $C_{\alpha },C_{\beta },C_{\gamma }$ be the relative compression bodies obtained by attaching 3D 2-handles to $I\times \mathrm{\Sigma }$ along the $\alpha ,\beta ,$ and $\gamma$ curves, with $\mathrm{\Sigma }$ canonically identified with $\left\{0\right\}\times \mathrm{\Sigma }$ in the boundary of each of these compression bodies. Thus, $C_{\alpha }$ is a relative cobordism from $\mathrm{\Sigma }$ to ${\mathrm{\Sigma }}_{\alpha }$, the result of surgering $\mathrm{\Sigma }$ along the $\alpha$ curves (similarly for $\beta$ and $\gamma$). A relative trisection diagram $\left(\mathrm{\Sigma },\alpha ,\beta ,\gamma \right)$ specifies an identification space $X_{\alpha \beta \gamma }$ defined by gluing $e_{\alpha }\times C_{\alpha }$, $e_{\beta }\times C_{\beta }$, and $e_{\gamma }\times C_{\gamma }$ to $\mathrm{\Delta }\times \mathrm{\Sigma }$ along $e_{\alpha }\times \mathrm{\Sigma }$, $e_{\beta }\times \mathrm{\Sigma }$, and $e_{\gamma }\times \mathrm{\Sigma }$, respectively. The corners in $X_{\alpha \beta \gamma }$ can be rounded to obtain a smooth manifold with boundary decomposed into three types of pieces: (1) $M_{\alpha \beta }⨿M_{\beta \gamma }⨿M_{\gamma \alpha }$, (2) $e_{\alpha }\times {\mathrm{\Sigma }}_{\alpha }⨿e_{\beta }\times {\mathrm{\Sigma }}_{\beta }⨿e_{\gamma }\times {\mathrm{\Sigma }}_{\gamma }$, and (3) $\left(\mathrm{\Delta }\cup \left(e_{\alpha }\times I\right)\cup \left(e_{\beta }\times I\right)\cup \left(e_{\gamma }\times I\right)\right)\times \partial \mathrm{\Sigma }$. Here, $M_{\alpha \beta }$ is the sutured manifold specified by the sutured Heegaard diagram $\left(\mathrm{\Sigma },\alpha ,\beta \right)$; that is, $M_{\alpha \beta }=C_{\alpha }\underset{\mathrm{\Sigma }}{\cup }{C}_{\beta }$. The manifolds $M_{\beta \gamma }⨿M_{\gamma \alpha }$ are defined similarly. Notice that the Heegaard sutured decompositions of $M_{\alpha \beta },M_{\beta \gamma },M_{\gamma \alpha }$ correspond to ${\partial }_g^{+}{Z}_k\cup {\partial }_g^{-}{Z}_k$ as in Section 2. To obtain a smooth 4-manifold with $m$ boundary components, we first “close up” these sutured portions of $\partial X_{\alpha \beta \gamma }$ with copies of a thickening of ${\partial }_m^0{Z}_k\cong {\partial }^{0}D\times P\hspace{0.17em}\cup \hspace{0.17em}D\times \partial P$. (The notations $D$ and ${\partial }^{0}D$ refer to Section 2.)

To understand the attachment, notice that $\partial M_{\alpha \beta }$ is the union of the horizontal boundary components of $C_{\alpha }$ and $C_{\beta }$ together with copies of ${\mathrm{\Sigma }}_{\alpha }$ and ${\mathrm{\Sigma }}_{\beta }$. Therefore, $D\times \partial P$ can be glued to $M_{\alpha \beta }$ by identifying ${\partial }^{+}D\times \partial P$ and ${\partial }^{-}D\times \partial P$ with the horizontal boundary components of $C_{\alpha }$ and $C_{\beta }$, respectively. Similarly, ${\partial }^{0}D\times P$ can be glued by identifying $\partial \left({\partial }^{0}D\right)\times P$ with ${\mathrm{\Sigma }}_{\alpha }⨿{\mathrm{\Sigma }}_{\beta }$. Analogous arguments hold for $M_{\beta \gamma },M_{\gamma \alpha }$. We refer the reader to ref. 3, corollary 14 for the proof that there is a unique way to attach each ${\partial }^{0}D\times P\hspace{0.17em}\cup \hspace{0.17em}D\times \partial P$.

This new 4-manifold $X_{\alpha \beta \gamma }\cup 3\left(I\times \left({\partial }^{0}D\times P\hspace{0.17em}\cup \hspace{0.17em}D\times \partial P\right)\right)$ has $m$ boundary components equipped with open books and three boundary components diffeomorphic to ${\#}^k{S}^1\times S^2$. We then use the result of Laudenbach and Poénaru (5), which states that there is a unique way to fill ${\#}^k{S}^1\times S^2$ with ${♮}^k{S}^1\times B^3$ to obtain a smooth 4-manifold $X$ with $m$ boundary components, each accompanied with an induced open book (Fig. 6). □

Fig. 6.

Schematic for identification space.

4. Gluing Relative Trisection Diagrams

Before discussing the gluing theorem, we first recall the algorithm described in ref. 3, theorem 5 for determining the monodromy of an open book determined by a relative trisection diagram; this works without modification in the case of multiple boundary components and is essential for drawing the diagram of a trisection obtained by gluing two relatively trisected 4-manifolds along diffeomorphic boundary components.

Recall that ${\mathrm{\Sigma }}_{\alpha }$ is the surface obtained from $\mathrm{\Sigma }$ by surgery along $\alpha$. A collection $a$ of properly embedded arcs in $\mathrm{\Sigma }$ disjoint from $\alpha$ is called a cut system of arcs for $\left(\mathrm{\Sigma },\alpha \right)$ if $a$ descends to a collection $a_{\alpha }$ of properly embedded arcs in ${\mathrm{\Sigma }}_{\alpha }$, which cut each component of ${\mathrm{\Sigma }}_{\alpha }$ into a disk.

The algorithm for obtaining the monodromy for the induced open book decomposition gives a well-defined procedure, given a cut system of arcs $a$ for $\alpha$, to produce cut systems of arcs $b$, $c$, and $â$ for $\beta$, $\gamma$, and $\alpha$, respectively, such that the monodromy map $\mu :{\mathrm{\Sigma }}_{\alpha }\to {\mathrm{\Sigma }}_{\alpha }$ is the unique map such that $\mu \left(a_{\alpha }\right)={â}_{\alpha }$; the monodromy $\mu$ actually factors as a map ${\mathrm{\Sigma }}_{\alpha }\to {\mathrm{\Sigma }}_{\beta }\to {\mathrm{\Sigma }}_{\gamma }\to {\mathrm{\Sigma }}_{\alpha }$ taking $a_{\alpha }$ to $b_{\beta }$ to $c_{\gamma }$ to ${â}_{\alpha }$.

(Technically, the algorithm in ref. 3 produces a cut system of arcs $b$ for a system of curves that is handleslide equivalent to $\beta$, not for $\beta$ itself, and similarly for $c$ in relation to $\gamma$ and $â$ in relation to $\alpha$. However, each arc system can be turned back into an arc system for the original system of curves, since any time one slides a closed curve $x$ over another curve $y$, if an arc $z$ from the corresponding arc system gets in the way, then $z$ can first be slid over $y$ to get it out of the way.)

Suppose that $\left(\mathrm{\Sigma },\alpha ,\beta ,\gamma \right)$ and $\left(\mathrm{\Sigma }\prime ,{\alpha }^{\prime },\beta \prime ,{\gamma }^{\prime }\right)$ are relative trisection diagrams that correspond to 4-manifolds $X$ and $X\prime$ with diffeomorphic boundaries. It is natural to ask how these relative trisection diagrams relate to a relative trisection diagram of the closed 4-manifold $X\underset{\partial }{\cup }X\prime$. Of course, the induced structures on the bounding 3-manifold must be compatible. That is, if $\left(P,\mu \right)$ and $\left(P\prime ,\mu \prime \right)$ are the open books induced by $\left(\mathrm{\Sigma },\alpha ,\beta ,\gamma \right)$ and $\left(\mathrm{\Sigma }\prime ,\alpha \prime ,\beta \prime ,\gamma \prime \right)$, respectively, there must exist an orientation reversing diffeomorphism $f:P\to P\prime$ such that $f○\mu ○f^{-1}=\mu \prime$. The function $f$ gives an identification between $\partial \mathrm{\Sigma }$ and $\partial \mathrm{\Sigma }\prime$, which we use to define the closed genus $G=g+g\prime +b-1$ surface $S=\mathrm{\Sigma }\underset{\partial }{\cup }\mathrm{\Sigma }\prime$. After we glue, we omit the separating curves $\tilde{\alpha },\tilde{\beta },\tilde{\gamma }$. The resulting collections $\alpha \cup \alpha \prime ,\beta \cup \beta \prime ,\gamma \cup \gamma \prime$ each contain $G-l$ curves. To account for the missing curves, we use the cut systems of arcs. Since we have the freedom to choose our initial cut systems of arcs $a\subset \mathrm{\Sigma }$ and $a\prime \subset {\mathrm{\Sigma }}^{\prime }$, we choose $a\prime$, so that $\partial a_i^{\prime }=f\left(\partial a_i\right)$. We denote the newly formed curves ${\alpha }_i^{*}=a_i\underset{\partial }{\cup }{a}_i^{\prime }$. From $a$, we obtain $b$ and $c$ by using the algorithm mentioned above in ref. 3, theorem 5; we similarly obtain $b\prime$ and $c\prime$, and we define ${\beta }_i^{*}=b_i\cup b_i^{\prime }$ and ${\gamma }_i^{*}=c_i\underset{\partial }{\cup }{c}_i^{\prime }$. We now have three collections of $G$ curves on $S$:

$$\overline{\alpha }=\alpha \cup \alpha \prime \cup {\alpha }^{*},\overline{\beta }=\beta \cup \beta \prime \cup {\beta }^{*},\overline{\gamma }=\gamma \cup \gamma \prime \cup {\gamma }^{*}.$$ [4.1]

In this case, when we glue along every induced open book decomposition, we obtain a $\left(G,K\right)$-trisection diagram, where $K=k+k\prime -l$.

The gluing theorem applies in greater generality, allowing us to glue together trisection (diagrams) along any number of boundary components (possibly less than $m$) and resulting in another relative trisection (diagram). Although the indexing becomes much more involved, the general idea remains the same. We glue the surfaces $\mathrm{\Sigma }$ and $\mathrm{\Sigma }\prime$ together along boundary components, which correspond to the bindings of compatible open book decomposition, using the associated cut system of arcs to account for the missing number of curves on the resulting surface.

Theorem 4.

The tuple $\left(S,\overline{\alpha },\overline{\beta },\overline{\gamma }\right)$, with $\overline{\alpha },\overline{\beta },\overline{\gamma }$ as in [4.1], is a trisection diagram for $X\cup X\prime$.

For details in the diagrammatic case of $m=1$, the reader is referred to ref. 4. Information on gluing relatively trisected manifolds with $m>1$ is in ref. 2. Here, we sketch the general proof.

Proof:

We first show that $\left(S,\overline{\alpha },\overline{\beta },\overline{\gamma }\right)$ is a trisection diagram. This follows from the fact that the Heegaard diagram with arcs $\left(\mathrm{\Sigma },\alpha \cup a,\beta \cup b\right)$ is handleslide equivalent to a standard diagram with arcs (where closed curves slide over closed curves and arcs slide over closed curves), and the same is true for the other two diagrams with arcs on $\mathrm{\Sigma }$ and the three diagrams with arcs on $\mathrm{\Sigma }\prime$. The standardizing slides then fit together to give standardizing slides for the three pairs $\left(S,\overline{\alpha },\overline{\beta }\right)$, $\left(S,\overline{\beta },\overline{\gamma }\right)$, and $\left(S,\overline{\gamma },\overline{\alpha }\right)$. Having established that we have a legitimate diagram, the fact that it contains as subdiagrams relative diagrams for $X$ and $X\prime$ shows that the closed 4-manifold is in fact $X\cup X\prime$. □

5. Trisecting Product Cobordisms

Note that the gluing theorem allows us to define a more refined bordism category, $\mathbf{T}\mathbf{R}\mathbf{I}$, with objects that are 3-manifolds equipped with open book decompositions, and morphisms are relatively trisected cobordisms. A curious feature, however, is that a product $\left[-\mathrm{1,1}\right]\times M^3$ does not have an obvious “product” trisection. Here, we show how to draw (unbalanced) trisection diagrams for such products. Given a closed, oriented 3-manifold $M$ with an open book decomposition with page $P$ and monodromy $\mu :P\to P$, fix a 2D handle decomposition of $P$ with one 0-handle and $l=1-\chi \left(P\right)\hspace{0.17em}1$-handles. Let $\left\{a_i\right\}$ be the cocores of the 1-handles (properly embedded arcs in $P$), and let $b_i=\mu \left(a_i\right)$ be their images under the monodromy. The diagram will be described in terms of these data.

Let $P\prime$ and $P^{″}$ be copies of $P$ in the interior of $P$ obtained as deformation retracts along collar neighborhoods of $\partial P$, with the property that the 0-handle of P″ is contained in the interior of the 0-handle of P′, but the 1-handles of P″ are disjoint from the 1-handles of P′. Let $D$ be the 0-handle of P″. Let $b_i^{\prime }$ be a copy of $b_i$ properly embedded in $P\prime$ and isotoped rel. boundary so as to avoid $D$. Let $c_i$ be the cores of the 1-handles of P″, with end points on $\partial D$ (Fig. 7); in this example, the 3-manifold $M$ is $S^3$.

Fig. 7.

Setting up the page.

We will first draw a Kirby diagram for $\left[-\mathrm{1,1}\right]\times M$ seen as a cobordism from $\left\{-\mathrm{1,1}\right\}\times \left[\mathrm{0,1}\right]\times P$ to $\left\{-\mathrm{1,1}\right\}\times \left[\mathrm{0,1}\right]\times P$. This diagram is drawn on $\left\{-\mathrm{1,1}\right\}\times P=-P⨿P$. The diagram has two $B^3$ for the feet of a single 4D 1-handle, a framed link of $2l$ components, and a 3-handle, which is not drawn. The two $B^3$ intersect $-P$ and $P$ as the disk $D$, the 0-handle of P″. Of the $2l$ components of the 2-handle attaching link, $l$ of them appear as two copies of each of the cores $c_1,\dots ,c_l$ of the 1-handles of P″, with each component running over the 4D 1-handle twice. This much is indicated in Fig. 8. The remaining $l$ components are obtained by completing each monodromy arc $b_i^{\prime }$ so as to wrap around the $i$th 1-handle of P″. Specifically, we take the union of $b_i^{\prime }$ in $P\prime \subset P\subset -P⨿P$ and an arc going around and behind the corresponding core curve $c_i$ as drawn in Fig. 9.

Fig. 8.

The beginning of the Kirby diagram.

Fig. 9.

The complete Kirby diagram.

Before continuing to the trisection diagram, we should discuss the meaning of this diagram briefly. This is a handle-attaching diagram for a 4-manifold built by attaching one 1-handle, $2l\hspace{0.17em}2$-handles, and one 3-handle to $\left(\left[-2,-1\right]⨿\left[\mathrm{1,2}\right]\right)\times \left[\mathrm{0,1}\right]\times P$ along $\left\{-\mathrm{1,1}\right\}\times \left[\mathrm{0,1}\right]\times P$, and the diagram is projected onto $\left\{-\mathrm{1,1}\right\}\times P=-P⨿P$. This will become a trisection diagram by turning $-P⨿P$ into $-P\#P$, with the connected sum occurring along a tube running over the 4D 1-handle and then adding extra genus (extra torus summands) to this surface so as to accommodate the crossings in the 2-handle attaching link. The connected sum tube gets an $\alpha$ and a $\beta$ curve parallel, since this records the 4D 1-handle. This much is shown in Fig. 10. Each extra torus summand, coming from a stabilization of a Heegaard splitting, gets an $\alpha$ meridian and a $\beta$ longitude curve. Then, the $2l$ components of the 2-handle attaching link become $\gamma$ curves as shown in Fig. 11. The last part is to find the remaining $\gamma$ curves. In the example drawn, we only need one extra $\gamma$ curve. In general, each $\gamma$ curve that we already have is dual to some $\beta$ curves, and the remaining $\gamma$ curves should be parallel to the remaining $\beta$ curves. However, this may not be possible with the $\beta$ curves as drawn, and therefore, we need to slide the remaining $\beta$ curves over other $\beta$ curves until they become disjoint from given $\gamma$ curves and then turn the resulting $\beta$ curves into $\gamma$ curves. In our example, the $\gamma$ curve obtained as two copies of the core $c_1$ is “dual” to both ${\beta }_A$ and ${\beta }_B$. The remaining $\gamma$ curve is obtained as a parallel copy of the result of sliding ${\beta }_A$ over ${\beta }_B$ twice with opposite signs. The final result is shown in Fig. 12.

Fig. 10.

First step toward the trisection diagram.

Fig. 11.

Second step toward the trisection diagram; we have some but not all of the $\gamma$ curves.

Fig. 12.

The complete trisection diagram.