Generalized trisections in all dimensions

Significance

Decomposing a manifold into handles was introduced by Smale, from the study of the critical points of smooth real-valued functions. Here we study combinatorial functions from a manifold to a simplex and use them to decompose the manifold into simple building blocks. Given a description of a manifold as the quotient space of a union of $n$-dimensional simplices, this paper constructs multisections, which describe an $n$-dimensional manifold as a union of $k+1$ $n$-dimensional handlebodies, where $n=2k$ or $2k+1.$ These handlebodies have disjoint interiors and subcollections intersect in submanifolds with spines of small dimension. The intersection of all of the handlebodies is the central submanifold $\mathrm{\Sigma }.$ This submanifold $\mathrm{\Sigma }$ can be chosen to have a special structure called a nonnegatively curved cubing.

Abstract

This paper describes a generalization of Heegaard splittings of 3-manifolds and trisections of 4-manifolds to all dimensions, using triangulations as a key tool. In particular, every closed piecewise linear $n$-manifold can be divided into $k+1$ $n$-dimensional 1-handlebodies, where $n=2k+1$ or $n=2k$, such that intersections of the handlebodies have spines of small dimensions. Several applications, constructions, and generalizations of our approach are given.

David Gay and Rob Kirby (1) introduced a beautiful decomposition of an arbitrary smooth, oriented closed $4$-manifold, called trisection, into three handlebodies glued along their boundaries as follows. Each handlebody is a boundary-connected sum of copies of $S^1\times B^3$ and has a boundary-connected sum of copies of $S^1\times S^2.$ The triple intersection of the handlebodies is a closed orientable surface $\mathrm{\Sigma },$ which divides the boundary of each handlebody into two 3D handlebodies (and hence is a Heegaard surface). These 3D handlebodies are precisely the intersections of pairs of the 4-dimensional handlebodies.

In dimensions $\le 4,$ there is a bijective correspondence between isotopy classes of smooth and piecewise linear (PL) structures (2, 3), but this breaks down in higher dimensions. This paper generalizes Gay and Kirby’s concept of a trisection to higher dimensions in the PL category. All manifolds, maps, and triangulations are therefore assumed to be PL unless stated otherwise. Our definition and results apply to any compact smooth manifold by passing to its unique PL structure (4).

1. Multisections

The definition of a multisection, which generalizes both that of a Heegaard splitting of a 3-manifold and that of a trisection of a 4-manifold, focuses on properties of spines. Let $N$ be a compact manifold with nonempty boundary. The subpolyhedron $P$ is a spine of $N$ if $P\subset \text{int}\left(N\right)$ and $N$ PL collapses onto $P.$

Definition 1 (multisection of closed manifold):

Let $M$ be a closed, connected, PL $n$-manifold. A multisection of $M$ is a collection of $k+1$ PL submanifolds $H_i\subset M,$ where $0\le i\le k$ and $n=2k$ or $n=2k+1,$ subject to the following four conditions:

• i)Each $H_i$ has a single $0$-handle and a finite number, $g_i$, of $1$-handles and is PL homeomorphic to a standard PL $n$-dimensional 1-handlebody of genus $g_i.$
• ii)The handlebodies $H_i$ have a pairwise disjoint interior, and $M={\bigcup }_i{H}_i.$
• iii)The intersection $H_{i_1}\cap H_{i_2}\cap \dots \cap H_{i_r}$ of any proper subcollection of the handlebodies is a compact submanifold with boundary and of dimension $n-r+1.$ Moreover, it has a spine of dimension $r,$ except if $n=2k$ and $r=k;$ then there is a spine of dimension $r-1.$
• iv)The intersection $H_0\cap H_1\cap \dots \cap H_k$ of all handlebodies is a closed submanifold of $M^n$ of dimension $n-k$ and called the central submanifold.

It follows from our definitions that the first condition in the above definition is equivalent to each $H_i$ having a spine graph with Euler characteristic $1-g_i.$ Moreover, each intersection $H_{i_1}\cap H_{i_2}\cap \dots \cap H_{i_r}$ is connected, where $1\le r\le k+1.$

Nomenclature.

A multisection of a 1-manifold is just the 1-manifold. The study of multisections in dimension 2 is the study of separating, simple, closed curves. A multisection of a 3-manifold is a Heegaard splitting. A trisection in the sense of Gay and Kirby (1) is a multisection of an orientable 4-manifold with the additional property that the handlebodies $H_j$ have the same genus. It is shown in ref. 5 that a multisection of an orientable 4-manifold can be modified to a trisection in the sense of ref. 1, by stabilizing the handlebodies of lower genus to achieve the same genus as the handlebody of highest genus. We therefore use the term trisection to apply to all multisections in dimension 4—if necessary, we will say that they are balanced if all handlebodies have the same genus and unbalanced otherwise. This allows us to talk about bisections ($n=\mathrm{2,3}$), trisections ($n=\mathrm{4,5}$), quadrisections ($n=\mathrm{6,7}$), etc., without further qualification.

Existence.

We recall the classical existence proof of Heegaard splittings (for instance, ref. 6), which motivates our definition in higher dimensions and provides a model for the existence proofs. Suppose that $M$ is a triangulated, closed, connected $3$-manifold and there is a partition $\left\{P_0,P_1\right\}$ of the set of all vertices in the triangulation, such that

• (1₃)for each set $P_k,$ every tetrahedron has a pair of vertices in the set, and
• (2₃)the union of all edges with both ends in $P_k$ is a connected graph ${\mathrm{\Gamma }}_k$ in $M.$

We can form regular neighborhoods of each of these graphs ${\mathrm{\Gamma }}_k,$ which are handlebodies $H_0,$ $H_1$, respectively, such that the handlebodies meet along their common boundary $\mathrm{\Sigma },$ which is a surface consisting entirely of quadrilateral disks, one in each tetrahedron, separating the vertices in $P_0,$ $P_1$ (Fig. 1). Hence $\mathrm{\Sigma }$ is a Heegaard surface in $M.$ A triangulation with the desired properties is obtained as follows. Suppose $\left|K\right|\to M$ is a triangulation of $M,$ and take the first barycentric subdivision $K\prime$ of $K.$ Let $P_0$ be the set of all vertices of $K$ and barycenters of edges of $K;$ and let $P_1$ be the set of all barycenters of the triangles and the tetrahedra of $K.$ Then $\left\{P_0,P_1\right\}$ is a partition of the vertices of $K\prime$ satisfying $\left(1_3\right)$ and $\left(2_3\right).$ Moreover, the vertices of the cubulated surface $\mathrm{\Sigma }$ have degrees $4$ or $6,$ and hence $\mathrm{\Sigma }$ is a nonpositively curved cube complex (cf. ref. 7).

Fig. 1.

Partition map for $n=3$.

Examples of triangulations of manifolds that satisfy $\left(1_3\right)$ and $\left(2_3\right),$ but are not barycentric subdivisions, are the standard 2-vertex triangulations of lens spaces (8). See 4. Constructing Multisections for a strategy to identify triangulations dual to multisections.

The partition $\left\{P_0,P_1\right\}$ defines a PL map $\varphi :M\to \left[\mathrm{0,1}\right]$ by $\varphi \left(P_0\right)=0$ and $\varphi \left(P_1\right)=1$. This is often called a height function and we refer to it as a partition map. The preimage ${\varphi }^{-1}\left(\frac{1}{2}\right)$ is a Heegaard surface $\mathrm{\Sigma }$ for $M$ as described above. The inverse image of any point in the interior of $\left[\mathrm{0,1}\right]$ is a surface isotopic to $\mathrm{\Sigma }$. The intersection of this inverse image with any tetrahedron of $\mathcal{T}$ is a quadrilateral disk ($2$-cube). The inverse image of either endpoint $0$ or $1$ is a graph and its intersection with any tetrahedron is an edge ($1$-cube). The division of the closed interval ($1$-simplex) into two half intervals is the dual decomposition into $1$-cubes. An analogous decomposition is exactly what we will use in arbitrary odd dimensions.

In even dimensions, one encounters the problem that a simplex has an odd number of vertices. In this case, one needs to add an additional modification. This is given in detail in ref. 9 in this paper. Again, this approach generalizes to all even dimensions.

An outline of the general existence result can thus be given as follows. The complete details can be found in ref. 5.

Theorem 2.

Every closed piecewise linear manifold has a multisection.

Sketch of proof:

Suppose $M$ is a closed, connected, PL manifold of dimension $n.$ Our strategy is to construct a PL map $\varphi :M\to \sigma ,$ where $\sigma$ is a $k$-simplex for $k$ satisfying $n=2k$ or $n=2k+1,$ and to obtain the multisection as the pullback of the dual cubical structure of $\sigma$ to $M.$ Our map $\varphi$ will have the property that each vertex of $\sigma$ pulls back to a connected graph, and each top-dimensional cube pulls back to a regular neighborhood of this graph, a 1-handlebody.

We use triangulations to define $\varphi .$ Since a PL manifold admits a PL triangulation $\left|K\right|\to M$ (where the link of each simplex in the simplicial complex $K$ is equivalent to a standard PL sphere), we can and will assume that such a triangulation of $M$ is fixed. Since $M$ is closed, there are a finite number of simplices in the triangulation, and $\varphi$ is uniquely determined by a partition of the vertices of the triangulation into $k+1$ sets and a bijection between the sets in this partition and the vertices of $\sigma .$ We call such a map $M\to \sigma$ a partition map. To ensure that the dual cubical structure of $\sigma$ pulls back to submanifolds with the required properties, we determine suitable combinatorial properties on the triangulation. In the odd-dimensional case, we show in ref. 5 that the first barycentric subdivision of any triangulation has a suitable partition. Moreover the $r$-dimensional spine of the intersection of $r$ handlebodies meets each top-dimensional simplex in $M$ in exactly one $r$-cube. In even dimensions, we obtain an analogous result in ref. 5 after performing bistellar moves on this subdivision.$\square$

We say that a triangulation supports a multisection if there is a partition of the vertices defining a partition map $\varphi :M\to \sigma$ with the property that the pullback of the cubical structure is a multisection. Special properties of triangulations may imply special properties of the supported multisections and vice versa. For instance, special properties of a Heegaard splitting of a 3-manifold are shown in ref. 10 to imply special properties of the dual triangulation. The cornerstone of the modern development of Heegaard splittings is the work of Casson and Gordon (11), and it is a tantalizing problem to generalize this to higher dimensions.

2. Examples

An extended set of examples of trisections of 4-manifolds can be found in ref. 1 and of multisections of higher-dimensional manifolds in ref. 5. The recent work of Gay (12); Meier, Schirmer, and Zupan (13); and Meier and Zupan (14, 15) gives some applications and constructions arising from trisections of 4-manifolds and relates them to other structures.

We begin with the “standard” tropical multisection of complex projective space and then give examples of multisections of a large class of spherical space forms. A spherical space form is a quotient space of a finite group of orthogonal transformations acting freely on a sphere. They admit a Riemannian metric of constant positive curvature. See ref. 16 for more information on spherical space forms.

The Tropical Picture of Complex Projective Space.

Consider the map $\mathbb{C}{P}^n\to {\mathrm{\Delta }}^n$ defined by

$$\left[z_0:\dots :z_n\right]\mapsto \frac{1}{\sum \left|z_k\right|}\left(\left|z_0\right|,\dots ,\left|z_n\right|\right).$$

The dual spine ${\mathrm{\Pi }}^n$ in ${\mathrm{\Delta }}^n$ is the subcomplex of the first barycentric subdivision of ${\mathrm{\Delta }}^n$ spanned by the 0-skeleton of the first barycentric subdivision minus the 0-skeleton of ${\mathrm{\Delta }}^n.$ This is shown for $n=\mathrm{2,3}$ in Fig. 2. Decomposing along ${\mathrm{\Pi }}^n$ gives ${\mathrm{\Delta }}^n$ a natural cubical structure with $n+1$ $n$-cubes, and the lower-dimensional cubes that we focus on are the intersections of nonempty collections of these top-dimensional cubes. Each $n$-cube pulls back to a $2n$-ball in $\mathbb{C}{P}^n,$ and the collection of these balls is a multisection. For example, if $n=2,$ the 2-cubes pull back to 4-balls, each 1-cube pulls back to $S^1\times D^2$ and the 0-cube pulls back to $S^1\times S^1$.

Fig. 2.

Dual cubical structure of the 2-simplex and the 3-simplex.

Lens Spaces.

Lens spaces form an important special subclass of spherical space forms.

A lens space $L\left(m:k_1,\dots k_n\right)$ is obtained as the quotient space of a free linear action of a finite cyclic group on $S^{2n-1}$, the unit sphere in ${\mathbb{C}}^n$. To be more specific, the action of ${\mathbb{Z}}_m$ on $S^{2n-1}$ is given by

$$\left(z_1,\dots ,z_n\right)\mapsto \left(z_1{e}^{\frac{2\pi i{k}_1}{m}},\dots ,z_n{e}^{\frac{2\pi i{k}_n}{m}}\right),$$

where $k_i,m$ are relatively prime positive integers, and $\left(z_1,\dots z_n\right)\in S^{2n-1}$.

Note there is an associated $S^1$-action on $L\left(m:k_1,\dots k_n\right)$. Here $\left[z_1,\dots ,z_n\right]\mapsto \left[z_{1}z,\dots ,z_{n}z\right]$, where $\left[z_1,\dots ,z_n\right]$ denotes the orbit of $\left(z_1,\dots ,z_n\right)$ under the action of ${\mathbb{Z}}_m$ on $S^{2n-1}$ and $z\in S^1\subset \mathbb{C}$.

Exceptional fibers ${\mathrm{\Gamma }}_j,1\le j\le n$ of this $S^1$-action are obtained as the sets of points $\left[0,\dots ,0,z_j,0,\dots ,0\right]$, where only one coordinate is nonzero. To find a natural multisection, we use a Dirichlet construction, based on the loops ${\mathrm{\Gamma }}_j,1\le j\le n$ as cores of the $n$ handlebodies $H_j,1\le j\le n$, each of which will then be of the form $S^1\times B^{2n-2}$. Hence

$$H_j=\left\{x\in L\left(m:k_1,\dots k_n\right):d\left(x,{\mathrm{\Gamma }}_j\right)\le d\left(x,{\mathrm{\Gamma }}_l\right),\forall l\ne j\right\},$$

where $d$ is the standard locally spherical metric induced on the lens space.

This construction is closely related to the tropical multisection of $\mathbb{C}{P}^n$ in the previous section and the multisections of $\mathbb{R}{P}^n$ in ref. 5. In particular, the central submanifold is again the $n$-torus $S^1\times \cdots \times S^1$ given by

$$\left\{\left[z_1,z_2,\dots z_n\right]:\left|z_j\right|=\frac{1}{\sqrt{n}},1\le j\le n\right\}.$$

This is an orbit of the action of the $n$-torus on the lens space by coordinate multiplication. The spine of each nonempty intersection of subsets of the handlebodies is a lower-dimensional torus, which is a singular orbit of this torus action.

3. Structure Results

Nonpositively Curved Cubings from Multisections.

We work with the combinatorial definition of a nonpositively curved cubing (ref. 7, section 2.1). A flag complex is a simplicial complex with the property that each subgraph in the 1-skeleton that is isomorphic to the 1-skeleton of a $k$-dimensional simplex is in fact the 1-skeleton of a $k$-dimensional simplex. A cube complex is nonpositively curved if the link of each vertex is a flag complex. Here, the link of a vertex in a cube complex is the simplicial complex whose $h$-simplices are the corners of $\left(h+1\right)$-cubes adjacent to the vertex. Basic facts are that the barycentric subdivision of any complex is flag and that the link (in the sense of simplicial complexes) of any simplex in a flag complex is a flag complex.

The partition map $\varphi :M\to \sigma$ can be used to pull back the dual cubical structure of the target simplex. This gives a natural cell decomposition of the submanifolds in a multisection, with cells of very simple combinatorial types. Each multisection submanifold that is the intersection of $r<k+1$ handlebodies has a spine with a cubing by $r$-cubes, and the closed, central submanifold $\mathrm{\Sigma }=H_0\cap H_1\cap \cdots \cap H_k$ has a cubing.

In ref. 5 we show that $\varphi$ can be chosen such that the cubing of $\mathrm{\Sigma }$ satisfies the Gromov link conditions (17) and hence is nonpositively curved:

Theorem 3.

Every PL manifold has a triangulation supporting a multisection, such that the central submanifold has a nonpositively curved cubing.

Since a $\left(2k+1\right)$-manifold has a central submanifold of dimension $k+1,$ this result produces manifolds with nonpositively curved cubings in each dimension. We also remark that our construction yields cubings with precisely one top-dimensional cube in the central submanifold for each top-dimensional simplex in the triangulation of the manifold.

Question 4.

What conditions does a nonpositively curved cubed $k$-manifold need to satisfy so that it is PL homeomorphic to the central submanifold in a multisection of a $\left(2k+1\right)$-manifold or a $2k$-manifold?

Uniqueness.

There is a natural stabilization procedure of multisections. In dimension 3, this increases the genus of both 3D handlebodies, while in higher dimensions, this increases the genus of just one of the top-dimensional handlebodies. The Reidemeister–Singer theorem (18, 19) states that any two Heegaard splittings of a 3-manifold have a common stabilization. Using in an essential way the uniqueness up to isotopy of genus $g$ Heegaard splittings of ${\#}^k\left(S^2\times S^1\right)$ due to Waldhausen (20), Gay and Kirby (1) show that any two trisections of a 4-manifold have a common stabilization up to isotopy. This implies that any two multisections of a 4-manifold also have a common stabilization up to isotopy.

Question 5.

Under what conditions is there a common stabilization for two given multisections of a manifold of dimension at least 5?

Our existence proof constructs multisections dual to triangulations. Conversely, up to possibly stabilizing the multisection, one can build triangulations dual to multisections. However, stabilization in higher dimensions adds summands of $S^1\times S^{n-k-1}$ to the central submanifold, and hence we expect the equivalence relation generated by stabilization to be finer than PL equivalence of the dual triangulations.

Recursive Structure of Multisections and Generalizations.

An important part of multisections is their recursive structure. By this we mean that inside a multisection of an $n$-dimensional manifold, we see a stratification of the boundary of each handlebody into lower-dimensional manifolds. For example, for a trisection where $n=\mathrm{4,5}$, we see partition functions on the boundaries of each handlebody, dividing the boundary into two pieces. For a quadrisection, where $n=\mathrm{6,7}$, the boundaries of the handlebodies are divided into three pieces. However, the top-dimensional pieces are not necessarily handlebodies, whereas all of the pieces have spines of low dimension. So these are not multisections in the sense defined above.

The same works in all dimensions. Namely for $n$-manifolds, with $n=2k$ or $n=2k+1$, the boundaries of the handlebodies have natural divisions into $k$ regions. Each of these regions has a spine of dimension at most 2. However, these regions are not necessarily 1-handlebodies.

From the viewpoint of the complexity theory of Martelli (21), such generalized multisections may be a fruitful approach to the study of classes of examples. For instance, a decomposition of a 4-manifold into 4-dimensional 1- or 2-handlebodies is a decomposition into 4-manifolds of complexity 0.

We note a useful result giving a relationship between the properties of having low-dimensional spines and connectivity of the intersection submanifolds. This requires the following definition, which applies to all subdivisions of manifolds in the recursive structure. Note that there is no relationship assumed between $n$ and $k.$

Definition 6 (generalized multisection of closed manifold):

Let $M$ be a closed, connected, PL $n$-manifold. A generalized multisection of $M$ is a collection of $k+1$ PL $n$-dimensional submanifolds $H_i\subset M,$ where $0\le i\le k,$ subject to the following three conditions:

• i)Each $H_i$ is nonempty and has a spine of codimension at least 2.
• ii)The submanifolds $H_i$ have a pairwise disjoint interior, and $M={\bigcup }_i{H}_i.$
• iii)The intersection $H_{i_1}\cap H_{i_2}\cap \dots \cap H_{i_r}$ of any proper subcollection of the submanifolds ($r\le k$) is a compact submanifold with boundary and of dimension $n-r+1.$ Moreover, it has a spine of codimension at least 2.

Proposition 7.

Suppose that a closed connected manifold $M$ has a generalized multisection into submanifolds $H_i$ for $0\le i\le k.$ Then the intersection $H_{i_1}\cap H_{i_2}\cap \dots \cap H_{i_r}$ of any collection of the submanifolds is nonempty and connected. In particular, each $H_i$ and the intersection $H_0\cap H_1\cap \dots \cap H_k$ are connected. Moreover, $H_0\cap H_1\cap \dots \cap H_k$ is a closed manifold of dimension $n-k.$

Proof:

The argument is by complete induction on $k$. To start the induction, suppose $M$ is a manifold of dimension $n$ and $k=1$, whence $M=H_0\cup H_1.$ Each component $X$ of $H_0$ has a spine of codimension 2 and hence a connected boundary. Since $M=H_0\cup H_1$ and $\text{int}\left(H_0\right)\cap \text{int}\left(H_1\right)=\varnothing ,$ we have $\partial H_0=\partial H_1.$ Therefore, there is a component $Y$ of $H_1$ such that $\partial X=\partial Y.$ But then $X\cup Y$ is a closed $n$-manifold and hence $M=X\cup Y.$ Moreover, $X\cap Y=\partial X$ is a nonempty, closed, and connected manifold of dimension $n-1.$ This proves the result for all manifolds $M$ and all generalized multisections into two submanifolds.

Before giving the general induction step, we move to $k=2.$ So assume $M$ is a manifold of dimension $n$ and that it has a generalized multisection into three submanifolds, whence $M=H_0\cup H_1\cup H_2.$ Again, each component $X$ of $H_0$ has connected boundary. If $X\cap H_2=\varnothing ,$ then there is a component $Y$ of $H_1$ such that $\partial X=\partial Y.$ But then, as above, $M=X\cup Y.$ This contradicts the fact that $H_2\ne \varnothing .$ Hence $X\cap H_2\ne \varnothing ,$ and by symmetry $X\cap H_1\ne \varnothing .$ Now $\partial X$ is a closed $\left(n-1\right)$-dimensional manifold and $\partial X=\left(X\cap H_1\right)\cup \left(X\cap H_2\right).$ Each of $X\cap H_1$ and $X\cap H_2$ is nonempty and $\left(n-1\right)$-dimensional and has a spine of codimension at least 2. Moreover, $\left(X\cap H_1\right)\cap \left(X\cap H_2\right)=X\cap H_1\cap H_2,$ which has dimension $n-2,$ and so $X\cap H_1$ and $X\cap H_2$ have a disjoint interior. Hence by induction, each of $X\cap H_1,$ $X\cap H_2$, and $X\cap H_1\cap H_2$ is nonempty and connected and the latter is a closed manifold of dimension $n-2.$ It follows that there is a unique component $Y$ of $H_1$ and a unique component $Z$ of $H_2$ such that $X\cap H_1=X\cap Y$ and $X\cap H_2=X\cap Z.$ In particular, $X\cap Y\cap Z$ is nonempty, connected, and closed. It follows that $\partial Y=\left(X\cap Y\right)\cup \left(Y\cap Z\right)$ and $\partial Z=\left(X\cap Z\right)\cup \left(Y\cap Z\right),$ whence $X\cup Y\cup Z$ has an empty boundary and hence $M=X\cup Y\cup Z.$ This finishes the proof for $k=2.$

Hence assume the conclusion holds for all manifolds and all multisections into at most $k_0$ submanifolds. Assume that we are given a generalized multisection of an $n$-manifold $M$ with $k_0+1$ submanifolds $H_0,\dots ,H_{k_0}.$ Each component $X_0$ of $H_0$ has $\partial X_0$ connected. As above, we have

$$\partial X_0=\left(X_0\cap H_1\right)\cup \dots \cup \left(X_0\cap H_{k_0}\right).$$

This is a generalized multisection of the closed manifold $\partial X_0$ with at most $k_0$ submanifolds. By the induction hypothesis, all components in the above decomposition are connected; hence there are unique components $X_i$ of $H_i$ such that $X_0\cap H_i=X_0\cap X_i,$ where we put $X_i=\varnothing$ if $X_0\cap H_i=\varnothing ,$ whence

$$\partial X_0=\left(X_0\cap X_1\right)\cup \dots \cup \left(X_0\cap X_{k_0}\right).$$

Since the nonempty submanifold $X_0\cap X_i\cap X_j$ is contained in $X_i\cap X_j,$ it follows by uniqueness that

$$\partial X_i=\left(X_i\cap X_0\right)\cup \left(X_i\cap X_1\right)\cup \dots \cup \left(X_i\cap X_{k_0}\right),$$

where we omit $X_i\cap X_i$ from the union. It follows that $X_0\cup X_1\cup \dots \cup X_{k_0}$ has an empty boundary and hence equals $M.$ In particular, each $X_i\ne \varnothing .$ This completes the proof.$\square$

4. Constructing Multisections

We first explain how the symmetric representations of ref. 22 can be used to construct multisections. A number of applications of this approach are given in ref. 5. Here we highlight generalized multisections and twisted multisections. The interested reader can find multisections of connected sums and products, a Dirichlet construction, and the case of manifolds with a nonempty boundary in ref. 5.

As a second construction, we describe another type of twisted multisections. These arise as generalizations of certain one-sided Heegaard splittings.

Constructing Multisections Using Symmetric Representations.

Given a triangulated $n$-manifold $\left|K\right|\to M$ with the property that the degree of each $\left(n-2\right)$-simplex is even, the authors defined a symmetric representation ${\pi }_1\left(M\right)\to \text{Sym}\left(n+1\right)$ in ref. 22 as follows. Pick one $n$-simplex as a base, choose a bijection between its corners and $\left\{1,\dots ,n+1\right\}$, and then reflect this labeling across its codimension-1 faces to the adjacent $n$-simplices. This induced labeling is propagated further and if one returns to the base simplex, one obtains a permutation of the vertex labels. Since the dual 1-skeleton carries the fundamental group, it can be shown that this gives a homomorphism ${\pi }_1\left(M\right)\to \text{Sym}\left(n+1\right).$ See ref. 22, section 2.3 for details. For example, the symmetric representation associated to any barycentric subdivision is trivial, since the labels correspond to the dimension of the simplex containing that vertex in its interior, but there may be more efficient even triangulations with this property.

The symmetric representation can also be used to propagate partitions of the vertices of the base simplex; this is done in ref. 22, section 2.5 for partitions into two sets, but extends to arbitrary partitions. One then obtains an induced representation, usually into a symmetric group of larger degree. The aim in ref. 22 was to obtain information on the topology of a manifold from a nontrivial symmetric representation arising from a triangulation with few vertices. Our needs in this paper are different: We wish to use the symmetric representations to identify triangulations to which we can apply our constructions without barycentric subdivision. So either we want the orbits of the vertices under the symmetric representation to give a partition satisfying the conditions in our constructions or we ask for partitions of the vertices with the property that the induced representation is trivial.

The main properties to check for a given partition of the vertices are that the graphs spanned by the partition sets are connected and, in even dimensions, that the dimension of the spine drops when intersecting all but one of the handlebodies.

The following are two applications of this approach.

Generalized Multisections.

Suppose that $M$ is a triangulated $n$-manifold with an even triangulation with trivial symmetric representation. As above, given any triangulation, the first barycentric subdivision has this property. We can define some special generalized multisections as follows.

Suppose that $n=3k+2$. Assume that we have partition sets $P_0,P_1,\dots P_k$ where the sets meet every $n$-simplex in three vertices. We then map each $n$-simplex to the $k$-simplex by mapping each partition set to a vertex of this $k$-simplex. It is then easy to verify that we obtain a division of $M$ into $k+1$ regions, and each region has a two-dimensional spine, given by the union of all of the $2$-simplices in each $n$-simplex with all vertices in the same partition set. In this case, the manifold $\mathrm{\Sigma }$, which is the intersection of all of the handlebodies, is closed of dimension $2k+2$. Again we can arrange that the induced cubing of $\mathrm{\Sigma }$ is nonpositively curved and each intersection of a proper subcollection has a spine of low dimension.

Another interesting example is to have two partition sets of size $k\prime ,k^{\star }$ of the vertices of each $n$-simplex, so that $k\prime +k^{\star }=n+1$. We assume that both $k\prime >1,k^{\star }>1$. The induced decomposition is a bisection into two regions with spines of dimension $k\prime ,k^{\star }$. Given a handle decomposition of $M$, this is similar to a hypersurface which is the boundary of the region containing all of the $i$-handles for $0\le i\le k\prime$.

Finally, a very specific example is a $6$-manifold $M$ with three partition sets of respective sizes $\mathrm{2,2,3}$. This induces a trisection of $M$ into three regions, where two are handlebodies and the third one has a spine of dimension $2$.

Twisted Multisections.

Suppose a closed PL $n$-manifold has an even triangulation with a nontrivial symmetric representation. Assume also that the symmetry preserves our standard partition of the vertices; i.e., every symmetry mapping produces a permutation of the partition sets of vertices. Then there is an associated “twisted” multisection, which we illustrate with a simple example—the general construction then becomes clear.

Assume $M$ is a 5-manifold that admits an even triangulation with a symmetric representation with image ${\mathbb{Z}}_3$. Also assume this symmetry is a permutation of the form $\left(012\right)\left(345\right)$ of the labeling of the vertices. In this case, we choose as partition sets $\left\{\mathrm{0,3}\right\},\left\{\mathrm{1,4}\right\},\left\{\mathrm{2,5}\right\}.$ Then these are permuted under the action of the symmetric mapping. The edges joining these three pairs of vertex sets form a connected graph $\mathrm{\Gamma }$.

A regular neighborhood of $\mathrm{\Gamma }$ then forms a single handlebody $H$ whose boundary is glued to itself to form $M$. The handlebody $H$ lifts to three handlebodies in a regular threefold covering space $\tilde{M}$ of $M$ and these give a standard trisection of $\tilde{M}$. The covering transformation group ${\mathbb{Z}}_3$ permutes the handlebodies and preserves the central submanifold. If the initial triangulation is flag, then the lifted triangulation is flag and hence the central submanifold has a nonpositively curved cubing on which the covering transformation group acts isometrically. Hence the quotient, which embeds in $\partial H,$ also has a nonpositively curved cubing.

Twisted Multisections of Some Other Spherical Space Forms.

We define a natural generalization of the one-sided Heegaard splittings of 3D spherical space forms which have fundamental groups that are either dihedral or binary dihedral by cyclic groups, discussed in ref. 23. Such splittings are given by embedded Klein bottles with complements open solid tori. One-sided Heegaard splittings are examples of twisted multisections in dimension 3.

Consider the unit sphere $S^{4n-1}$ in ${\mathbb{H}}^n$, where $\mathbb{H}$ is the quaternions. Let $G$ be a suitable direct product of a finite subgroup of the unit quaternions that is dihedral or binary dihedral and a relatively prime order cyclic subgroup. In particular we require that joint left and right multiplication, respectively, of these factors of $G$ on the unit quaternions defines a free action.

Now use the above to define a diagonal action of $G$ on the $n$ quaternionic factors in $S^{4n-1}\subset {\mathbb{H}}^n$. We claim there is a twisted multisection of the spherical space form $S^{4n-1}/G$, consisting of $n$ copies of $S^1\times B^{4n-2}$ glued together.

An easy way to see how this multisection is constructed is to pass to the twofold cover of $S^{4n-1}/G$ by a lens space, using a normal cyclic subgroup of $G$ of index $2$. The multisection of this lens space as described in the previous subsection is easily seen to be invariant under the covering transformation. In fact, this covering involution interchanges pairs of exceptional fibers, and hence in the quotient space there are $n$ loops which are projections of the $2n$ exceptional fibers.

The multisection of the lens space is obtained by a Dirichlet construction from the exceptional fibers and hence all of the components, i.e., intersections of families of neighborhoods of exceptional fibers, are invariant or interchanged under the involution. So it is easy to verify that there is an induced twisted multisection of $S^{4n-1}/G$ as claimed.

5. Category of $\left(k+1\right)$-Colored Structures

A manifold $M$ admits a $\left(k+1\right)$-coloring if it has a triangulation where the vertices are partitioned into $k+1$ sets $P_0,P_1,\dots P_k$ so that every top-dimensional simplex has either one or two vertices in each set $P_i$.

If two manifolds $M,N$ have triangulations which both admit $\left(k+1\right)$-colorings, then a color-preserving mapping $\varphi :M\to N$ is a simplicial map which takes the partition sets $P_0,P_1,\dots P_k$ of the vertices of $M$ to the partition sets ${P\prime }_0,{P\prime }_1,\dots {P\prime }_k$ of the vertices of $N$.

There is clearly a category of $\left(k+1\right)$-colored structures defined this way. Note that the basic construction of a multisection arises from a color-preserving mapping $\varphi :M\to \sigma$ where $\sigma$ is the $\left(k+1\right)$-simplex with the trivial partition consisting of one vertex in each partition set.

We can also specialize to $\left(k+1\right)$-colored structures which induce multisections. The corresponding color-preserving map then takes the multisection of the domain to the multisection of the range.

Waldhausen (24) showed that given any degree-one mapping $\varphi$ between 3-manifolds $M,N$ and a Heegaard splitting $\mathrm{\Sigma }$ for $N$, the map $\varphi$ can be homotoped so that ${\varphi }^{-1}\left(\mathrm{\Sigma }\right)$ is a Heegaard splitting for $M$. Moreover after the homotopy, the map $\varphi$ can be put into a standard form. This implies there are 2-colored triangulations of both $M,N$ and a color-preserving map between them in the homotopy class of $\varphi$.

Question 8.

Given a degree-one mapping $\varphi$ between 4-manifolds, $M,N$ and a trisection of $N$ is there a trisection of $M$ so that $\varphi$ can be homotoped to a map taking one trisection to the other? In particular, is there a color-preserving map from $M$ to $N$ with respect to 3-colored triangulations of $M,N$, where the 3-colored triangulation of $N$ induces the given trisection of $N$?