Functoriality of group trisections

Significance

In three dimensions, it has been known for some time that, by using the fact that all three-manifolds admit Heegaard splittings, three-manifold topology can be understood to be in some sense equivalent to understanding maps from surface groups to free groups. Recently, a decomposition for smooth four-manifolds analogous to Heegaard splittings has been discovered and used to establish a similar group-theoretic framework for studying smooth four-manifolds. We review these constructions and show how they are in fact functorial.

Abstract

Building on work by Stallings, Jaco, and Hempel in three dimensions and a more recent four-dimensional analog by Abrams, Kirby, and Gay, we show how the splitting homomorphism and group trisection constructions can be extended to functors between appropriate categories. This further enhances the bridge between smooth four-dimensional topology and the group theory of free and surface groups.

We begin by reviewing research done by Stallings, Jaco, and Hempel that, by way of taking ${\pi }_1$ of Heegaard splittings, creates a group-theoretic framework for three-manifolds. In particular, they found purely algebraic statements that are equivalent to the Poincaré conjecture. Recently, following the introduction of the notion of trisections of four-manifolds by Gay and Kirby (1), a similar story has unfolded for four-manifolds (2). We explain how these constructions can be made functorial by introducing the appropriate categories and constructing categorical equivalences. We end with some open questions that arise from setting up this framework.

Functoriality of Heegaard Splitting

All manifolds will be closed and orientable unless stated otherwise. Let $M$ be a three-manifold together with a Heegard splitting $M=H_1\cup H_2$, where $H_1$ and $H_2$ are genus $g$ handlebodies and $\mathrm{\Sigma }=H_1\cap H_2=\partial H_1=\partial H_2$. Fixing a basepoint $x\in \mathrm{\Sigma }$, we have the group homomorphism induced by the Heegaard splitting (3)

$${\pi }_1\left(\mathrm{\Sigma },x\right)\to {\pi }_1\left(H_1,x\right)\times {\pi }_1\left(H_2,x\right)$$

resulting from the inclusion of the surface into the two handlebodies that is surjective on each of the factors. Let $S_g=⟨a_1,b_1,\dots a_g,b_g|{\prod }_{i=1}^g\left[a_i,b_i\right]⟩$ denote the fundamental group of a closed, orientable surface of genus $g$, which we will call the surface group of genus $g$. Let $F_g$ denote the free group of rank $g$. Let ${\mathrm{\Sigma }}_g$ denote the closed, orientable surface of genus $g$ and for each $g$ fix a basepoint that we will denote $x$ (we could write $x_g$ as we will deal with surfaces of different genera, but we will abuse notation instead). After fixing isomorphisms ${\pi }_1\left(\mathrm{\Sigma },x\right)\cong S_g,{\pi }_1\left(H_1,x\right)\cong F_g$ and ${\pi }_1\left(H_2,x\right)\cong F_g$, we have a homomorphism

$$S_g\to F_g\times F_g.$$

We will call such homomorphisms from a genus $g$ surface group to a product of rank $g$ free groups that are surjective onto each factor splitting homomorphisms.

In refs. 4 and 5, it is shown how to construct a three-manifold $M$ together with a genus $g$ Heegaard spltting of $M$ from a given splitting homomorphism. We now review this construction. Let

$$g_1\times g_2:S_g\to F_g\times F_g$$

be a splitting homomorphism. Let $W_g$ be the wedge of $g$ circles where the wedge point is denoted $y$ and assume the wedge point on each circle is not the north pole. Fix once and for all isomorphsms $S_g\cong {\pi }_1\left({\mathrm{\Sigma }}_g,x\right)$ and $F_g\cong {\pi }_1\left(W_g,y\right)$. We then obtain

$$g_1\times g_2:{\pi }_1\left(\mathrm{\Sigma },x\right)\to {\pi }_1\left(W_g,y\right)\times {\pi }_1\left(W_g,y\right).$$

Since ${\mathrm{\Sigma }}_g$ and $W_g$ are both $K\left(\pi ,1\right)$ spaces, the group homomorphisms $g_i:{\pi }_1\left(\mathrm{\Sigma },x\right)\to {\pi }_1\left(W_g,y\right)$ can be induced by continuous maps $G_i:{\mathrm{\Sigma }}_g\to W_g$ for $i=\mathrm{1,2}$. Let $n_i\in W_g$ for $1\le i\le g$ be the north pole of the $i^{\text{th}}$ circle in $W_g$, and let $N=\left\{n_1,\dots ,n_g\right\}$. Jaco shows how we can in fact choose the continuous maps $G_i$ so that $\left\{G_i^{-1}\left(n_j\right):1\le j\le g\right\}$ is a collection of simple closed curves in ${\mathrm{\Sigma }}_g$ whose union is nonseparating. In addition, Jaco arranges the maps so that the mapping cylinders $M_{G_i}$ are handlebodies. We then have the commutative diagrams

where the parameterization $F_g\cong {\pi }_1\left(W_g,y\right)$ yields a parameterization $F_g\cong {\pi }_1\left(M_{G_i},x\right)$ and the induced map on the fundamental groups

$$S_g\cong {\pi }_1\left({\mathrm{\Sigma }}_g,x\right)\to {\pi }_1\left(M_{G_i},x\right)\cong F_g$$

is just $g_i$.

We now have inclusions ${\mathrm{\Sigma }}_g\to M_{G_i}$, and by using these to glue $M_{G_1}$ to $M_{G_2}$ along $\mathrm{\Sigma }$, we obtain a three-manifold $M$ with a Heegaard splitting $M=M_{G_1}\cup M_{G_2}$. In addition, $M$ has a natural base point $x$—namely, the chosen base point of ${\mathrm{\Sigma }}_g$—and we have parameterizations ${\pi }_1\left(M_{G_i},x\right)\cong F_g$ and ${\pi }_1\left({\mathrm{\Sigma }}_g,x\right)\cong S_g$ coming from our chosen parameterizations earlier. We will now address the uniqueness of $M$. First, we will need a topological lemma that we will state and prove in a level of generality that will be useful later on.

Lemma 2.1: Let $H$ and $H\prime$ be handlebodies. Any map $f:\partial H\to \partial H\prime$ extends to a map $F:H\to H\prime$ if and only if every curve $\gamma$ in $\partial H$ that bound a disk in $H$, $f\left(\gamma \right)$ bounds a disk in $H\prime$. If the map $f$ is a homeomorphism, then $F$ can be chosen to be a homeomorphism.

Proof: Note that by Dehn’s lemma, if an embedded curve in $\partial H$ or $\partial H\prime$ bounds a disk, then it will bound an embedded disk. Henceforth, we will assume that the disks are embedded. If $f$ extends to a map on the entire handlebody and $\gamma$ is a curve in $\partial H$ that bound a disk $D$ in $H$, then $f\left(\gamma \right)$ bounds the disk $F\left(D\right)$. Conversely, let $D_1,\dots ,D_g$ be a minimal system of disks for $H$. We will construct the map $F$ in pieces. By assumption, the curves $f\left(\partial D_1\right),\dots ,f\left(\partial D_g\right)$ all bound disks in $H\prime$, and therefore, we can extend the map $f$ from $\partial H$ to $H\cup \left\{D_1,\dots ,D_g\right\}$. Now $H\backslash \left\{\partial H,D_1,\dots D_g\right\}$ is an open ball, and we have defined the map $F$ on the boundary already. Since ${\pi }_2\left(H\prime \right)=0$, the map $F$ extends to all of $H$. If $f$ is a homeomorphism, then at each step we can arrange for $F$ to also be a homeomorphism.

A map between manifolds with boundary is called proper if the image of the boundary of the domain is contained in the boundary of the codomain.

Lemma 2.2: Let $H$ and $H\prime$ be handlebodies, and let $f_1,f_2:H\to H\prime$ be proper maps. Then, any homotopy $\partial H\times I\to \partial H\prime$ between $f_1{|}_{\partial H}$ and $f_2{|}_{\partial H}$ extends to a homotopy $H\times I\to H\prime$ between $f_1$ and $f_2$.

Proof: Fix a regular cell decomposition for $H$ with only two cells and three cells in $\mathrm{I}\mathrm{n}\mathrm{t}H$. This then gives a regular cell decomposition of $H\times I$. Consider the map

$$\partial H\times I\cup H\times \left\{\mathrm{0,1}\right\}\to H\prime$$

given by the homotopy $\partial H\times I\to \partial H^{\prime }$ together with $f_1$ and $f_2$ on $H\times \left\{0\right\}$ and $H\times \left\{1\right\}$, respectively. This map can be extended over the three cells and four cells in $\mathrm{I}\mathrm{n}\mathrm{t}H\times \left(\mathrm{0,1}\right)$ since ${\pi }_2\left(H\prime \right)={\pi }_3\left(H\prime \right)=0$.

We will need the following result to prove the uniqueness of Jaco’s construction (see ref. 6).

Theorem 2.3: [Dehn–Nielsen–Baer] Given an isomorphism $f:{\pi }_1\left(\mathrm{\Sigma },x\right)\to {\pi }_1\left({\mathrm{\Sigma }}^{\prime },x\prime \right)$, there exists a homeomorphism $F:\left(\mathrm{\Sigma },x\right)\to \left(\mathrm{\Sigma }\prime ,x\prime \right)$ such that ${\pi }_1\left(F\right)=f$.

Proposition 2.4: Let $M$ and $M\prime$ be closed, orientable three-manifolds with Heegaard splittings $M=H_1\cup H_2$, $M\prime =H_1^{\prime }\cup H_2^{\prime }$. Denote the splitting surfaces $\mathrm{\Sigma }=H_1\cap H_2$ and $\mathrm{\Sigma }\prime =H_1^{\prime }\cap H_2^{\prime }$. Given basepoints $x\in \mathrm{\Sigma }$ and $x\prime \in \mathrm{\Sigma }\prime$ and a commutative diagram of isomorphisms

there is a homeomorphism $\left(M,H_1,H_2,x\right)\to \left(M\prime ,H_1^{\prime },H_2^{\prime },x\prime \right)$ such that the isomorphisms in the above commutative diagram are the maps induced by ${\pi }_1$. Such a map is unique up to base point-preserving homotopy.

Proof: We construct a map $f:M\to M\prime$ first on the splitting surface and then use the commutativity of the above diagram to extend over the handlebodies. We are given an isomorphism

$${\pi }_1\left(\mathrm{\Sigma },x\right)\to {\pi }_1\left(\mathrm{\Sigma }\prime ,x\prime \right),$$

and therefore, by theorem 2.3, there is some homeomorphism $f:\left(\mathrm{\Sigma },x\right)\to \left(\mathrm{\Sigma }\prime ,x\prime \right)$ that induces this map, and since $\mathrm{\Sigma }$ and $\mathrm{\Sigma }\prime$ are $K\left(\pi ,1\right)$ spaces, $f$ is unique up to base point-preserving homotopy. Now consider the commutative diagram

Let $\gamma$ be a curve in $\mathrm{\Sigma }$ that bounds a disk in $H_1$. If $x\in \gamma$, then since $\gamma$ bounds a disk in $H_1$, $\left[\gamma \right]=0\in {\pi }_1\left(H_1,x\right)$, and therefore, by the commutativity of the diagram, $\left[f○\gamma \right]=0\in {\pi }_1\left(H_1^{\prime },x\prime \right)$ so $f\left(\gamma \right)$ bounds a disk in $H_1^{\prime }$. Thus, by lemma 2.1, we can extend $f$ over $H_1$ as a homeomorphism onto $H\prime$. If $x\notin \gamma$, after choosing an arc connecting $x$ to $\gamma$, a similar argument goes through.

By lemma 2.2, the resulting homeomorphism is unique up to base point-preserving homotopy since any two choices will be homotopic when restricted to the splitting surface, and we can then extend the homotopy over each of the handlebodies.

Using this result, we see that if $M$ and $M\prime$ are the three-manifolds that result from applying the above construction to the same splitting homomorphism $g_1\times g_2$, where for $M$ we use the maps $G_1,G_2$ and for $M\prime$ we use the maps $G_1^{\prime },G_2^{\prime }$, then for $M$ and $M\prime$ we have

and therefore, $M$ and $M\prime$ are homeomorphic with a homeomorphism as in the preceding proposition. We will use the notation $M\left(g_1\times g_2\right)$ to represent a result of Jaco’s construction applied to some splitting homomorphism $g_1\times g_2$ (thus sweeping under the rug the choice of $G_1$ and $G_2$, as these do not affect the homeomorphism class of the result). Since every three-manifold admits a Heegaard splitting, every three-manifold can be obtained by applying Jaco’s construction to some splitting homomorphism.

Let SplitHom be the category where the objects are splitting homomorphisms

$$S_g\to F_g\times F_g,$$

and the morphisms are commutative diagrams

Given such a commutative diagram, we can construct a map $M\left(g_1\times g_2\right)\to M\left(g_1^{\prime }\times g_2^{\prime }\right)$ that preserves the base points and Heegaard splitting. The construction of this map is the same construction as the proof of proposition 2.4, however we cannot make all of the maps homeomorphisms and therefore will not invoke theorem 2.3. The same argument also shows that the resulting map is unique up to base point-preserving homotopy.

Letting $M\left(g_1\times g_2\right)=H_1\cup H_2$ and $M\left(h_1\times h_2\right)=H_1^{\prime }\cup H_2^{\prime }$ and applying the parameterizations of the fundamental groups, we have the commutative diagram

which we show is just the original morphism between the splitting homomorphisms. The map $H_1\cap H_2\to H_1^{\prime }\cap H_2^{\prime }$ is constructed so that upon applying ${\pi }_1$ we obtain the given map $S_g\to S_h$. Given surjections of the surface groups onto the free groups, the dotted map in

is uniquely determined if it exists. Therefore, our map $H_1\to H_1^{\prime }$ must induce the original map $F_g\to F_h$ after applying ${\pi }_1$ and similarly for $H_2$.

Let Split3Man be the category whose objects are Heegaard split three-manifolds $M=H_1\cup H_2$ with a specified base point $x\in H_1\cap H_2$ and parameterizations ${\pi }_1\left(H_1,x\right)\cong F_g\cong {\pi }_1\left(H_2,x\right)$, ${\pi }_1\left(H_1\cap H_2,x\right)\cong S_g$, and the morphisms are maps preserving the splitting and base point considered up to homotopy preserving the Heegaard splittings and the base point. Then, there is a functor ${\pi }_1:\mathbf{Split3Man}\to \mathbf{SplitHom}$ given by taking ${\pi }_1$. The above construction describes a functor $\mathcal{M}:\mathbf{SplitHom}\to \mathbf{Split3Man}$ once a particular choice of representative of $M\left(g_1\times g_2\right)$ has been made for every splitting homomorphism $g_1\times g_2$.

Theorem 2.5: The functor $\mathcal{M}$ above is an equivalence of categories

$$\mathcal{M}:\mathbf{SplitHom}\to \mathbf{Split3Man}.$$

Proof: Faithfulness follows from lemma 2.2, together with the observation that surfaces other than $S^2$ are $K\left(\pi ,1\right)$ spaces. Fullness follows from applying the ${\pi }_1$ functor followed by $\mathcal{M}$.

Essential surjectivity follows by taking an object $\left(M,H_1,H_2,x\right)$ in $\mathbf{Split3Man}$, applying ${\pi }_1$ and then $\mathcal{M}$, and noting that by proposition 2.4 the result is isomorphic in $\mathbf{Split3Man}$ to $\left(M,H_1,H_2,x\right)$.

Corollary 2.6: If $\left(M,H_1,H_2,x\right)$ and $\left(M\prime ,H_1^{\prime },H_2^{\prime },x\prime \right)$ are two based Heegaard split three-manifolds and $f:\left(M,H_1,H_2,x\right)\to \left(M\prime ,H_1^{\prime },H_2^{\prime },x^{\prime }\right)$ is a based homotopy equivalence that restricts to based homotopy equivalences on $H_1,H_2$ and $H_1\cap H_2$, then $M$ and $M\prime$ are homeomorphic.

Proof: The map $f$ is exactly an isomorphism in the category $\mathbf{Split3Man}$. Therefore, after choosing parameterizations and taking ${\pi }_1$, we see that the resulting splitting homomorphisms are isomorphic in $\mathbf{SplitHom}$. Therefore, the original manifolds must in fact be homeomorphic since an isomorphism of the splitting homomorphisms can be used as above to construct such a homeomorphism.

Using Waldhausen’s theorem that all Heegaard splittings of $S^3$ are homeomorphic (7), together with the observation that a splitting homomorphism corresponds to a simply connected three-manifold if and only if the splitting homormorphism is surjective (3), it has been shown that the three-dimensional Poincaré conjecture is equivalent to a purely algebraic statement involving splitting homomorphisms (8). A purely algebraic operation on splitting homomorphisms has been introduced that corresponds to stabilizations of Heegaard splittings (4). Using this operation together with the Reidemeister–Springer theorem that all Heegaard splittings of the same three-manifold are stably homemorphic (9, 10), a bijection between stable isomorphism classes of splitting homomorphisms and closed, connected three-manifolds has been given (5).

Functoriality of Trisections

All manifolds that we consider will be smooth, closed, and orientable unless stated otherwise. In ref. 1, the notion of a $\left(g,k\right)$-trisection of a four-manifold $M$ is introduced, and it is shown that all four-manifolds admit a trisection for some choice of $g$ and $k$. Trisections are a four-dimensional analog of Heegaard splittings. In ref. 2, the notion of a trisection of a group $G$ is given, and in a result analogous to Jaco’s construction, it is shown how given a group trisection one can obtain a trisected based parameterized four-manifold $M$ with ${\pi }_1\left(M,x\right)\cong G$. We review this construction now and show how it can be extended to a functor in a way that is completely analogous to how Jaco’s construction was extended to a functor in the previous section. We will use the notation and results in ref. 2. By a four-dimensional handlebody, we will mean a smooth four-manifold $X$ diffeomorphic to ${♮}^k{S}^1\times B^3$. Fix once and for all a base point $x\in \partial \left({♮}^k{S}^1\times D^3\right)$ and an isomorphism ${\pi }_1\left({♮}^k{S}^1\times D^3,x\right)\cong F_k$.

We note that since all of the faces in a group trisection are pushouts (for the definition, see ref. 11), all of the data of a group trisection are actually just contained in the one corner of the cube that can be written as a map

$$g_1\times g_2\times g_3:S_g\to F_g\times F_g\times F_g$$

that is surjective onto each factor and has the property that the pushout of each pair

$$g_i\times g_j:S_g\to F_g\times F_g$$

is isomorphic to $F_k$ for some $k$ and for all $1\le i<j\le 3$. We will call such maps splitting homomorphisms. As in the previous section, we can find maps $G_1,G_2,G_3$ that realize the maps $g_1,g_2$ and $g_3$, respectively, and then construct three-manifolds $M\left(G_1,G_2\right),M\left(G_2,G_3\right)$ and $M\left(G_3,G_1\right)$. Since these three-manifolds have free fundamental group $F_k$, by Perelman’s proof of the geometrization conjecture (12), they are all homeomorphic to ${♯}^k{S}^1\times S^2$. All of these come with Heegaard splittings, which we will denote $M\left(G_i,G_j\right)=H_{ij}^i\cup H_{ij}^j$, and we have identifications $H_{ij}^j\cong H_{jk}^j$. After gluing together the three three-manifolds along the handlebodies, we have three handlebodies, which we denote $H_{12},H_{23}$, and $H_{31}$ all bounding a common surface $\mathrm{\Sigma }$. We then choose diffeomorphisms

$${\varphi }_1:\partial \left({♮}^k{S}^1\times D^3\right)\to H_{12}\cup H_{31},$$

$${\varphi }_2:\partial \left({♮}^k{S}^1\times D^3\right)\to H_{12}\cup H_{23},$$

$${\varphi }_3:\partial \left({♮}^k{S}^1\times D^3\right)\to H_{23}\cup H_{31},$$

each preserving the base points.

Then, after gluing together $M\left(G_1,G_2\right),M\left(G_2,G_3\right)$ and $M\left(G_3,G_1\right)$ and then gluing in the four-handlebodies using ${\varphi }_1,{\varphi }_2$ and ${\varphi }_3$, the result is a smooth four-manifold $M\left(G_1,G_2,G_3,{\varphi }_1,{\varphi }_2,{\varphi }_3\right)$. To prove that this result in fact does not depend on any choices, the following theorem is required (see ref. 13). We remark that this whole discussion could be carried through with unbalanced trisections where we allow for three numbers $k_1,k_2,$ and $k_3$ in place of just $k$. All of the theory that we discuss will go through in this case, however in the interest of simplifying notation, we will just work with balanced trisections. We will make use of the following result in ref. 13.

Theorem 3.1: Let $X$ and $X\prime$ be four-dimensional handlebodies. Then, any diffeomorphism $f:\partial X\to \partial X\prime$ can be extended to a diffeomorphism $F:X\to X\prime$.

Using the same idea as the proof proposition 2.4, we can show that the choices of the maps $G_i$ are irrelevant, and using theorem 3.1, we see that the ${\varphi }_j$s are also. Thus, as before in the three-dimensional setting, we will denote the resulting four-manifold by just $M\left(g_1\times g_2\times g_3\right)$. Let $X_i$ denote the image in $M\left(g_1\times g_2\times g_3\right)$ of ${\varphi }_i$. Then, from the construction, $M\left(g_1\times g_2\times g_3\right)$ has a natural choice of base point, and we have parameterizations

$${\pi }_1\left(H_{12}\cap H_{23}\cap H_{31},x\right)\cong S_g,$$

$${\pi }_1\left(H_{12},x\right)\cong {\pi }_1\left(H_{23},x\right)\cong {\pi }_1\left(H_{31},x\right)\cong F_g,$$

and

$${\pi }_1\left(X_1,x\right)\cong {\pi }_1\left(X_2,x\right)\cong {\pi }_1\left(X_3,x\right)\cong F_k.$$

The inclusions therefore induce a commutative cube where all maps are surjective and all faces are pushouts

We now state and prove the necessary topological lemmas that are the four-dimensional analogues of lemma 2.1 and lemma 2.2 that will allow us to make the above construction into a functor.

Lemma 3.2: Let $X$ and $X\prime$ be four-dimensional handlebodies. Any map $f:\partial X\to \partial X\prime$ extends to a map $F:X\to X\prime$.

Proof: Notice that $X$ and $X\prime$ are both diffeomorphic to spaces that result from attaching one-handle to a zero-handle (and then smoothing corners). Therefore, we can find cell decompositions of $X$ and $X\prime$ that have all of the zero-, one-, and two-cells in the boundary. Then, the map $f$ extends over the three-cells and four-cells because ${\pi }_2\left(X\prime \right)={\pi }_3\left(X\prime \right)=0$.

Lemma 3.3: Let $X$ and $X\prime$ be four-dimensional handlebodies, and let $f,g:X\to X\prime$ be proper maps. Then, any homotopy $\partial X\times I\to \partial X\prime$ between ${f|}_{\partial X}$ and ${g|}_{\partial X}$ extends to a homotopy $X\times I\to X\prime$ between $f$ and $g$.

Proof: Using the cell decomposition from the preceding proof, this follows just as in the proof of lemma 2.2 since ${\pi }_3\left(X\prime \right)={\pi }_4\left(X\prime \right)=0$.

As before, we have a category that we will also call SplitHom (since from context, it will be clear if we are dealing with three-manifolds or four-manifolds) where the objects are splitting homomorphisms

$$S_g\to F_g\times F_g\times F_g$$

and the morphisms are commutative diagrams

Note that given such a commutative diagram, if we instead consider the commutative cubes corresponding to each splitting homomorphism, then since each side is a pushout, by the universal property of pushouts, we get unique maps between the other vertices making the diagram commute. In particular, at the vertex of the cube opposite $S_g$ is the fundamental group of the resulting four-manifold, and therefore, we have a map between the fundamental groups of the corresponding four-manifolds. Given such a commutative diagram, we construct a map $M\left(g_1\times g_2\times g_3\right)\to M\left(g_1^{\prime }\times g_2^{\prime }\times g_3^{\prime }\right)$ that preserves the base points and trisection decompositions. The construction of this map starts as in the same construction of the maps between three-manifolds in the previous section. Letting $M\left(g_1\times g_2\times g_3\right)=X_1\cup X_2\cup X_3$ and $M\left(g_1^{\prime }\times g_2^{\prime }\times g_3^{\prime }\right)=X_1^{\prime }\cup X_2^{\prime }\cup X_3^{\prime }$ denote the splittings into four-dimensional handlebodies, we have $H_{ij}=X_i\cap X_j$ and $H_{ij}^{\prime }=X_i^{\prime }\cap X_j^{\prime }$. As in the construction of the maps between the Heegaard split three-manifolds in the preceding section, we can construct the map

$$H_{12}\cup H_{23}\cup H_{31}\to H_{12}^{\prime }\cup H_{23}^{\prime }\cup H_{31}^{\prime }$$

by first constructing a map on the surfaces inducing the map $S_g\to S_h$ and then extending over all of the handlebodies using the commutativity of the squares

as before. Using lemma 3.2, we can then extend this map over each of the four-dimensional handlebodies. The constructed map preserves the base points and the trisections of the involved four-manifolds. Applying the parameterizations to the fundamental groups of the pieces, we can recover the morphism between the splitting homomorphisms. As in the preceding section, using lemma 2.2 together with lemma 3.3, we find that any two ways of constructing the map $M\left(g_1\times g_2\times g_3\right)\to M\left(g_1^{\prime }\times g_2^{\prime }\times g_3^{\prime }\right)$ are homotopic, with homotopies preserving the base points and the trisections.

Let $\mathbf{Trisect4Man}$ be the category whose objects are trisected four-manifolds $M=X_1\cup X_2\cup X_3$ with a specified base point $x\in X_1\cap X_2\cap X_3$ and parameterizations ${\pi }_1\left(X_1\cap X_2,x\right)\cong {\pi }_1\left(X_2\cap X_3,x\right)\cong {\pi }_1\left(X_3\cap X_1,x\right)\cong F_g$, ${\pi }_1\left(X_1\cap X_2\cap X_3,x\right)\cong S_g$, and the morphisms are maps preserving the splitting and base point considered up to homotopy preserving the Heegaard splittings and the base point. Then, there is a functor ${\pi }_1:\mathbf{Trisect4Man}\to \mathbf{SplitHom}$ given by taking ${\pi }_1$. The above construction describes a functor $\mathcal{M}:\mathbf{SplitHom}\to \mathbf{Trisect4Man}$ once a particular choice of representative of $M\left(g_1\times g_2\times g_3\right)$ has been made for every splitting homomorphism $g_1\times g_2\times g_3$.

Theorem 3.4: The functor $\mathcal{M}:\mathbf{SplitHom}\to \mathbf{Trisect4Man}$ above is an equivalence of categories.

Proof: Faithfulness follows from lemma 2.2, together with the observation that surfaces are $K\left(\pi ,1\right)$ spaces. Fullness follows from applying the ${\pi }_1$ functor followed by $\mathcal{M}$. Essential surjectivity follows by starting with an object $\left(X,X_1,X_2,X_3,x\right)$ in $\mathbf{Trisect4Man}$, applying ${\pi }_1$ and then $\mathcal{M}$, and noting that as in 2.4 (but adapted to triples) together with 3.1, the result is isomorphic in $\mathbf{Trisect4Man}$ to $\left(X,X_1,X_2,X_3,x\right)$.

Corollary 3.5: If $\left(X,X_1,X_2,X_3,x\right)$ and $\left(X\prime ,X_1^{\prime },X_2^{\prime },X_3^{\prime },x\prime \right)$ are two based trisected four-manifolds, and $f:\left(X,X_1,X_2,X_3,x\right)\to \left(X\prime ,X_1^{\prime },X_2^{\prime },X_3^{\prime },x\prime \right)$ is a based homotopy equivalence that restricts to based homotopy equivalences on $X_1\cap X_2,X_2\cap X_3,X_3\cap X_1$ and $X_1\cap X_2\cap X_3$, then $X$ and $X\prime$ are diffeomorphic.

Proof: As before, the map $f$ is exactly an isomorphism in the category $\mathbf{Split4Man}$. Therefore, after choosing parameterizations and taking ${\pi }_1$, we see that the resulting splitting homomorphisms are isomorphic in $\mathbf{SplitHom}$. Therefore, the original manifolds must in fact be diffeomorphic since an isomorphism of the splitting homomorphisms can be used as above to construct such a diffeomorphism from proposition 2.4 and theorem 3.1.

Questions

In this section, we discuss some questions and further directions motivated by the above results. Since all three- and four-manifolds admit Heegaard splittings and trisections, respectively, we have seen how all of the objects in the category of (smooth, closed, orientable) three- and four-manifolds are represented algebraically by splitting homomorphisms. However, it is not clear to the author to what extent all maps between three-manifolds (respectively, four-manifolds) are seen in this algebraic picture. Namely, given a continuous map $f:M\to M\prime$ between closed, connected, orientable three-manifolds, do there exist Heegaard splittings $M=H_1\cup H_2$ and $M\prime =H_1^{\prime }\cup H_2^{\prime }$ and a map $g$ that is homotopic to $f$ with the property that $g\left(H_1\right)\subset H_1^{\prime }$ and $g\left(H_2\right)\subset H_2^{\prime }$? This appears to have been answered affirmatively for degree 1 maps due to Waldhausen (14) (see theorem 2.1). A similar question could be formulated for four-manifolds. These questions address to what extent the above categories $\mathbf{Split3Man}$ and $\mathbf{Trisct4Man}$ actually contain all of the morphisms in the categories of three- and four-manifolds (with maps considered up to homotopy).

There is also the question of formulating an $n$-dimensional notion of splitting homomorphisms and thus creating a group-theoretic framework for smooth manifold topology in any dimension. This should ideally correspond to taking ${\pi }_1$ of a higher dimensional generalization of trisections. Finally, we should mention that to the author’s knowledge, no application of this algebraic framework for three- or four-manifolds has ever been given. It would be interesting to see if a group-theoretic understanding of maps from surface groups to free groups could be used to understand Heegaard splittings and trisections. Additionally, it would be interesting to see how various invariants of three- and four-manifolds could be seen at the level of splitting homomorphisms.