On locally constructible spheres and balls

Introduction

Ambjørn, Boulatov, Durhuus, Jonsson, and others have worked to develop a three-dimensional analogue of the simplicial quantum gravity theory, as provided for two dimensions by Regge . (See and for surveys.) The discretized version of quantum gravity considers simplicial complexes instead of smooth manifolds; the metric properties are artificially introduced by assigning length $a$ to any edge. (This approach is due to Weingarten and known as “theory of dynamical triangulations”.) A crucial path integral over metrics, the “partition function for gravity”, is then defined via a weighted sum over all triangulated manifolds of fixed topology. In three dimensions, the whole model is convergent only if the number of triangulated $3$-spheres with $N$ facets grows not faster than $C^{N}$, for some constant $C$. But does this hold? How many simplicial spheres are there with $N$ facets, for $N$ large?

Without the restriction to “local constructibility” this crucial question still represents a major open problem, which was put into the spotlight also by Gromov [19, pp. 156-157]. Its 2D-analogue, however, was answered long time ago by Tutte , who proved that there are asymptotically fewer than \big{(}\frac{16}{3\sqrt{3}}\big{)}^{N} combinatorial types of triangulated $2$-spheres. (By Steinitz’ theorem, cf. [49, Lect. 4], this quantity equivalently counts the maximal planar maps on $n\geq 4$ vertices, which have $N=2n-4$ faces, and also the combinatorial types of simplicial $3$-dimensional polytopes with $N$ facets.)

In the following, the adjective “simplicial” will often be omitted when dealing with balls, spheres, or manifolds, as all the regular cell complexes and polyhedral complexes that we consider are simplicial.

Why are $2$-spheres “not so many”? Every combinatorial type of triangulation of the $2$-sphere can be generated as follows (Figure 1): First for some even $N\geq 4$ build a tree of $N$ triangles (which combinatorially is the same thing as a triangulation of an ($N+2$)-gon), and then glue edges according to a complete matching of the boundary edges. A necessary condition in order to obtain a $2$-sphere is that such a matching is planar. Planar matchings and triangulations of ($N+2$)-gons are both enumerated by a Catalan number $C_{N+2}$, and since the Catalan numbers satisfy a polynomial bound $C_{N}=\frac{1}{N+1}\binom{2N}{N}<4^{N}$, we get an exponential upper bound for the number of triangulations.

Neither this simple argument nor Tutte’s precise count can be easily extended to higher dimensions. Indeed, we have to deal with three different problems when trying to extend results or methods from dimension two to dimension three:

Many combinatorial types of simplicial $3$-spheres are not realizable as boundaries of convex $4$-polytopes; thus, even though we observe below that there are only exponentially-many simplicial $4$-polytopes with $N$ facets, the $3$-spheres could still be more numerous.

The counts of combinatorial types according to the number $n$ of vertices and according to the number $N$ of facets are not equivalent any more. We have $3n-10\leq N\leq\frac{1}{2}n(n-3)$ by the lower resp. upper bound theorem for simplicial $3$-spheres. We know that there are more than $2^{n\sqrt{n}}$ $3$-spheres , but less than $2^{20n\log n}$ types of $4$-polytopes with $n$ vertices , yet this does not answer the question for a count in terms of the number $N$ of facets.

While it is still true that there are only exponentially-many “trees of $N$ tetrahedra”, the matchings that can be used to glue $3$-spheres are not planar any more; thus, they could be more than exponentially-many. If, on the other hand, we restrict ourselves to “local gluings”, we generate only a limited family of $3$-spheres, as we will show below.

In the early nineties, new finiteness theorems by Cheeger and Grove et al. yielded a new approach, namely, to count $d$-manifolds of “fluctuating topology” (not necessarily spheres) but “bounded geometry” (curvature and diameter bounded from above, and volume bounded from below). This allowed Bartocci et al. to bound for any $d$-manifold the number of triangulations with $N$ or more facets, under the assumption that no vertex had degree higher than a fixed integer. However, for this it is crucial to restrict the topological type: Already for $d=2$, there are more than exponentially many triangulated $2$-manifolds of bounded vertex degree with $N$ facets.

In 1995, the physicists Durhuus and Jonsson introduced the class of “locally constructible” (LC) $3$-spheres. An LC $3$-sphere (with $N$ facets) is a sphere obtainable from a tree of $N$ tetrahedra, by identifying pairs of adjacent triangles in the boundary. “Adjacent” means here “sharing at least one edge”, and represents a dynamic requirement. Clearly, every $3$-sphere is obtainable from a tree of $N$ tetrahedra by matching the triangles in its boundary; according to the definition of LC, however, we are allowed to match only those triangles that are adjacent – or that have become adjacent by the time of the gluing.

Durhuus and Jonsson proved an exponential upper bound on the number of combinatorially distinct LC spheres with $N$ facets. Based also on computer simulations (, see also and ) they conjectured that all $3$-spheres should be LC. A positive solution of this conjecture would have implied that spheres with $N$ facets are at most $C^{N}$, for a constant $C$ – which would have been the desired missing link to implement discrete quantum gravity in three dimensions.

In the present paper, we show that the conjecture of Durhuus and Jonsson has a negative answer: There are simplicial $3$-spheres that are not LC. (With this, however, we do not resolve the question whether there are fewer than $C^{N}$ simplicial $3$-spheres on $N$ facets, for some constant $C$.)

On the way to this result, we provide a characterization of LC simplicial $d$-complexes which relates the “locally constructible” spheres defined by physicists to concepts that originally arose in topological combinatorics.

A simplicial $d$-sphere, $d\geq 3$, is LC if and only if the sphere after removal of one facet can be collapsed down to a complex of dimension $d-2$. Furthermore, there are the following inclusion relations between families of simplicial $d$-spheres:

We use the hierarchy in conjunction with the following extension and sharpening of Durhuus and Jonsson’s theorem (who discussed only the case $d=3$).

For fixed $d\geq 2$, the number of combinatorially distinct simplicial LC $d$-spheres with $N$ facets grows not faster than $2^{d^{2}\cdot N}$.

We will give a proof for this theorem in Section 4; the same type of upper bound, with the same type of proof, also holds for LC $d$-balls with $N$ facets.

Already in 1988 Kalai constructed for every $d\geq 4$ a family of more than exponentially many $d$-spheres on $n$ vertices; Lee later showed that all of Kalai’s spheres are shellable. Combining this with Theorem 4.4 and Theorem 2.1, we obtain the following asymptotic result:

For fixed $d\geq 4$, the number of shellable simplicial $d$-spheres grows more than exponentially with respect to the number $n$ of vertices, but only exponentially with respect to the number $N$ of facets.

The hierarchy of Main Theorem 1 is not quite complete: It is still not known whether constructible, non-shellable $3$-spheres exist (see ). A shellable $3$-sphere that is not vertex-decomposable was found by Lockeberg in his 1977 Ph.D. work (reported in [33, p. 742]; see also ). Again, the $2$-dimensional case is much simpler and completely solved: All $2$-spheres are vertex decomposable (see ).

In order to show that not all spheres are LC we study in detail simplicial spheres with a “knotted triangle”; these are obtained by adding a cone over the boundary of a ball with a knotted spanning edge (as in Furch’s 1924 paper ; see also Bing ). Spheres with a knotted triangle cannot be boundaries of polytopes. Lickorish had shown in 1991 that

a $3$-sphere with a knotted triangle is not shellable if the knot is at least $3$-complicated.

Here “at least $3$-complicated” refers to the technical requirement that the fundamental group of the complement of the knot has no presentation with less than four generators. A concatenation of three or more trefoil knots satisfies this condition. In 2000, Hachimori and Ziegler demonstrated that Lickorish’s technical requirement is not necessary for his result:

a $3$-sphere with any knotted triangle is not constructible.

In the present work, we re-justify Lickorish’s technical assumption, showing that this is exactly what we need if we want to reach a stronger conclusion, namely, a topological obstruction to local constructibility. Thus, the following result is established in order to prove that the last inclusion of the hierarchy in Theorem 2.1 is strict.

A $3$-sphere with a knotted triangle is not LC if the knot is at least $3$-complicated.

The knot complexity requirement is now necessary, as non-constructible spheres with a single trefoil knot can still be LC (see Example 2.26).

The combinatorial topology of $d$-balls and that of $d$-spheres are of course closely related – our study builds on the well-known connections and also adds new ones.

A simplicial $d$-ball is LC if and only if after the removal of a facet it collapses down to the union of the boundary with a complex of dimension at most $d-2$. We have the following hierarchy:

All the inclusions of Main Theorem 4 hold with equality for simplicial $2$-balls. In the case of $d=3$, collapsibility onto a $(d-2)$-complex is equivalent to collapsibility. In particular, we settle a question of Hachimori (see e.g. [23, pp. 54, 66]) whether all constructible $3$-balls are collapsible.

Furthermore, we show in Corollary 3.24 that some collapsible $3$-balls do not collapse onto their boundary minus a facet, a property that comes up in classical studies in combinatorial topology (compare ). In particular, a result of Chillingworth can be restated in our language as “if for any geometric simplicial complex $\Delta$ the support (union) $|\Delta|$ is a convex $3$-dimensional polytope, then $\Delta$ is necessarily an LC $3$-ball”, see Theorem 3.27. Thus any geometric subdivision of the $3$-simplex is LC.

In the following, we present the notion of “local constructibility” (due to Durhuus and Jonsson). Although in the end we are interested in this notion as applied to finite simplicial complexes, the iterative definition of locally constructible complexes dictates that for intermediate steps we must allow for the greater generality of finite “simplicial regular CW complexes”. A CW complex is regular if the attaching maps for the cells are injective on the boundary (see e.g. ). A regular CW-complex is simplicial if for every proper face $F$, the interval $[0,F]$ in the face poset of the complex is boolean. Every simplicial complex (and in particular, any triangulated manifold) is a simplicial regular CW-complex.

The $k$-dimensional cells of a regular CW complex $C$ are called $k$-faces; the inclusion-maximal faces are called facets, and the inclusion-maximal proper subfaces of the facets are called ridges. The dimension of $C$ is the largest dimension of a facet; pure complexes are complexes where all facets have the same dimension. All complexes that we consider in the following are finite, most of them are pure. A $d$-complex is a $d$-dimensional complex. Conventionally, the 0-faces are called vertices, and the $1$-faces edges. (In the discrete quantum gravity literature, the $(d-2)$-faces are sometimes called “hinges” or “bones”, whereas the edges are sometimes referred to as “links”.) If the union $|C|$ of all simplices of $C$ is homeomorphic to a manifold $M$, then $C$ is a triangulation of $M$; if $C$ is a triangulation of a $d$-ball or of a $d$-sphere, we will call $C$ simply a $d$-ball (resp. $d$-sphere). The dual graph of a pure $d$-dimensional simplicial complex $C$ is the graph whose nodes correspond to the facets of $C$: Two nodes are connected by an arc if and only if the corresponding facets share a $(d-1)$-face.

1.2 Knots

All the knots we consider are tame, that is, realizable as $1$-dimensional subcomplexes of some triangulated $3$-sphere. A knot is $m$-complicated if the fundamental group of the complement of the knot in the $3$-sphere has a presentation with $m+1$ generators, but no presentation with $m$ generators. By “at least $m$-complicated” we mean “$k$-complicated for some $k\geq m$”. There exist arbitrarily complicated knots: Goodrick showed that the connected sum of $m$ trefoil knots is at least $m$-complicated.

Another measure of how tangled a knot can be is the bridge index (see e.g. [32, p. 18] for the definition). If a knot has bridge index $b$, the fundamental group of the knot complement admits a presentation with $b$ generators and $b-1$ relations [32, p. 82]. In other words, the bridge index of a $t$-complicated knot is at least $t+1$. As a matter of fact, the connected sum of $t$ trefoil knots is $t$-complicated, and its bridge index is exactly $t+1$ .

1.3 The combinatorial topology hierarchy

In the following, we review the key properties from the inclusion

valid for all simplicial complexes, and the inclusion

applicable only for contractible simplicial complexes, both known from combinatorial topology (see [10, Sect. 11] for details).

Shellability can be defined for pure simplicial complexes as follows:

a $d$-dimensional pure simplicial complex $C$ which is not a simplex is shellable if and only if it can be written as $C=C_{1}\cup C_{2}$, where $C_{1}$ is a shellable $d$-complex, $C_{2}$ is a $d$-simplex, and $C_{1}\cap C_{2}$ is a shellable $(d-1)$-complex.

Constructibility is a weakening of shellability, defined by:

a $d$-dimensional pure simplicial complex $C$ which is not a simplex is constructible if and only if it can be written as $C=C_{1}\cup C_{2}$, where $C_{1}$ and $C_{2}$ are constructible $d$-complexes, and $C_{1}\cap C_{2}$ is a constructible $(d-1)$-complex.

Let $C$ be a $d$-dimensional simplicial complex. An elementary collapse is the simultaneous removal from $C$ of a pair of faces $(\sigma,\Sigma)$ with the following prerogatives:

$\sigma$ is not a proper face of any other face of $C$.

(The three conditions above are usually abbreviated in the expression “$\sigma$ is a free face of $\Sigma$”; some complexes have no free face). If $C^{\prime}:=C-\Sigma-\sigma$, we say that the complex $C$ collapses onto the complex $C^{\prime}$. We also say that the complex $C$ collapses onto the complex $D$, and write $C\searrow D$, if $C$ can be reduced to $D$ by a finite sequence of elementary collapses. Thus a collapse refers to a sequence of elementary collapses. A collapsible complex is a complex that can be collapsed onto a single vertex.

Since $C^{\prime}:=C-\Sigma-\sigma$ is a deformation retract of $C$, each collapse preserves the homotopy type. In particular, all collapsible complexes are contractible. The converse does not hold in general: For example, the so-called “dunce hat” is a contractible $2$-complex without free edges, and thus with no elementary collapse to start with. However, the implication “contractible $\Rightarrow$ collapsible” holds for all $1$-complexes, and also for shellable complexes of any dimension.

A connected $2$-dimensional complex is collapsible if and only if it does not contain a $2$-dimensional complex without a free edge. In particular, for $2$-dimensional complexes, if $C\searrow D$ and $D$ is not collapsible, then $C$ is also not collapsible. This holds no more for complexes $C$ of dimension larger than two .

1.4 LC pseudomanifolds

By a $d$-pseudomanifold [possibly with boundary] we mean a finite regular CW-complex $P$ that is pure $d$-dimensional, simplicial, and such that each $(d-1)$-dimensional cell belongs to at most two $d$-cells. The boundary of the pseudomanifold $P$, denoted $\partial P$, is the smallest subcomplex of $P$ containing all the $(d-1)$-cells of $P$ that belong to exactly one $d$-cell of $P$.

According to our definition, a pseudomanifold needs not be a simplicial complex; it might be disconnected; and its boundary might not be a pseudomanifold.

For $d\geq 2$, let $C$ be a pure $d$-dimensional simplicial complex with $N$ facets. A local construction for $C$ is a sequence $T_{1},T_{2},\ldots,T_{N},\ldots,T_{k}$ ($k\geq N$) such that $T_{i}$ is a $d$-pseudomanifold for each $i$ and

if $i\leq N-1$, then $T_{i+1}$ is obtained from $T_{i}$ by gluing a new $d$-simplex to $T_{i}$ alongside one of the $(d-1)$-cells in $\partial T_{i}$;

if $i\geq N$, then $T_{i+1}$ is obtained from $T_{i}$ by identifying a pair $\sigma,\tau$ of $(d-1)$-cells in the boundary $\partial T_{i}$ whose intersection contains a $(d-2)$-cell $F$;

We say that $C$ is locally constructible, or LC, if a local construction for $C$ exists. With a little abuse of notation, we will call each $T_{i}$ an LC pseudomanifold. We also say that $C$ is locally constructed along $T$, if $T$ is the dual graph of $T_{N}$, and thus a spanning tree of the dual graph of $C$.

The identifications described in item (3) above are operations that are not closed with respect to the class of simplicial complexes. Local constructions where all steps are simplicial complexes produce only a very limited class of manifolds, consisting of $d$-balls with no interior $(d-3)$-faces. (When in an LC step the identified boundary facets intersect in exactly a $(d-2)$-cell, no $(d-3)$-face is sunk into the interior, and the topology stays the same.)

However, since by definition the local construction in the end must arrive at a pseudomanifold $C$ that is a simplicial complex, each intermediate step $T_{i}$ must satisfy severe restrictions: for each $t\leq d$,

distinct $t$-simplices that are not in the boundary of $T_{i}$ share at most one $(t-1)$-simplex;

distinct $t$-simplices in the boundary of $T_{i}$ that share more than one $(t-1)$-simplex will need to be identified by the time the construction of $C$ is completed.

if $\sigma,\tau$ are the two $(d-1)$-cells glued together in the step from $T_{i}$ to $T_{i+1}$, $\sigma$ and $\tau$ cannot belong to the same $d$-simplex of $T_{i}$; nor can they belong to two $d$-simplices that are already adjacent in $T_{i}$.

For example, in each step of the local construction of a $3$-sphere, no two tetrahedra share more than one triangle. Moreover, any two distinct interior triangles either are disjoint, or they share a vertex, or they share an edge; but they cannot share two edges, nor three; and they also cannot share one edge and the opposite vertex. If we glued together two boundary triangles that belong to adjacent tetrahedra, no matter what we did afterwards, we would not end up with a simplicial complex any more. Roughly speaking,

a locally constructible $3$-sphere is a triangulated $3$-sphere obtained from a tree of tetrahedra $T_{N}$ by repeatedly identifying two adjacent triangles in the boundary.

As we mentioned, the boundary of a pseudomanifold need not be a pseudomanifold. However, if $P$ is an LC $d$-pseudomanifold, then $\partial P$ is automatically a $(d-1)$-pseudomanifold. Nevertheless, $\partial P$ may be disconnected, and thus, in general, it is not LC.

All LC $d$-pseudomanifolds are simply connected; in case $d=3$, their topology is controlled by the following result.

Every LC $3$-pseudomanifold $P$ is homeomorphic to a $3$-sphere with a finite number of “cacti of $3$-balls” removed. (A cactus of $3$-balls is a tree-like connected structure in which any two $3$-balls share at most one point.) Thus the boundary $\partial P$ is a finite disjoint union of cacti of $2$-spheres. In particular, each connected component of $\partial P$ is a simply-connected $2$-pseudomanifold.

Thus every closed $3$-dimensional LC pseudomanifold is a sphere, while for $d>3$ other topological types such as products of spheres are possible (see Benedetti ).

On LC Spheres

In this section, we establish the following hierarchy announced in the introduction.

For all $d\geq 3$, we have the following inclusion relations between families of simplicial $d$-spheres:

The first two inclusions, and strictness of the first one, are known; the third one will follow from Lemma 2.23 and will be shown to be strict by Example 2.26 together with Lemma 2.24; finally, Corollary 2.22 will establish the strictness of the fourth inclusion for all $d\geq 3$.∎

Let $S$ be a simplicial $d$-sphere ($d\geq 2$), and $T$ a spanning tree of the dual graph of $S$. We denote by $K^{T}$ the subcomplex of $S$ formed by all the $(d-1)$-faces of $S$ that are not intersected by $T$.

Let $S$ be any $d$-sphere with $N$ facets. Then for every spanning tree $T$ of the dual graph of $S$,

$K^{T}$ is a contractible pure $(d-1)$-dimensional simplicial complex with $\frac{dN-N+2}{2}$ facets;

for any facet $\Delta$ of $S$, $\;\;S-\Delta\;\searrow K^{T}$.

Any collapse of a $d$-sphere $S$ minus a facet $\Delta$ to a complex of dimension at most $d-1$ proceeds along a dual spanning tree $T$. To see this, fix a collapsing sequence. We may assume that the collapse of $S-\Delta$ is ordered so that the pairs $((d-1)\textrm{-face},\;d\textrm{-face})$ are removed first. Whenever both the following conditions are met:

$\sigma$ is the $(d-1)$-dimensional intersection of the facets $\Sigma$ and $\Sigma^{\prime}$ of $S$;

the pair $(\sigma,\Sigma)$ is removed in the collapsing sequence of $S-\Delta$,

draw an oriented arrow from the center of $\Sigma^{\prime}$ to the center of $\Sigma$. This yields a directed spanning tree $T$ of the dual graph of $S$, where $\Delta$ is the root. Indeed, $T$ is spanning because all $d$-simplices of $S-\Delta$ are removed in the collapse; it is connected, because the only free $(d-1)$-faces of $S-\Delta$, where the collapse can start at, are the proper $(d-1)$-faces of the “missing simplex” $\Delta$; it is acyclic, because the center of each $d$-simplex of $S-\Delta$ is reached by exactly one arrow. We will say that the collapsing sequence acts along the tree $T$ (in its top-dimensional part). Thus the complex $K^{T}$ appears as intermediate step of the collapse: It is the complex obtained after the $(N-1)$st pair of faces has been removed from $S-\Delta$.

By a facet-killing sequence for a $d$-dimensional simplicial complex $C$ we mean a sequence $C_{0},C_{1},\ldots,C_{t-1},C_{t}$ of complexes such that $t=f_{d}(C)$, $C_{0}=C$, and $C_{i+1}$ is obtained by an elementary collapse that removes a free $(d-1)$-face $\sigma$ of $C_{i}$, together with the unique facet $\Sigma$ containing $\sigma$.

If $C$ is a $d$-complex, and $D$ is a lower-dimensional complex such that $C\searrow D$, there exists a facet-killing sequence $C_{0}$, $\ldots$, $C_{t}$ for $C$ such that $C_{t}\searrow D$. In other words, the collapse of $C$ onto $D$ can be rearranged so that the pairs $\left((d-1)\textrm{-face},\,d\textrm{-face}\right)$ are removed first. In particular, for any $d$-complex $C$, the following are equivalent:

there exists a facet-killing sequence for $C$;

there exists a $k$-complex $D$ with $k\leq d-1$ such that $C\searrow D$.

What we argued before can be rephrased as follows:

Let $S$ be a $d$-sphere, and $\Delta$ a $d$-simplex of $S$. Let $C$ be a $k$-dimensional simplicial complex, with $k\leq d-2$. Then,

The right-hand side in the equivalence of Proposition 2.4 does not depend on the $\Delta$ chosen. So, for any $d$-sphere $\Delta$, either $S-\Delta$ is collapsible for every $\Delta$, or $S-\Delta$ is not collapsible for any $\Delta$.

One more convention: by a natural labeling of a rooted tree $T$ on $n$ vertices we mean a bijection $b:V(T)\longrightarrow\{1,\ldots,n\}$ such that if $v$ is the root, $b(v)=1$, and if $v$ is not the root, there exists a unique vertex $w$ adjacent to $v$ such that $b(w)<b(v)$.

We are now ready to link the LC concept with collapsibility. Take a $d$-sphere $S$, a facet $\Delta$ of $S$, and a rooted spanning tree $T$ of the dual graph of $S$, with root $\Delta$. Since $S$ is given, fixing $T$ is really the same as fixing the manifold $T_{N}$ in the local construction of $S$; and at the same time, fixing $T$ is the same as fixing $K^{T}$.

Once $T$, $T_{N}$, and $K^{T}$ have been fixed, to describe the first part of a local construction of $S$ (that is, $T_{1},\ldots,T_{N}$) we just need to specify the order in which the tetrahedra of $S$ have to be added, which is the same as to give a natural labeling of $T$. Besides, natural labelings of $T$ are in bijection with collapses $S-\Delta\searrow K^{T}$ (the $i$-th facet to be collapsed is the node of $T$ labeled $i+1$; see Proposition 2.4).

What if we do not fix $T$? Suppose $S$ and $\Delta$ are fixed. Then the previous reasoning yields a bijection among the following sets:

the set of all facet-killing sequences of $S-\Delta$;

the set of “natural labelings” of spanning trees of $S$, rooted at $\Delta$;

the set of the first parts $(T_{1},\ldots,T_{N})$ of local constructions for $S$, with $T_{1}=\Delta$.

Can we understand also the second part of a local construction “combinatorially”? Let us start with a variant of the “facet-killing sequence” notion.

A pure facet-massacre of a pure $d$-dimensional simplicial complex $P$ is a sequence $P_{0},P_{1},\ldots,P_{t-1},P_{t}$ of (pure) complexes such that $t=f_{d}(P)$, $P_{0}=P$, and $P_{i+1}$ is obtained by $P_{i}$ removing:

(a) a free $(d-1)$-face $\sigma$ of $P_{i}$, together with the unique facet $\Sigma$ containing $\sigma$, and

(b) all inclusion-maximal faces of dimension smaller than $d$ that are left after the removal of type (a) or, recursively, after removals of type (b).

In other words, the (b) step removes lower-dimensional facets until one obtains a pure complex. Since $t=f_{d}(P)$, $P_{t}$ has no facets of dimension $d$ left, nor inclusion-maximal faces of smaller dimension; hence $P_{t}$ is empty. The other $P_{i}$’s are pure complexes of dimension $d$. Notice that the step $P_{i}\longrightarrow P_{i+1}$ is not a collapse, and does not preserve the homotopy type in general. Of course $P_{i}\longrightarrow P_{i+1}$ can be “factorized” in an elementary collapse followed by a removal of a finite number of $k$-faces, with $k<d$. However, this factorization is not unique, as the next example shows.

Let $P$ be a full triangle. $P$ admits three different facet-killing collapses (each edge can be chosen as free face), but it admits only one pure facet-massacre, namely $P,\emptyset$.

Let $P$ be a pure $d$-dimensional simplicial complex. Every facet-killing sequence of $P$ naturally induces a unique pure facet-massacre of $P$. All pure facet-massacres of $P$ are induced by some (possibly more than one) facet-killing sequence.

The map consists in taking a facet-killing sequence $C_{0}$, $\ldots$, $C_{t}$, and “cleaning up” the $C_{i}$ by recursively killing the lower-dimensional inclusion-maximal faces. As the previous example shows, this map is not injective. It is surjective essentially because the removed lower-dimensional faces are of dimension “too small to be relevant”. In fact, their dimension is at most $d-1$, hence their presence can interfere only with the freeness of faces of dimension at most $d-2$; so the list of all removals of the form $((d-1)\hbox{-face},\,d\hbox{-face})$ in a facet-massacre yields a facet-killing sequence. ∎

Let $S$ be a $d$-sphere; fix a spanning tree $T$ of the dual graph of $S$. The second part of a local construction for $S$ along $T$ corresponds bijectively to a facet-massacre of $K^{T}$.

Fix $S$ and $T$; $T_{N}$ and $K^{T}$ are determined by this. Let us start with a local construction $\left(T_{1},\ldots,T_{N-1},\right)T_{N},\ldots,T_{k}$ for $S$ along $T$. Topologically, $S=T_{N}/{\sim}$, where ${\sim}$ is the equivalence relation determined by the gluing (two distinct points of $T_{N}$ are equivalent if and only if they will be identified in the gluing). Moreover, $K^{T}=\partial T_{N}/{\sim}$, by the definition of $K^{T}$.

Define $P_{0}:=K^{T}=\partial T_{N}/{\sim}$, and $P_{j}:=\partial T_{N+j}/{\sim}$. We leave it to the reader to verify that $k-N$ and $f_{d}(K^{T})$ are the same integer (see Lemma 2.2), which from now on is called $D$. In particular $P_{D}=\partial T_{k}/{\sim}=\partial S/{\sim}=\emptyset$.

In the first LC step, $T_{N}\rightarrow T_{N+1}$, we remove from the boundary a free ridge $r$, together with the unique pair $\sigma^{\prime},\sigma^{\prime\prime}$ of facets of $\partial T_{N}$ sharing $r$. At the same time, $r$ and the newly formed face $\sigma$ are sunk into the interior. This step $\partial T_{N}\longrightarrow\partial T_{N+1}$ naturally induces an analogous step $\partial T_{N+j}/{\sim}\longrightarrow\partial T_{N+j+1}/{\sim}$, namely, the removal of $r$ and of the (unique!) $(d-1)$-face $\sigma$ containing it.

In the $j$-th LC step, $\partial T_{N+j}\longrightarrow\partial T_{N+j+1}$, we remove from the boundary a ridge $r$ together with a pair $\sigma^{\prime},\sigma^{\prime\prime}$ of facets sharing $r$; moreover, we sink into the interior a lower-dimensional face $F$ if and only if we have just sunk into the interior all faces containing $F$. The induced step from $\partial T_{N+j}/{\sim}$ to $\partial T_{N+j+1}/{\sim}$ is precisely a “facet-massacre” step.

For the converse, we start with a “facet-massacre” $P_{0}$, …, $P_{D}$ of $K^{T}$, and we have $P_{0}=K_{T}=\partial T_{N}/{\sim}$. The unique $(d-1)$-face $\sigma_{j}$ killed in passing from $P_{j}$ to $P_{j+1}$ corresponds to a unique pair of (adjacent!) $(d-1)$-faces $\sigma_{j}^{\prime}$, $\sigma_{j}^{\prime\prime}$ in $\partial T_{N+j}$. Gluing them together is the LC move that transforms $T_{N+j}$ into $T_{N+j+1}$. ∎

The first part of a local construction along a tree $T$ corresponds to a facet-killing collapse of $S-\Delta$ (that ends in $K^{T}$).

The second part of a local construction along a tree $T$ corresponds to a pure facet-massacre of $K^{T}$.

A single facet-massacre of $K^{T}$ corresponds to many facet-killing sequences of $K^{T}$.

By Proposition 2.4, there exists a facet-killing sequence of $K^{T}$ if and only if $K^{T}$ collapses onto some $(d-2)$-dimensional complex $C$. This $C$ is necessarily contractible, like $K^{T}$.

So $S$ is locally constructible along $T$ if and only if $K^{T}$ collapses onto some $(d-2)$-dimensional contractible complex $C$, if and only if $K^{T}$ has a facet-killing sequence. What if we do not fix $T$?

Let $S$ be a $d$-sphere ($d\geq 3$). Then the following are equivalent:

for some spanning tree $T$ of $S$, $K^{T}$ is collapsible onto some $(d-2)$-dimensional (contractible) complex $C$;

there exists a $(d-2)$-dimensional (contractible) complex $C$ such that for every facet $\Delta$ of $S$, $S-\Delta\searrow C$;

for some facet $\Delta$ of $S$, $S-\Delta$ is collapsible onto a $(d-2)$-dimensional contractible complex $C$.

$S$ is LC if and only if it is LC along some tree $T$; thus $(1)\Leftrightarrow(2)$ follows from Remark 2.9. Besides, $(2)\Rightarrow(3)$ follows from the fact that $S-\Delta\;\searrow K^{T}$ (Lemma 2.2), where $K^{T}$ is independent of the choice of $\Delta$. $(3)\Rightarrow(4)$ is trivial. To show $(4)\Rightarrow(2)$, take a collapse of $S-\Delta$ onto some $(d-2)$-complex $C$; by Lemma 2.4, there exists some tree $T$ (along which the collapse acts) so that $S-\Delta\searrow K^{T}$ and $K^{T}\searrow C$. ∎

Let $S$ be a $3$-sphere. Then the following are equivalent:

$K^{T}$ is collapsible, for some spanning tree $T$ of the dual graph of $S$;

$S-\Delta$ is collapsible for every facet $\Delta$ of $S$;

$S-\Delta$ is collapsible for some facet $\Delta$ of $S$.

This follows from the previous theorem, together with the fact that all contractible $1$-complexes are collapsible. ∎

We are now in the position to exploit results by Lickorish about collapsibility.

Let $\mathfrak{L}$ be a knot on $m$ edges in the $1$-skeleton of a simplicial $3$-sphere $S$. Suppose that $S-\Delta$ is collapsible, where $\Delta$ is some tetrahedron in $S-\mathfrak{L}$. Then $|S|-|\mathfrak{L}|$ is homotopy equivalent to a connected cell complex with one 0-cell and at most $m$ $1$-cells. In particular, the fundamental group of $|S|-|\mathfrak{L}|$ admits a presentation with $m$ generators.

Now assume that a certain sphere $S$ containing a knot $\mathfrak{L}$ is LC. By Corollary 2.11, $S-\Delta$ is collapsible, for any tetrahedron $\Delta$ not in the knot $\mathfrak{L}$. Hence by Lickorish’s criterion the fundamental group $\pi_{1}\left(|S|-|\mathfrak{L}|\right)$ admits a presentation with $m$ generators.

Any $3$-sphere with a $3$-complicated $3$-edge knot is not LC. More generally, a $3$-sphere with an $m$-gonal knot cannot be LC if the knot is at least $m$-complicated.

As in the construction of the classical “Furch–Bing ball” [16, p. 73] [9, p. 110] , we drill a hole into a finely triangulated $3$-ball along a triple pike dive of three consecutive trefoils; we stop drilling one step before destroying the property of having a ball (see Figure 3). If we add a cone over the boundary, the resulting sphere has a three edge knot which is a connected sum of three trefoil knots. By Goodrick the connected sum of $m$ copies of the trefoil knot is at least $m$-complicated. So, this sphere has a knotted triangle, the fundamental group of whose complement has no presentation with $3$ generators. Hence $S$ cannot be LC.

From this we get a negative answer to the Durhuus–Jonsson conjecture:

Lickorish proved also a higher-dimensional statement, basically by taking successive suspensions of the 3-sphere in Example 2.14.

For each $d\geq 3$, there exists a PL $d$-sphere $S$ such that $S-\Delta$ is not collapsible for any facet $\Delta$ of $S$.

To exploit our Theorem 2.10 we need a sphere $S$ such that $S-\Delta$ is not even collapsible to a $(d-2)$-complex. To establish that such a sphere exists, we strengthen Lickorish’s result.

Let $K$ be a $d$-manifold, $A$ an $r$-simplex in $K$, and $\hat{A}$ the barycenter of $A$. Consider the barycentric subdivision $sd(K)$ of $K$. The dual $A^{*}$ of $A$ is the subcomplex of $sd(K)$ given by all flags

where $r=\dim A$, and $\dim A_{i+1}=\dim A_{i}+1$ for each $i$.

$A^{*}$ is a cone with apex $\hat{A}$, and thus collapsible. If $K$ is PL (see e.g. Hudson for the definition), we can say more:

Let $K$ be a PL $d$-manifold (without boundary), and let $A$ be a simplex in $K$ of dimension $r$. Then

if $A$ is a face of an $(r+1)$-simplex $B$, then $B^{*}$ is a $(d-r-1)$-subcomplex of $\partial\,A^{*}$.

We have observed in Lemma 2.2 that for any $d$-sphere $S$ and any facet $\Delta$ the ball $S-\Delta$ is collapsible onto a $(d-1)$-complex: In other words, via collapses one can always get one dimension down. To get two dimensions down is not so easy: Our Theorem 2.10 states that $S-\Delta$ is collapsible onto a $(d-2)$-complex precisely when $S$ is LC.

This “number of dimensions down you can get by collapsing” can be related to the minimal presentations of certain homotopy groups. The idea of the next theorem is that if one can get $k$ dimensions down by collapsing a manifold minus one facet, then the $(k-1)$-th homotopy group of the complement of any $(d-k)$-subcomplex of the manifold cannot be too complicated to present.

Let $t$, $d$ with $0\leq t\leq d-2$, and let $K$ be a PL $d$-manifold (without boundary). Suppose that $K-\Delta$ collapses onto a $t$-complex, for some facet $\Delta$ of $K$. Then, for each $t$-dimensional subcomplex $\mathfrak{L}$ of $K$, the homotopy group

has a presentation with $f_{t}(\mathfrak{L})$ generators, while $\pi_{i}(|K|-|\mathfrak{L}|)$ is trivial for $i<d-t-1$.

As usual, we assume that the collapse of $K-\Delta$ is ordered so that:

first all pairs $((d-1)\textrm{-face},\;d\textrm{-face})$ are collapsed;

then all pairs $((d-2)\textrm{-face},\;(d-1)\textrm{-face})$ are collapsed;

finally, all pairs $(t\textrm{-face},\;(t+1)\textrm{-face})$ are collapsed.

Let us put together all the faces that appear above, maintaining their order, to form a single list of simplices

In such a list $A_{1}$ is a free face of $A_{2}$; $A_{3}$ is a free face of $A_{4}$ with respect to the complex $K-A_{1}-A_{2}$; and so on. In general, $A_{2i-1}$ is a face of $A_{2i}$ for each $i$, and in addition, if $j>2i$, $A_{2i-1}$ is not a face of $A_{j}$.

We set $X_{0}=A_{0}:=\hat{\Delta}$ and define a finite sequence $X_{1},\ldots,X_{M}$ of subcomplexes of $sd(K)$ as follows:

None of the $A_{2i}$’s can be in $\mathfrak{L}$, because $\mathfrak{L}$ is $t$-dimensional and $\dim A_{2i}\geq\dim A_{2M}=t+1$. However, exactly $f_{t}(\mathfrak{L})$ of the $A_{2i-1}$’s are in $\mathfrak{L}$. Consider how $X_{j}$ differs from $X_{j-1}$. There are two cases:

By Lemma 2.18, setting $r=\dim A_{2j-1}$, $A^{*}_{2j-1}$ is a $(d-r)$-ball that contains in its boundary the $(d-r-1)$-ball $A^{*}_{2j}$. Thus $|X_{j}|$ is just $|X_{j-1}|$ with a $(d-r)$-cell attached via a cell in its boundary, and such an attachment does not change the homotopy type.

As this occurs only when $\dim A_{2j-1}=t$, we have that $\dim A_{2j}=t+1$ and $\dim A^{*}_{2j}=d-t-1$; hence $|X_{j}|$ is just $|X_{j-1}|$ with a $(d-t-1)$-cell attached via its whole boundary.

Only in the second case the homotopy type of $|X_{j}|$ changes at all, and this second case occurs exactly $f_{t}(\mathfrak{L})$ times. Since $X_{0}$ is one point, it follows that $X_{M}$ is homotopy equivalent to a bouquet of $f_{t}(\mathfrak{L})$ many $(d-t-1)$-spheres.

Now let us list by (weakly) decreasing dimension the faces of $K$ that do not appear in the previous list $A_{1},A_{2},\ldots,A_{2M-1},A_{2M}$. We name the elements of this list

(where $\sum_{i=1}^{d}f_{i}(K)=F+1$ because all faces appear in $A_{0},\ldots,A_{F}$).

Correspondingly, we recursively define a new sequence of subcomplexes of $sd(K)$ setting $Y_{0}:=X_{M}$ and

Since $\dim A_{2M+h}\leq\dim A_{2M+1}=t$, we have that $|Y_{h}|$ is just $|Y_{h-1}|$ with possibly a cell of dimension at least $d-t$ attached via its whole boundary. Let us consider the homotopy groups of the $Y_{h}$ ’s : Recall that $Y_{0}$ was homotopy equivalent to a bouquet of $f_{t}(\mathfrak{L})$ $(d-t-1)$-spheres. Clearly, for all $h$,

Moreover, the higher-dimensional cell attached to $|Y_{h-1}|$ to get $|Y_{h}|$ corresponds to the addition of relators to a presentation of $\pi_{d-t-1}(Y_{h-1})$ to get a presentation of $\pi_{d-t-1}(Y_{h})$. This means that for all $h$ the group $\pi_{d-t-1}(Y_{h})$ is generated by (at most) $f_{t}(\mathfrak{L})$ elements.

The conclusion follows from the fact that, by construction, $Y_{F-2M}$ is the subcomplex of $sd(K)$ consisting of all simplices of $sd(K)$ that have no vertex in $sd(\mathfrak{L})$; and one can easily prove (see [36, Lemma 1]) that such a complex is a deformation retract of $|K|-|\mathfrak{L}|$. ∎

Let $S$ be a PL $d$-sphere with a $(d-2)$-dimensional subcomplex $\mathfrak{L}$. If the fundamental group of $|S|-|\mathfrak{L}|$ has no presentation with $f_{d-2}(\mathfrak{L})$ generators, then $S$ is not LC.

Set $t=d-2$ in Theorem 2.19, and apply Theorem 2.10. ∎

Fix an integer $d\geq 3$. Let $S$ be a $3$-sphere with an $m$-gonal knot in its 1-skeleton, so that the knot is at least $(m\cdot 2^{d-3})$-complicated. Then the $(d-3)$-rd suspension of $S$ is a PL $d$-sphere that is not LC.

Let $S^{\prime}$ be the $(d-3)$-rd suspension of $S$, and let $\mathfrak{L}^{\prime}$ be the subcomplex of $S^{\prime}$ obtained taking the $(d-3)$-rd suspension of the $m$-gonal knot $\mathfrak{L}$. Since $|S|-|\mathfrak{L}|$ is a deformation retract of $|S^{\prime}|-|\mathfrak{L}^{\prime}|$, they have the same homotopy groups. In particular, the fundamental group of $|S^{\prime}|-|\mathfrak{L}^{\prime}|$ has no presentation with $m\cdot 2^{d-3}$ generators. Now $\mathfrak{L}^{\prime}$ is $(d-2)$-dimensional, and

whence we conclude via Corollary 2.20, since all $3$-spheres are PL (and the PL property is maintained by suspensions). ∎

For every $d\geq 3$, not all PL $d$-spheres are LC.

Theorem 2.19 can be used in connection with the existence of $2$-knots, that is, $2$-spheres embedded in a $4$-sphere in a knotted way (see Kawauchi [32, p. 190]), to see that there are many non-LC $4$-spheres beyond those that arise by suspension of $3$-spheres. Thus, being “non-LC” is not simply induced by classical knots.

2 Many spheres are LC

Next we show that all constructible manifolds are LC.

Let $C$ be a $d$-pseudomanifold. If $C$ can be split in the form $C=C_{1}\cup C_{2}$, where $C_{1}$ and $C_{2}$ are LC $d$-pseudomanifolds and $C_{1}\cap C_{2}$ is a strongly connected $(d-1)$-pseudomanifold, then $C$ is LC.

Notice first that $C_{1}\cap C_{2}=\partial C_{1}\cap\partial C_{2}$. In fact, every ridge of $C$ belongs to at most two facets of $C$, hence every $(d-1)$-face $\sigma$ of $C_{1}\cap C_{2}$ is contained in exactly one $d$-face of $C_{1}$ and in exactly one $d$-face of $C_{2}$.

Each $C_{i}$ is LC; let us fix a local construction for each of them, and call $T_{i}$ the tree along which $C_{i}$ is locally constructed. Choose some $(d-1)$-face $\sigma$ in $C_{1}\cap C_{2}$, which thus specifies a $(d-1)$-face in the boundary of $C_{1}$ and of $C_{2}$. Let $C^{\prime}$ be the pseudomanifold obtained attaching $C_{1}$ to $C_{2}$ along the two copies of $\sigma$. $C^{\prime}$ can be locally constructed along the tree obtained by joining $T_{1}$ and $T_{2}$ by an edge across $\sigma$: Just redo the same moves of the local constructions of the $C_{i}$’s. So $C^{\prime}$ is LC.

If $C_{1}\cap C_{2}$ consists of one simplex only, then $C^{\prime}\equiv C$ and we are already done. Otherwise, by the strongly connectedness assumption, the facets of $C_{1}\cap C_{2}$ can be labeled $0,1,\ldots,m$, so that:

each facet labeled by $k\geq 1$ is adjacent to some facet labeled $j$ with $j<k$.

Now for each $i\geq 1$, glue together the two copies of the facet $i$ inside $C^{\prime}$. All these gluings are local because of the labeling chosen, and we eventually obtain $C$. Thus, $C$ is LC. ∎

Since all constructible simplicial complexes are pure and strongly connected , we obtain for simplicial $d$-pseudomanifolds that

The previous containment is strict: Let $C_{1}$ and $C_{2}$ be two LC simplicial $3$-balls on $7$ vertices consisting of $7$ tetrahedra, as indicated in Figure 4. (The $3$-balls are cones over the subdivided triangles on their fronts.)