Proof The first fact follows from the fact that $Y_i$ is a handlebody with boundary $\Sigma _i \cup \Sigma _{i+1}$ , where the surfaces are identified along their boundaries. The second fact is obtained by solving the simultaneous equations in the first fact.

Proof If a trisection is trivial, Heegaard stabilizing once produces a non-trivial trisection. This means that applying at most one Heegaard stabilization, the trisections can be assumed to be non-trivial. The trisections are now equivalent up to stabilization, by Theorem 2.8.

Proof For each i, the cut system on $\Sigma _i \cup \Sigma _{i+1}$ are instructions for gluing $p_i + p_{i+1} + b - 1$ disks into $\Sigma _i \cup \Sigma _{i+1}$ . A 3-ball now glues into the remaining cavity to produce a handlebody $Y_i$ of genus $y_i = p_i + p_{i+1} + b - 1$ . The handlebodies $Y_i$ and $Y_{i+1}$ glue along $\Sigma _{i+1}$ . Gluing all three handlebodies together in this way produces a trisected closed 3-manifold. To see that the resulting trisected 3-manifold is unique up to diffeomorphism, we use that $\text {Diff}(B^n \text { rel } \partial )$ is contractible for $n=2$ and $n=3$ [Reference Hatcher9, Reference Smale23]. Suppose Y and $Y'$ are built from a given triple Heegaard diagram as above. We will incrementally build a diffeomorphism $Y \to Y'$ preserving the trisection structure. First, consider the identity map on $\cup _i \Sigma _i$ . The first steps in building Y and $Y'$ was to glue disks along curves in the cut system. Since $\text {Diff}(B^2 \text { rel } \partial )$ is contractible, the aforementioned map extends to $\cup _i \Sigma _i \cup \text {\{disks\}}$ independent of how these disks are glued in. The final step is to glue in three 3-balls, and again, the map extends, since $\text {Diff}(B^3 \text { rel } \partial )$ is contractible. This guarantees that we can build a diffeormophism $Y \to Y'$ , and moreover that the diffeomorphism preserves the trisection structure. That is, Y and $Y'$ are diffeomorphic as trisected 3-manifolds.

Proof Every 3-manifold admits a trisection, as in Example 2.4. Moreover, every trisection is induced by a triple Heegaard diagram. This is because every sector of the trisection is a handlebody, and every handlebody is determined by a cut system on the boundary surface. A collection of three cut systems for each sector is precisely the data of a triple Heegaard diagram.

Proof Johannson [Reference Johannson11] defines a meridian-system to be a pair $(M,\mathcal D)$ where M is a 3-manifold with boundary and $\mathcal D$ is a collection of neatly embedded disks such that $M - D$ is homeomorphic to a 3-ball. Reference [Reference Johannson11, Corollary 1.6] shows that meridian-systems of handlebodies (which are considered up to isotopy) are unique up to handleslides. Given a sector $Y_i$ of the trisection, $(\Sigma _{i}, \Sigma _{i+1}, \boldsymbol {\delta }_i)$ induces a unique meridian-system. It follows that, up to some diffeomorphism of $(Y_1, Y_2, Y_3)$ , the triple Heegaard diagram is unique up to isotopy and handleslides. Finally, diffeomorphisms of the trisection descend to diffeomorphisms of triple Heegaard diagrams.

Proof By Lemma 2.21, any two triple Heegaard diagrams of a fixed trisection are diffeomorphic after a sequence of handleslides and isotopies. By Proposition 2.10, any two trisections of a given 3-manifold are equivalent up to stabilization and Heegaard stabilization. (Examples 2.18 and 2.19 show how stabilization and Heegaard stabilization descend to triple Heegaard diagrams.) The result follows by applying each of these results.

Proof Item (1) is immediate from Proposition 2.2. Item (2) follows from earlier observations about how stabilization and Heegaard stabilization affect each $y_i$ . Finally item (3) follows from the fact that a genus g Heegaard splitting induces a $(g, g, 0; 1)$ -trisection.

Proof The first fact is immediate from the definition,specifically, items (3), (5), and (6) are exactly necessary properties for $(Y_1, Y_2, Y_3)$ to be a trisection of Y. The second fact is from Proposition 2.2, restated for completeness. The third fact follows from the fact that $H_i$ has boundary $\Sigma _i \cup \Sigma _C$ . Finally, the Euler characteristic formula is derived by repeated applications of the difference formula $\chi (A\cup B) = \chi (A) + \chi (B) - \chi (A\cap B)$ . Note that $|\boldsymbol y| + b$ is odd because $|\boldsymbol y| + b = 2|\boldsymbol p| + 4b - 3$ , so in each of the last three properties, no half-integers appear.

Proof The reader is directed to [Reference A. Castro2] for a description of relative stabilization, and a proof that any two relative trisections of 4-manifolds are equivalent up to internal and relative stabilizations, provided the induced open books on the boundary are equivalent up to Hopf stabilization. Reference [Reference Piergallini and Zuddas20, Theorem 3.5] shows that all open books of a given 3-manifold are equivalent up to Hopf stabilization and the $\partial U$ move (described in [Reference Piergallini and Zuddas20]). The $\partial U$ move is extended to the relative double twist move of relative trisections in [Reference Castro, Islambouli, Miller and Tomova6], completing the proof.

Proof It must be shown that relative trisections as above satisfy all six of the defining properties of pseudo-trisections from Definition 3.2. Property 1 is immediate, since the definition of relative trisections requires that each sector $X_i$ is diffeomorphic to a standard piece $Z_k$ . Properties 4, 5, and 6 are exactly observations 1, 2, and 3 at the start of [Reference Gay, Pinzón-Caicedo and Castro24, Section 4]. This leaves properties 2 and 3.

In a relative trisection, $H_i = X_i \cap X_{i-1}$ is necessarily diffeomorphic to a standard piece $Y^+_{g,k;p,b}$ . This is a compression body from $\Sigma _{g,b}$ to $\Sigma _{p,b}$ obtained by compressing along $g-p$ simple closed curves. In the proposition statement, we require that the binding of the induced open book in the boundary is non-empty, which is equivalent to requiring that b is at least 1. A compression body of this form is diffeomorphic to a handlebody of genus $g+p+b-1$ , verifying that property 2 is satisfied.

Finally we must verify that property 3 is satisfied. In the notation of [Reference Gay, Pinzón-Caicedo and Castro24], we have that $X_i \cap \partial X$ is diffeomorphic to a standard piece $Y^0_{g,k;p,b}$ , which is itself equal to $(D \times \partial \Sigma _{p,b}) \cup (I \times \Sigma _{p,b})$ , where D is a wedge of a disk and I is the intersection of D with the boundary of the underlying disk. Topologically the union corresponds to gluing b solid tori to $I \times \Sigma _{p,b}$ along their longitudes, and thus $Y^0_{g,k;p,b}$ is diffeomorphic to $I \times \Sigma _{p,b}$ . Again since $b \geq 1$ , the latter is a handlebody (of genus $2p+b-1$ ).

Proof Every such 4-manifold admits a relative trisection [Reference A. Castro2]. In particular, the binding can be chosen to be non-empty. This relative trisection is a pseudo-trisection.

Proof There are two steps: first we stabilize the pseudo trisections to agree on the boundary, and then stabilize the interior.

Let $\mathcal T, \mathcal T'$ be pseudo-trisections of X. Their restrictions $\tau , \tau '$ to $Y = \partial X$ are trisections of Y. By Proposition 2.10, $\tau $ and $\tau '$ are stably equivalent. This means there is a trisection $\sigma $ of Y and arcs $\{\alpha \}, \{\alpha '\}$ such that stabilizations and Heegaard stabilizations of $\tau $ along $\{\alpha \}$ produces $\sigma $ , and similarly stabilizations and Heegaard stabilizations of $\tau '$ along $\{\alpha '\}$ produces $\sigma $ .

We now return to 4-dimensions: Boundary stabilizing and Heegaard stabilizing $\mathcal T$ along $\{\alpha \}$ produces a pseudo-trisection $\mathcal S$ which restricts to the 3-manifold trisection $\sigma $ on Y. On the other hand, boundary and Heegaard stabilizing $\mathcal T'$ along $\{\alpha '\}$ produces a pseudo-trisection $\mathcal S'$ which also restricts to $\sigma $ on Y. We have successfully stabilized our pseudo-trisections to agree on the boundary.

Next we turn to the interior. We proceed as in the proof of the analogous result for closed trisections in [Reference Gay and Kirby7]. To that end, we require a cobordism structure on X, which we obtain by first turning our boundary trisection into a Heegaard splitting.

Consider the sectors $X_1$ and $X_1'$ in $\mathcal S$ and $\mathcal S'$ respectively. These restrict to $Y_1 = Y_1'$ on the boundary. Choose a maximal collection of non-separating arcs in $\Sigma _3$ . Boundary stabilizing the two pseudo-trisections along these arcs produces two new pseudo-trisections (which we still call $\mathcal S$ and $\mathcal S'$ ). However, the restricted trisection on the boundary now has that $\Sigma _3$ is a disk. This means $Y_{2}\cup Y_{3}$ now glues along a disk, so it is itself a handlebody. In summary, the decomposition $Y_{1} \cup (Y_{2} \cup Y_{3})$ is a Heegaard splitting.

Next we view X as a cobordism from $Y_{2} \cup Y_{3}$ to $Y_{1}$ . Formally, we do this by considering a tubular neighborhood N of the Heegaard splitting surface $\Sigma _{1}\cup \Sigma _{2}$ , so that Y is of the form $(Y_1-N) \cup N \cup (Y_2 \cup Y_3 - N)$ . The theorem now essentially follows from [Reference Gay and Kirby7, Theorem 21]: there are relative handle decompositions (equivalently, Morse functions), induced by the pseudo-trisections and compatible with the cobordism structure. Cerf theory tells us that the handle decompositions are related by a series of moves which correspond to internal stabilization. In summary, after a sequence of internal stabilizations, our pseudo-trisections are isotopic.

Proof In item (1), the first equality is an application of the last item in Proposition 3.4. The second equality is an application of the second item in the aforementioned proposition.

For item (2), we use the observation that each of the three types of stabilization for pseudo-trisections increases one of the $k_i$ by 1, while leaving the other two unchanged.

For item (3), we require several steps. In step 1, we establish a general homological statement about 4-manifolds with boundary. In step 2 we establish bounds for $\beta _1$ in terms of $|\boldsymbol {k}|$ (and $|\boldsymbol {y}|$ ) for relative trisections (and pseudo-trisections).

Step 1. In the long exact sequence $H_4(X,\partial X) \xrightarrow {i} H_3(\partial X) \xrightarrow {j} H_3(X) \xrightarrow {k} H_3(X,\partial X)$ , the map k is necessarily injective because i is an isomorphism. Combining this fact with the isomorphism $H^1(X) \cong H_3(X,\partial X)$ , we have

$$ \begin{align*}\beta_1 = \operatorname{\mathrm{rank}} H^1(X) = \operatorname{\mathrm{rank}} H_3(X, \partial X) \geq \operatorname{\mathrm{rank}} H_3(X) = \beta_3.\end{align*} $$

On the other hand, $\chi (X) = 1 - \beta _1 + \beta _2 - \beta _3$ , so

$$ \begin{align*}\chi(X) - 1 \geq \beta_2 - 2\beta_1.\end{align*} $$

Step 2. Suppose $\mathcal {T}$ is either a relative trisection or pseudo-trisection. Let $\gamma $ be a neatly embedded loop in X. Then $\gamma $ meets each $X_i$ along a set of arcs whose endpoints lie in $\partial X_i$ . Since each $X_i$ is a handlebody, the arcs can be homotoped rel boundary to lie in $\partial X_i$ . We do this in every sector, to obtain a loop $\gamma '$ contained entirely within the 3-skeleton of $\mathcal {T}$ . Next we repeat this step a dimension lower: $\gamma '$ intersects each $H_i$ or $Y_i$ along an arc with endpoints in $\partial H_i$ or $\partial Y_i$ respectively. These arcs can again be homotoped rel boundary to lie within $\partial H_i$ and $\partial Y_i$ respectively. Doing this in every 3-dimensional sector, we obtain a loop $\gamma "$ contained entirely within the 2-skeleton of $\mathcal {T}$ .

The proof now diverges for pseudo-trisections and relative trisections. In the former case, notice that $\gamma "$ can be viewed as loop in $X_i \cup \Sigma _{i-1}$ for any i. This means $\pi _1(X_i \cup \Sigma _{i-1})$ surjects onto $\pi _1(X)$ . Taking into account how $\Sigma _{i-1}$ is glued to $X_i$ , it follows that $2p_{i-1} + b - 1 + k_i \geq \beta _1$ . This holds over each i, giving

$$ \begin{align*}|\boldsymbol k| + |\boldsymbol{y}| = 2|\boldsymbol{p}| + 3b - 3 + |\boldsymbol{k}| \geq 3\beta_1.\end{align*} $$

Combining this with the result from step 1 gives

$$ \begin{align*}c(\mathcal{T}) + c(\partial\mathcal{T}) = \chi(X) + |\boldsymbol{k}| - 1 + |\boldsymbol{y}| \geq 3\beta_1 + \beta_2 - 2\beta_1 = \beta_1 + \beta_2.\end{align*} $$

In the case of relative trisections, the boundary is an open book. Therefore the arcs $\gamma " \cap \Sigma _i$ can be homotoped rel boundary to any $\Sigma _j$ via the fibration defining the open book. It follows that $\gamma "$ can be homotoped to lie entirely within any one of the $X_i$ . It follows that $|\boldsymbol {k}| \geq 3\beta _1$ , and therefore that $c(\mathcal {T}) \geq \beta _1 + \beta _2$ .

Finally, for item (4), we have that

$$ \begin{align*} c(\mathcal{T}_1\natural \mathcal{T}_2) &= \chi(X_1\natural X_2) + |\boldsymbol{k}_{\mathcal{T}_1 + \mathcal{T}_2}| - 1\\ &= \chi(X_1) + \chi(X_2) - 1 + |\boldsymbol{k}_{\mathcal{T}_1}|+|\boldsymbol{k}_{\mathcal{T}_2}| - 1 = c(\mathcal{T}_1) + c(\mathcal{T}_2) \end{align*} $$

as required.

Proof The proof structure is to determine all possible pseudo-trisection diagrams with complexity 0 or 1 up to handleslides and isotopies. This proof depends on Section 4 and in particular Lemma 4.6, but we have placed it here for thematic reasons.

It is immediate that a pseudo-trisection with complexity $0$ is the trivial pseudo-trisection.

Next, suppose X is pseudo-trisected with complexity $1$ . By Proposition 3.27, $b=1$ is fixed, and exactly one of $\Sigma _C, \Sigma _1, \Sigma _2, \Sigma _3$ has genus 1 while the rest are disks.

Case 1: $\Sigma _C$ has genus 1, and each $\Sigma _i$ is a disk. Then identifying the boundary of X with the boundary of a trivially trisected $B^4$ produces a trisection of a closed manifold with trisection genus 1. By [Reference Gay and Kirby7] this trisection is the unique trisection of $\mathbb {CP}^2$ up to diffeomorphism, and consequently X is $\mathbb {CP}^2 - B^4$ , also trisected up to diffeomorphism with complexity $1$ .

Case 2: Without loss of generality, $\Sigma _1$ has genus 1, and each of $\Sigma _C, \Sigma _2, \Sigma _3$ is a disk. Then there is a pseudo-trisection diagram of the pseudo-trisection of X consisting of three simple closed curves $\alpha _1, \delta _1, \delta _3$ on $\Sigma _1$ . The triple Heegaard diagrams $(\Sigma _C, \Sigma _3,\Sigma _1,\varnothing , \delta _3,\alpha _1)$ and $(\Sigma _C, \Sigma _1, \Sigma _2, \alpha _1, \delta _1, \varnothing )$ must encode either $S^3$ or $S^1\times S^2$ , which corresponds to the intersection numbers $\alpha _1 \cdot \delta _1$ and $\delta _3 \cdot \alpha _1$ being $0$ or $1$ .

Suppose $\alpha _1 \cdot \delta _1 = 0$ and $\delta _3 \cdot \alpha _1 = 0$ . Then the resulting pseudo-trisection is of $S^1\times B^3$ , described in Example 3.13. Suppose $\alpha _1\cdot \delta _1 = 0$ and $\delta _3 \cdot \alpha _1 = 1$ . Then $\delta _3 \cdot \delta _1 = 1$ , and the resulting trisection is the Heegaard stabilization of the trivial trisection of $B^4$ . This is diffeomorphic to the trisection obtained when $\alpha _1 \cdot \delta _1 = 1$ and $\delta _3 \cdot \alpha _1 = 0$ . Finally, suppose $\alpha _1 \cdot \delta _1$ and $\delta _3 \cdot \alpha _1$ are both $1$ . Then $\delta _3 \cdot \delta _1 = e$ for any integer e, and for each e there is a unique configuration of $\alpha _1, \delta _1, \delta _3$ on $\Sigma _1$ (up to handleslides and isotopies) achieving the given intersections. Each such diagram encodes the disk bundle over $S^2$ of Euler number e. In particular, when $e=0$ , the corresponding pseudo-trisection is of $S^2\times B^2$ , described in Example 3.14.

Proof By Gompf’s generalization of Wall’s theorem [Reference Gompf8, Reference Wall27], for any two homeomorphic compact oriented smooth 4-manifolds $X, X'$ , there exists an integer k such that $X \natural ^k (S^2\times S^2 - B^4)$ and $X'\natural ^k (S^2\times S^2 - B^4)$ are diffeomorphic. Additivity of complexity would then imply that $c(X) = c(X')$ , but Proposition 3.35 lists all manifolds with complexity at most 1, and the list contains no exotic pairs.

Proof Suppose X and $X'$ are pseudo-trisected 4-manifolds constructed from a given pseudo-trisection diagram by $\mathcal R$ . Let $X_i$ and $X_i'$ be the sectors, W and $W'$ the 2-skeleta, and Z and $Z'$ be the 3-skeleta of X and $X'$ respectively. Notice that the 2-skeleta are exactly the union of the surfaces in the pseudo-trisection diagram, so we start with the identity map $W \to W'$ . Next, we extend this to a map $Z \to Z'$ by (the proof of) Proposition 2.13. This map restricts to diffeomorphisms $\partial X_i \to \partial X_i'$ as trisected 3-manifolds. Finally, we apply a theorem of Laudenbach and Poénaru [Reference Laudenbach and Poénaru15] which states that diffeomorphisms of $\#k(S^1\times S^2)$ extend to diffeomorphisms of $\natural ^k(S^1\times B^3)$ . Namely, the diffeomorphisms $\partial X_i \to \partial X_i'$ above each extend to maps $X_i \to X^{\prime }_i$ . These three extensions glue along Z to give a diffeomorphism $X \to X'$ .

Proof This essentially follows from the proof of Lemma 2.21. For each sector $Y_i$ or $H_i$ of the trisection, the corresponding meridian system (considered up to isotopy) [Reference Johannson11] is unique up to handleslides. A pseudo-trisection diagram is a collection of six overlapping meridian systems.

Proof Item (1) is a direct consequence of the definition of realiation. Specifically, the connected sum of some $\Sigma _j$ with a torus results in the 3-dimensional 1-handlebodies $H_j, Y_{j-1}$ , and $Y_j$ each gaining a handle $h_{H_j}, h_{Y_{j-1}}$ , and $h_{Y_j}$ respectively. The curves added in the type I torus stabilization ensure that $h_{H_j}$ and $h_{Y_{j-1}}$ are parallel, while the other two pairs cancel. Gluing $H_j \cup h_{H_j}, Y_{j-1} \cup h_{Y_{j-1}}$ , and $H_{j-1}$ , we obtain a 3-manifold differing from $\partial X_{j-1}$ by a connected sum with $S^1\times S^2$ . This corresponds to the boundary of the handle attached in Heegaard stabilization.

To prove item (2) one can choose any pseudo-trisection diagram $\mathcal D$ of $\mathcal T$ . The Heegaard stabilization involves an arc $\alpha $ in some $Y_j$ with endpoints $p, q$ in $\Sigma _j$ , with neighborhood $N(\alpha ) \subset Y_j$ . Let $\gamma _1$ be the arc obtained by pushing $\alpha $ to the boundary of $N(\alpha )$ , and $\gamma _2$ the arc obtained by isotoping $\alpha $ onto $\Sigma _j$ and pushing the endpoints to agree with $\gamma _1$ . Let $\gamma $ be the closed curve $\gamma _1 \cup \gamma _2$ . Next, let $\mu _1$ and $\mu _2$ be meridians of $N(\alpha )$ . Create the diagram $\mathcal D'$ by a surgery replacing disks around p and q with an annulus corresponding to the boundary of $N(\alpha )$ , and add new curves $\gamma , \mu _1, \mu _2$ to the diagram as in the definition of type I torus stabilizations. After handleslides and isotopies of $\gamma $ , the resulting diagram is a type I torus stabilization of $\mathcal D$ which has realiation $\mathcal T'$ .

Items (3) and (4) are analogous to (1) and (2) and left to the reader.

Proof The proof is similar to the proof of Proposition 4.8. The equivalence is established by viewing the surfaces of the diagram as the 2-skeleton of the pseudo-trisection, and equating the arc $\alpha $ in the definition of band stabilization with the arc used in the definition of boundary stabilization.

Proof This proof is analogous to that of Proposition 2.22. By Lemma 4.6, any two pseudo-trisection diagrams of a fixed pseudo-trisection are diffeomorphic after a sequence of handleslides and isotopies. By Theorem 1.2, any two pseudo-trisections of a given 4-manifold are equivalent up to boundary, Heegaard, and internal stabilization. By Propositions 4.10 and 4.8, band and torus stabilizations correspond to boundary, Heegaard, and internal stabilizations.

Proof Referring back to Definition 4.1, being well defined requires that each $\boldsymbol {\delta }_i$ and $\boldsymbol {\alpha }_i$ must be a cut system of its corresponding surface, and that each of the triple Heegaard diagrams induced on the three triples $(\Sigma _C', \Sigma _i', \Sigma _{i+1}')$ are specifically diagrams of connected sums of $S^1\times S^2$ .

For the first condition, observe that the only surface-pairs whose unions have changed in the operation are $\Sigma _2' \cup \Sigma _3'$ and $\Sigma _1' \cup \Sigma _C'$ . In the former, the genus is decreased by a surgery along a neighborhood of $\gamma $ , so $\boldsymbol {\delta }_2 - \{\gamma \}$ is a cut system of $\Sigma _2' \cup \Sigma _3'$ . In the latter, the genus is increased and $\mu $ is the corresponding meridian, so $\boldsymbol {\alpha }_1 \cup \{\mu \}$ is a cut system.

For the second condition, we observe that $(\Sigma _C', \Sigma _1', \Sigma _2', \boldsymbol {\alpha }_1 \cup \{\mu \}, \boldsymbol {\delta }_1, \boldsymbol {\alpha }_2)$ is a stabilisation of $(\Sigma _C, \Sigma _1, \Sigma _2, \boldsymbol {\alpha }_1, \boldsymbol {\delta }_1, \boldsymbol {\alpha }_2)$ . The latter is necessarily a diagram of $\#^{k_1}(S^1\times S^2)$ because our operation is on a valid pseudo-trisection diagram. Consequently, $(\Sigma _C', \Sigma _1', \Sigma _2', \boldsymbol {\alpha }_1\cup \{\mu \}, \boldsymbol {\delta }_1, \boldsymbol {\alpha }_2)$ is a diagram of $\#^{k_1}(S^1\times S^2)$ . Similarly, $(\Sigma _C', \Sigma _3', \Sigma _1', \boldsymbol {\alpha }_3, \boldsymbol {\delta }_3, \boldsymbol {\alpha }_1 \cup \{\mu \})$ is a diagram of $\#^{k_3}(S^1\times S^2)$ . The remaining check is that $(\Sigma _C', \Sigma _2', \Sigma _3', \boldsymbol {\alpha }_2, \boldsymbol {\delta }_2 \cup \{\gamma \}, \boldsymbol {\alpha }_3)$ is locally a destabilisation of $(\Sigma _C, \Sigma _2, \Sigma _3, \boldsymbol {\alpha }_2, \boldsymbol {\delta }_2, \boldsymbol {\alpha }_3)$ , so it is a diagram of $\#^{k_2}(S^1\times S^2)$ .

Proof Item (1). This is an immediate consequence of the definition, by restricting the diagram to $(\Sigma _i', \boldsymbol {\delta }_i')$ and observing that it stabilizes to $(\Sigma _i, \boldsymbol {\delta }_i).$

Item (2). The complexity of the pseudo-trisection is unchanged since each $k_i$ is unchanged (as was observed in the proof of Lemma 4.12.) The complexity of the boundary decreases by 1, by Proposition 2.26.

Item (3). This is proved diagrammatically, in Figure 18. The top row shows local regions of a pseudo-trisection diagram of $\mathcal T^*$ , and $\alpha $ denotes the curve along which the boundary stabilisation occurs. Applying a boundary stabilisation along $\alpha $ followed by a boundary stabilisation shift, the resulting pseudo-trisection diagram is shown in the second row. Applying diffeomorphisms to the $\Sigma _i'$ and $\Sigma _C'$ , and isotoping the curves, the bottom row is obtained. Applying handleslides and isotopies to $\mu $ , and a diffeomorphism rel boundary to $\Sigma _C'$ , the diagram of the bottom row can be expressed as a type II torus stabilisation of the initial diagram. The result now follows from Proposition 4.8.

Figure 18:

Proof of Proposition 4.13, item (3).

Figure 18: Long description

The diagram consists of three rows and four columns of panels.

Top Row:

- Panel 1: Sigma super star sub 1 shows two vertical lines with a shaded region between them and a horizontal curve alpha.

- Panel 2: Sigma super star sub 2 shows two vertical lines with shading on the outer sides.

- Panel 3: Sigma super star sub 3 is identical to the second panel.

- Panel 4: Sigma super star sub C is identical to the second and third panels.

Middle Row:

- Panel 1: Sigma prime sub 1 shows a surface with a narrow neck containing a blue horizontal line mu and a vertical arc.

- Panel 2: Sigma prime sub 2 shows the neck split into two separate rounded ends labeled gamma sub 2.

- Panel 3: Sigma prime sub 3 shows a similar split with labels gamma sub 3.

- Panel 4: Sigma prime sub C shows a combined view with labels gamma sub 2, gamma sub 3, and mu.

Bottom Row:

- Panel 1: Sigma prime sub 1 shows two vertical lines with a straight blue horizontal line mu.

- Panel 2: Sigma prime sub 2 shows two vertical lines with outer shading.

- Panel 3: Sigma prime sub 3 is identical to the second panel.

- Panel 4: Sigma prime sub C shows a three-dimensional cylinder extending from the left vertical line toward the right. A blue line mu runs through the cylinder. On the right side, three concentric colored loops (yellow, orange, and pink) are labeled gamma sub 2 and gamma sub 3.

Proof Case 1: $\tau $ is the trivial trisection of $Y = S^3$ . Fill the boundary Y with a 4-ball $B^4$ to obtain a closed manifold $X\cup B^4$ . There is a trisection of $X\cup B^4$ by the usual existence result in the closed setting [Reference Gay and Kirby7]. Choose any point in the central surface of the trisection, and there is a trivially trisected neighborhood N of the point. Now X is diffeomorphic to $(X \cup B^4) - N$ , and the latter is equipped with a pseudo-trisection with boundary $\tau $ .

Case 2: Y is equipped with any trisection $\tau $ other than the trivial trisection of $S^3$ . (Y may or may not be $S^3$ .) Equip X with a pseudo-trisection $\mathcal T$ by Corollary 3.20. By Proposition 2.10, there is a trisection $\tau '$ of Y such that there is a sequence of stabilisations and Heegaard stabilisations from $\mathcal T|_Y$ to $\tau '$ , and a sequence of stabilisations from $\tau $ to $\tau '$ . At the 4-dimensional level, there is an induced sequence of boundary stabilisations and Heegaard stabilisations from $\mathcal T$ to some pseudo-trisection $\mathcal T'$ with $\mathcal T'|_Y = \tau '$ . Further, there is a sequence of boundary stabilisation shifts from $\mathcal T'$ to a pseudo-trisection $\mathcal T_1$ with $\mathcal T_1|_Y = \tau $ .

Proof The data of the 3-skeleton describes how the triples $((H_i, \tau _i),(H_{i+1}, \tau _{i+1}),(Y_i, L_i))$ glue together to produce links in the boundaries of the sectors $X_i$ . These links are necessarily unlinks, which bound unique trivial disk-tangles in the sectors up to neat isotopy. Compare with [Reference Meier and Zupan19, Lemma 2.3 and Corollary 2.4].

Proof We must show that the map as defined truly sends a pseudo-shadow diagram to a pseudo-bridge trisection, and that any two pseudo-bridge trisections built from the same pseudo-shadow diagram by the map are neatly isotopic. (More precisely, after a diffeomorphism, the embedded surfaces differ by neat isotopy.)

Let $(X, \mathcal K)$ be the realization of a pseudo-shadow diagram $\mathcal D$ . Then each $\mathcal K \cap X_i$ is necessarily a trivial disk tangle, because it is obtained by isotoping a collection of disks rel boundary from $\partial X_i$ to $X_i$ . Similarly, each $\mathcal K \cap H_i$ and $\mathcal K \cap Y_i$ is necessarily a trivial tangle. It follows that $(X, \mathcal K)$ is a pseudo-bridge trisection.

Next, suppose $(X, \mathcal K)$ and $(X',\mathcal K')$ are realizations of the same pseudo-shadow diagram $\mathcal D$ . By Proposition 4.3, the underlying 4-manifolds X and $X'$ and their pseudo-trisections are diffeomorphic. Hereafter $\mathcal K, \mathcal K'$ are considered to both lie in X. The arcs $\tau _i$ and $L_i$ lift to the 2-skeleton of the pseudo-trisection of X, and are further isotoped into $H_i$ and $Y_i$ respectively to produce exactly the 3-skeleta of $(X, \mathcal K)$ and $(X, \mathcal K')$ . By Proposition 5.10, the 3-skeleta determine the surface in pseudo-bridge position up to neat isotopy.