Mogami manifolds, nuclei, and 3D simplicial gravity

Mogami introduced in 1995 a large class of triangulated 3-dimensional pseudomanifolds, henceforth called"Mogami pseudomanifolds". He proved an exponential bound for the size of this class in terms of the number of tetrahedra. The question of whether all 3-balls are Mogami has remained open since, a positive answer would imply a much-desired exponential upper bound for the total number of 3-balls (and 3-spheres) with N tetrahedra. Here we provide a negative answer: many 3-balls are not Mogami. On the way to this result, we characterize the Mogami property in terms of nuclei, in the sense of Collet-Eckmann-Younan:"The only three-dimensional Mogami nucleus is the tetrahedron".


Introduction
A long standing open question in discrete geometry (also highlighted by Gromov, cf. [Gro00,) is whether there are exponentially many simplicial complexes homeomorphic to the 3-sphere, or more than exponentially many. What is counted here is the number of combinatorial types, in terms of the number N of tetrahedra. This enumeration problem is crucial for the convergence of a certain model in discrete quantum gravity, called "dynamical triangulations"; see for example the book [ADJ97] or the survey [RW00] for an introduction.
By deleting one simplex from any (triangulated) 3-sphere, we obtain a (triangulated) 3-ball. Conversely, by coning off the boundary of any 3-ball, we get a 3-sphere. This close relation between 3-spheres and 3-balls is reflected in the asymptotic enumeration. In fact, it is not hard to see that 3-balls are more than exponentially many if and only if 3-spheres are. In other words, one can equivalently rephrase our enumeration problem by replacing "3-sphere" with "3-ball".
To tackle the problem, in 1995 Durhuus and Jónsson introduced the class of Locally Constructible ("LC") manifolds, for which they were able to prove an exponential upper bound [DJ95] [BZ11,Theorem 4.4]. The geometric idea is ingeniously simple. Let us agree to call tree of d-simplices any triangulated d-ball whose dual graph is a tree. Definitorially, LC manifolds are those triangulations of manifolds with boundary that can be obtained from some tree of d-simplices by repeatedly gluing together two adjacent boundary facets. This adjacency condition for the matching, together with the fact that trees are exponentially many, results in a global exponential upper bound.
With this, the problem of enumerating combinatorial types of 3-balls remains wide open. All the known strategies expected to succeed in showing an exponential bound (cf. e.g. [ADJ97,[295][296]) have currently failed. A combinatorial criterion that divides the entire family of triangulated 3-manifolds (or d-manifolds, for any fixed d) into nested subfamilies, each of exponential size, was introduced in [Ben12]; metric restrictions on triangulations that also give exponential bounds have been discovered in [AB13+].

Methods
Our proof is technical but the main idea is elementary, and best sketched with an example. In Figure 1, we show a portion of the boundary of some nicely triangulated 3-ball B; specifically, the star of a vertex v in ∂B. For brevity throughout the paper we say "boundary-link of v" instead of "link of v in the boundary of B". The green triangles are incident at v, but not adjacent. Their identification is a "Mogami gluing", but not an "LC gluing" (cf. Definitions 5, 6). As depicted in Figure 1, the gluing changes the topology of the boundary: v becomes a singularity, in the sense that its link is disconnected. Also, after the gluing, we no longer have a simplicial complex, because the pink triangles now share 2 edges out of 3. We call such configuration of two boundary triangles sharing exactly 2 edges a wound. Let us now perform a second identification, namely, let us glue the pink triangles together. As we do that, the topology changes back: The boundary-link of v returns to be a single circle, as one of its connected components, the red digon, is sunk into the interior. The step of gluing together two boundary triangles with exactly 2 edges in common is called a healing. (The same step was called "type-(iv) LC gluing" in [BZ11,Definition 3.17].) The healing makes the wound disappear, as the triangle resulting from the identification is sunk into the interior. Now, let us start back from B and let us perform the same two gluings in inverse order : Pink first, then green. There are two pleasant novelties with this reshuffling: (1) When the pink triangles are glued, they share 1 edge, not 2. So the gluing is not a healing.
(2) When the green triangles are glued, they share 1 edge, not just one vertex. As a result, the "green gluing" is now a legitimate LC gluing. By postponing the Mogami-non-LC move to after the healing move, topologically these two bizarre moves have 'canceled out'; and we have obtained a sequence in which all triangles that we match have exactly 1 edge in common at the moment of the gluing. The final complex is obviously the same ball as before.
Using this idea, we will prove that all Mogami 3-ball without interior vertices are LC (Theorem 34). The trick is to systematically rearrange the Mogami sequence to obtain a sequence that is also LC. This does not work for all pseudomanifolds; but if we focus on Mogami constructions of 3-balls without interior vertices, we know that the boundary-link of every vertex should eventually become a 1-sphere. Hence, all the extra components of a boundary-link created by Mogami non-LC gluings have to be suppressed throughout the construction. Now, the only way to suppress a component is via a "healing" step. By reshuffling, we will obtain a new sequence where the non-LC step and the healing step 'cancel out'.

Notation
Throughout this paper, d is always an integer ≥ 2. For the definitions of simplicial complex, regular CW complex, pure, shellable, cone..., we refer the reader to [BZ11]. Following [BZ11], by pseudomanifold we mean a finite regular CW complex which is pure d-dimensional, simplicial, and such that every (d − 1)-cell belongs to at most two d-cells. The boundary is the smallest subcomplex of the pseudomanifold containing all the (d − 1)-cells that belong to exactly one d-cell. We call "d-ball" (resp."d-sphere" ) any simplicial complex homeomorphic to the unit ball in R d (resp. to the unit sphere in R d+1 ). A tree of d-simplices is any d-ball whose dual graph is a tree.
Definition 1 (unite; split). Let P 1 , P 2 be two disjoint d-pseudomanifolds, d ≥ 2. The operation unite consists in identifying a (d − 1)-face ∆ in ∂P 1 with a (d − 1)-face ∆ in ∂P 2 . (For d = 3, this was called "step of type (i)" in [BZ11, Definition 3.17].) If the P i do not have interior vertices, neither does the obtained pseudomanifold Q; and if both P i 's are d-balls, so is Q. Note also that Q contains in its interior a (d − 1)-face ∆ with ∂∆ completely contained in ∂Q.
The inverse operation is called "split". (For d = 3, this goes under the name of "Cut-a-3-face" in [CEY14, p. 267] and of "Operation (I)" in [Hac00,p. 85].) It is defined whenever a pseudomanifold Q has some interior (d − 1)-face ∆ with ∂∆ ⊂ ∂Q. If Q is simply-connected, the effect of split is to divide Q (along the face ∆) into two disconnected pseudomanifolds. In general, the effect of split on the dual graph of the pseudomanifold is to delete one edge.
Trees of N d-simplices are characterized as the d-complexes obtainable from N disjoint dsimplices via exactly N − 1 unite steps.
Definition 2 (fold; spread). Let P be a d-pseudomanifold, d ≥ 2. The operation fold consists in identifying two boundary facets ∆ , ∆ that share exactly one (d − 2)-face e; compare Figure 2. (For d = 3, the operation was called "an LC step of type (ii)" in [BZ11, Definition 3.17].) If P is a d-ball, then the obtained pseudomanifold Q is homeomorphic to P . (This is false if P is an arbitrary pseudomanifold, cf. Example 30.) Moreover, if d ≥ 3 and P does not have interior vertices, neither does Q. The obtained pseudomanifold Q contains in its interior a (d − 1)-face ∆ with exactly d − 1 of its facets in ∂Q: in fact, the only facet of ∆ in the interior of Q is the (d − 2)-face e.
The inverse operation is called spread; compare Figure 2. (For d = 3, it goes under the name 'Open-a-2-face" in [CEY14,p. 267] and "Operation (II)" in [Hac00,p. 85]). It is defined whenever a pseudomanifold Q has some interior (d − 1)-face ∆ that has one of its (d − 2)-faces in the interior of Q, and all its other (d − 2)-faces in the boundary of Q. When spread is applied to a simplicial complex, it outputs a simplicial complex. In contrast, it is easy to see that fold moves may lead out of the world of simplicial complexes.
Next, we introduce nuclei, which were called "reduced balls" in [Hac00, p. 85]: Definition 3 (Nucleus). Let d ≥ 2. A nucleus is a d-ball where (1) every (d − 3)-face belongs to the boundary, and (2) every interior (d − 1)-face has at least d − 1 of its d facets in the interior of the ball. The d-simplex (for which condition (2) is void) is called the trivial nucleus.
Lemma 4 (Hachimori [Hac00], Collet-Eckmann-Younan [CEY14]). Every 3-ball without interior vertices can be reduced to a disjoint union of nuclei with some (greedy) sequence of split and spread moves. Without loss of generality, one can assume that all spread steps are performed before the split ones.
The next move can be viewed as a variation/expansion of fold.
Every fold is an LC gluing. The converse is false: for example, when d = 2, gluing together two boundary edges that have both endpoints in common is an LC gluing, but not a fold. The difference is topologically remarkable. It was proven in [BZ11] that the only manifolds obtainable from a tree of d-simplices with fold moves, are d-balls. In contrast, with LC gluings one can obtain all polytopal d-spheres, for example. It was proven in [Ben15] that except when d = 4, all simply-connected smooth d-manifolds (with or without boundary!) have a triangulation that can be obtained from some tree of simplices via LC gluings (cf. Theorem 10).
Here is a further generalization, potentially leading to a broader gauge of complexes: Definition 6 (Mogami gluing). Let P be a d-pseudomanifold, d ≥ 2. Identifying two boundary facets ∆ , ∆ whose intersection is nonempty is an operation called a Mogami gluing.
Clearly, every LC gluing is a Mogami gluing, while the converse is false (unless d = 2). We have arrived to the most important definition of the paper: Definition 7 (LC manifolds; Mogami manifolds). Let d ≥ 2. Let M be a pure d-dimensional simplicial complex with N facets that is also a pseudomanifold. M is called LC (resp. Mogami) if it can be obtained from a tree of N d-simplices via some sequence, possibly empty, of LC gluings (resp. of Mogami gluings). We refer to the sequence as "the LC construction" (respectively, "the Mogami construction"). With abuse of notation, the intermediate pseudomanifolds in the LC construction of an LC manifold are also called "LC pseudomanifolds"; same for Mogami.
Remark 8. The original definition of [Mog95], given only for d = 3, was slightly different. Mogami considered a class C of 3-pseudomanifolds obtained from a tree of tetrahedra by performing either (1) LC gluings, or (2) identifications of incident boundary edges, subject to a certain planarity condition. Now, identifying 2 boundary edges that share a vertex v creates new adjacencies between triangles that before were only incident at v. So it is clear that Mogami 3-pseudomanifolds (with our definition) all belong to the class C, since we could realize any Mogami gluing as a "combo" of an identification of adjacent boundary edges followed by an LC gluing.
Conversely, we claim that all manifolds in C are Mogami. (This is false for pseudomanifolds.) In fact, if we identify two boundary edges that share a vertex v in the boundary of an arbitrary pseudomanifold, we create an entire "singular edge". To get a manifold, we have to get rid of this singular edge; the only way to do so is by identifying two triangles ∆ , ∆ containing that edge, at some point in the Mogami construction. But then we can rearrange the sequence of gluings by performing the Mogami gluing ∆ ≡ ∆ before all other gluings.

General Aspects of Mogami Complexes
Let us start with a topological motivation to study the Mogami class.
Proof. By induction on the number of Mogami gluings. Any tree of simplices is topologically a ball, hence simply connected. Consider now the moment in which we glue together two incident boundary facets ∆ and ∆ of a simply-connected d-pseudomanifold P ; and suppose a new loop arises. This means that we have just identified two endpoints x ∈ ∆ and x ∈ ∆ of a path whose relative interior lies completely in the interior of P . Let v be a vertex in ∆ ∩ ∆ . By homotoping both x and x to v, one sees that the "new loop" is actually homotopy equivalent to an "old loop" already contained in P (hence homotopically trivial, by induction.) Not all triangulations of simply-connected manifolds are Mogami, as we will prove in Theorem 43. However, a partial converse to Proposition 9 can be derived from [Ben15]: Theorem 10 (Benedetti [Ben15]). For d = 4, any PL triangulation of any simply-connected d-manifold (with boundary) becomes an LC triangulation after performing a suitable number of consecutive barycentric subdivisions. In particular, every simply-connected smooth d-manifold d = 4, admits a Mogami triangulation.
Recall that a simplicial complex is called strongly-connected if it pure (i.e. all facets have the same dimension) and its dual graph is connected. By induction on the number of Mogami steps, one can easily prove: Proposition 11. Every Mogami d-pseudomanifold is strongly-connected, and all vertex links in it are strongly-connected as well.
The converse does not hold: any triangulation of an annulus is strongly-connected and has strongly-connected links, but it cannot be Mogami by Proposition 9.
For 2-dimensional pseudomanifolds, the LC property and the Mogami property are equivalent, because two boundary edges are adjacent if and only if they are incident. We show next that the two properties diverge from dimension 3 on.
In [BZ11, Lemma 2.23] it is shown that the union of two LC pseudomanifolds with a codimension-one strongly-connected intersection, is LC. Interestingly, an analogous result holds for the Mogami property, basically up to replacing "strongly-connected" with "connected": Proof. First of all, we observe that A ∩ B is contained in both ∂A and ∂B. In fact, since A ∪ B is a pseudomanifold, every (d − 1)-face of A ∩ B can be contained in at most two d-faces of A ∪ B, so it has to be contained in exactly one d-face of A and in exactly one d-face of B.
Since A ∩ B is connected, we can find a total order F 0 , . . . , F s of the facets of A ∩ B such that for each i ≥ 1, F i is incident to some F j , with j < i. Let us fix a Mogami construction for A and one for B. Let T A (resp. T B ) be the tree of d-simplices from which A (resp. B) is obtained. If we perform a unite move and join T A and T B "at F 0 ", we obtain a unique tree of tetrahedra T C containing all facets of C. Each F i (i ≥ 1) corresponds to two distinct (d − 1)-faces in the boundary of T C , one belonging to T A and one to T B ; we will call these two faces "the two copies of F i ". Now C admits a Mogami construction starting from T C , as follows: (a) first we perform all identifications of boundary facets of T C that belonged to T A , exactly as prescribed in the chosen Mogami construction of A from T A ; (b) then we perform the identifications given by the Mogami construction of B; (c) finally, for each i ≥ 1 (and in the same order!), we glue together the two copies of F i . Since each F i is incident to some F j , with j < i, the gluings of phase (c) are Mogami gluings.
Proof. Let C 1 and C 2 be two shellable simplicial 3-balls consisting of 4 tetrahedra, as indicated in Figure 3. (The 3-balls are cones over the subdivided squares on their front.) Since shellable Figure 3: Gluing the simplicial 3-balls along the shaded 2-dimensional subcomplex (which is connected, but not strongly-connected) gives a Mogami 3-pseudomanifold that is not LC. Note that the resulting 3-pseudomanifold is a cone over an annulus (the vertex v is the apex of the cone).
implies LC [BZ11] and LC implies Mogami, both C 1 and C 2 are Mogami. Glue them together in the shaded subcomplex in their boundary (which uses 5 vertices and 2 triangles.) Note that such subcomplex is connected, but not strongly-connected. Let P be the resulting 3-dimensional pseudomanifold. By Proposition 12 the pseudomanifold P is Mogami. It remains to prove that P cannot be LC. For this we use a topological result by Durhuus and Jónsson [DJ95]: If L is any LC 3-dimensional pseudomanifold, then any strongly-connected component of ∂L is a 2-sphere; in addition, any two strongly-connected components of ∂L intersect in at most one point. Yet our ∂P has a different topology: It is a "pinched sphere", i.e. the space obtained by identifying two antipodal points of a 2-sphere. Hence, P cannot be LC. (Alternatively, one can also observe that P is a cone over an annulus; an annulus is not simply connected and therefore not LC; via [BZ11, Proposition 3.25], this implies that P cannot be LC either.) We have arrived to another crucial difference between the LC and the Mogami notion, namely, the behavior with respect to taking cones. In [BZ11, Proposition 3.25] it is proven that for any pseudomanifold P and for any vertex v not in P , the cone v * P is LC if and only if P is LC. It turns out that cones tend to be Mogami more often.
Proof. The "only if" part follows from Proposition 11, since the link of v in v * A is A itself. As for the "if": Since the dual graph of A is connected, we may choose a spanning tree, which uniquely determines a tree of d-simplices T N inside A. Since every (d − 1)-face of A belongs to at most two d-simplices, the complex A can be obtained from T N via identifications of pairs of (not necessarily incident!) boundary facets. Now let us take a new vertex v. Clearly v * T N is a tree of (d + 1)-simplices. Let us 'mimic' the construction of A from T N , to obtain a construction of v * A from v * T N . (By this we mean that if the construction of A from T N started by gluing two faces σ and σ of ∂T N , then we should start the new construction of v * A from taking v * T N by gluing v * σ with v * σ ; and so on.) Clearly, v * A is obtained from v * T N via identifications of pairs of boundary facets that contain v, and therefore are incident.
Proof. Let k be any integer such that 2 ≤ k ≤ d − 1. Let A be any k-pseudomanifold that is strongly-connected, but not LC. (They exists; for example, for k = 2 one can choose any triangulation of an annulus; compare Figure 3, which illustrates the case d = 3.) Take d − k consecutive cones over C. The resulting d-complex is Mogami by Proposition 14 and not LC by [BZ11, Proposition 3.25].
2 Intermezzo: Planar matchings and extensively-LC manifolds Here we show that all 2-spheres and 2-balls are Mogami and even LC independently from which tree of triangles one starts with. These results are not new; they essentially go back to Durhuus, cf. [Dur90] [DJ95, p. 184], but we include them to showcase some proof mechanisms that will later be needed in the 3-dimensional case. We also discuss a higher-dimensional extension of this phenomenon of "irrelevance of the chosen tree", called "extensively-LC" property. The reader eager for new theorems may skip directly to the next Section. We need some additional notation. By a cycle we mean from now on a simple cycle; that is, any closed path in which all vertices are distinct, except for the first and last one. A graph (resp. a multigraph) is for us a 1-dimensional simplicial complex (resp. a 1-dimensional cell complex). In other words, graphs are multigraphs that do not have loops or double edges. Given any simplicial complex, we call "free" any face that is properly contained in only one other face. The free faces in a graph are called leaves; some complexes have no free face. An elementary collapse is the deletion of a single free face (and of the other face containing it).
Definition 16 (Extensively collapsible). A complex C is called extensively-collapsible if any sequence of elementary collapses reduces C to a complex that is itself collapsible. In other words, C is extensively collapsible if and only if by performing elementary collapses, we never get stuck. We also say that C is extensively-collapsible onto D if any sequence of elementary collapses that does not delete faces of D, reduces C to a complex that is itself collapsible to D.
For example, trees are extensively collapsible; in fact, every tree is extensively collapsible onto any of its subtrees. It is well-known that all collapsible 2-complexes are also extensivelycollapsible, cf. e.g. [HAM93]. However, an 8-vertex example of a collapsible but not extensivelycollapsible complex (in fact, a 3-ball) was given in [BL13].
Lemma 17. Let C be a cycle. Let M be any planar matching, partial or complete, of the edges of C. Let G be the multigraph obtained from C by pairwise identifying the edges according to M (preserving orientation). The following are equivalent: (1) G contains at most one cycle; (2) G can be obtained from C via some sequence of LC gluings.
Proof. "(2) ⇒ (1)": Let C be a cycle. Any LC gluing of two edges of C either -preserves the number of cycles (if the edges share only one vertex), or -'kills' one cycle (in case the edges have both endpoints in common). So when we perform local gluings on a multigraph, the total number of cycles can only decrease. Since we started with a cycle, G contains at most one cycle.
"(1) ⇒ (2)": Fix a planar matching M of C. If M is a complete matching, the resulting multigraph G will be a tree; if instead it is partial, G will be a cycle with some trees attached. We proceed by induction on the number n of edges of C, the case 3 ≤ n ≤ 6 being easy. Let Figure 4: A 12-gon (left) and the (multi)graph obtained from it by identifying only the two blue edges (right). Note that this graph cannot be obtained from the 12-gon via LC gluings, because the only two edges we are supposed to glue in the 12-gon are far apart from one another.
e , e be two edges of C that are matched in M. If e , e are adjacent in C, their identification is an LC gluing, and there is nothing to show; so we shall assume they are not adjacent. The effect of the gluing e ≡ e is to squeeze C into a left cycle L and a right cycle R, bridged by a single edge e (as in Figure 4). Moreover, M restricts to planar matchings on both L and R. Of these two "submatchings", at least one has to be complete, otherwise the final multigraph G would contain at least two cycles. We will assume the submatching on L is complete, the other case being symmetric. Let G L be the subtree of G corresponding to the edges of L. Let v = e ∩ G L . Choose a collapsing sequence of the tree G L onto v. This yields a natural ordering e 0 , e 1 , . . . , e k−1 of the k edges of the tree G L , where e i is the i-th edge to be collapsed and e k−1 contains v. Observe that e 0 must be a leaf of G L ; it corresponds therefore to a pair of adjacent edges e 0 and e 0 of L matched under M. Recursively, for each i, the edges e i and e i become adjacent once we have identified e j with e j , for all j < i. In other words, the identifications (e i ≡ e i ) 0≤i≤k−1 , performed in this order, are legitimate LC gluings. Now we are ready to rearrange the sequence, by postponing the initial step e ≡ e . So let us set e k := e, e k := e and e k := e . Starting from the initial cycle C, let us perform (e k ≡ e k ) after all of the gluings (e i ≡ e i ) 0≤i≤k−1 have been carried out. The advantage is that e k ≡ e k is now an LC step, because e k−1 and e k both contained the vertex v (so after e k−1 and e k−1 are identified, the edges e k ≡ e k become incident at v).
We are eventually left with the right cycle R. The subgraph G R of G corresponding to the edges of R contains at most one cycle. By inductive assumption, G R can be obtained from R via a sequence of LC gluings. The latter sequence, performed after (e i ≡ e i ) 0≤i≤k , forms a longer sequence of LC gluings that constructs G from C.
Remark 18. Topologically, the proof above can be recapped as follows. Initially, we have an "unwanted" non-LC gluing e ≡ e that increases the number of cycles from 1 to 2. Since in the end the graph G produced has at most 1 cycle, at some point the extra cycle has to be suppressed. The only way to suppress a cycle with a planar matching, is to identify some pair of edges f , f that have both endpoints in common. Our proof strategy was: • to postpone the gluing e ≡ e , so that it is becomes an LC gluing; and also • to anticipate f ≡ f , so that these two edges are glued when they only share one of their endpoints, not both. We did not change the matching; we only changed the order in which the matching is performed. But in the rearranged sequence, no step increases the number of cycles by one. (There is also one less step that decreases the number of cycles by one; these two steps 'canceled out'.) Here is a variation we will need in the next Section. Given a graph G, we say that a vertex v of G is active if it belongs to a cycle. For example, every vertex of a cycle C is active. If we perform an LC gluing of two adjacent edges of C, the vertex between the two edges gets "de-activated". In a tree, no vertex is active.
Lemma 19. Let C be a cycle. Let M be any complete planar matching of the edges of C. Let G be the tree obtained from C by pairwise identifying the edges according to M, as in the previous Lemma. Given an arbitrary vertex c 0 of C, there is a sequence of LC gluings that produced G from C and in which the vertex c 0 is active until the very last gluing.
Proof. Since every tree is simplicially collapsible onto any of its vertices, we may choose a collapsing sequence of G onto the vertex corresponding to c 0 . Now, every pair of adjacent edges in C matched by M corresponds to a leaf in the tree G; and elementary collapses in G (which are just leaf deletions) correspond to LC gluings on C. Hence, our collapse of G onto c 0 induces a sequence of LC gluings, the last of which identifies two edges sharing both endpoints (one of the endpoints being c 0 ). Definition 21 (Extensively LC). Let P be a d-dimensional pseudomanifold. We say that P is extensively LC if, for any spanning tree T of the dual graph of B, (a complex combinatorially equivalent to) P can be obtained via LC gluings from the tree of d-simplices T N dual to T .
If we replace "any" with "some" in the definition above, we recover the classical definition of LC. Hence, "extensively-LC" trivially implies LC. See Remark 26 below for the difference.
Proof. Let B be an arbitrary 2-sphere or 2-ball. Let T and T N be as in the definition of extensively-LC. By construction, we know that B is obtained from T N by some matching M of the edges of ∂T N , which is a 1-dimensional sphere (or in other words, a cycle). Note that the matching is uniquely determined once the tree T N is chosen. If B is a 2-sphere, the matching is complete; if B is a ball, ∂B is a cycle, the matching is partial, and the edges left unmatched are precisely the edges of ∂B. In both cases, the multigraph obtained from ∂T N via the identifications in M contains at most one cycle. Using Lemma 17, we conclude.
If T is a spanning tree of the dual graph of a (connected) d-manifold, following [BZ11, p. 214] we denote by K T the (d − 1)-dimensional subcomplex of the manifold determined by all the (d − 1)-faces that are not intersected by T . When d = 3, K T is 2-dimensional. Recall that for 2-complexes collapsibility and extensive-collapsibility are equivalent notions. Using this, it is an easy exercise to adapt the original proofs of [BZ11, Corollary 2.11] and of [BZ11, Corollary 3.11], respectively, to derive the following results: Theorem 23. Let S be a triangulated 3-sphere. The following are equivalent: (i) S is extensively-LC; (ii) for every spanning tree T of the dual graph of S, the complex K T is collapsible; (iii) for every tetrahedron ∆ of S, the 3-ball S − ∆ is extensively collapsible; (iv) for some tetrahedron ∆ of S, the 3-ball S − ∆ is extensively collapsible.
Theorem 24. Let B be a triangulated 3-ball. The following are equivalent: (i) B is extensively-LC; (ii) for some tetrahedron ∆, the 3-ball B − ∆ is extensively collapsible to ∂B; (iii) for every tetrahedron ∆, the 3-ball B − ∆ is extensively collapsible to ∂B; (iv) for every spanning tree T of the dual graph of S, the complex K T collapses to ∂B.
Corollary 25. Every triangulated d-ball or d-sphere with less than 8 vertices is extensively-LC.
Proof. By a result of Bagchi and Datta [BD05], all acyclic 2-complexes with less than 8 vertices are collapsible; it follows that all collapsible 2-complexes with less than 8 vertices are extensively collapsible [BL13].
Remark 26. Some 3-sphere with 8 vertices that is LC, but not extensively, is presented in [BL13]. After we remove a tetrahedron from such sphere, we obtain a collapsible ball B; but there is also a sequence of elementary collapses that from B gets us stuck in an 8-vertex triangulation of the Dunce Hat [BL13].

The only Mogami nucleus is the simplex
Let us now focus on d = 3. We wish to study how LC or Mogami steps in a construction of a 3-manifold affect the boundary-link of a single vertex. The four examples we present will be crucial in the proof of our Main Theorems. First we need one additional notation.
Definition 27 (Merging). Let C, D be two cycles with an edge e in common. The merging operation produces a new cycle as follows: we take the union C ∪ D, and we delete the edge e.
Example 28. Let B be a 3-ball. Let e be an edge in ∂B. Let v and w be the two vertices in link (e, ∂B). If we identify the two triangles v * e and w * e, this is a legitimate LC gluing -in fact, a fold. Let Q be the pseudomanifold obtained. Topologically, Q is also a 3-ball. With slight abuse of notation, let us keep calling v be the vertex of Q resulting from the identification of v and w. It is easy to see that link (v, ∂Q) is the cycle obtained by merging link (v, ∂B) and link (w, ∂B).
Example 29. Let B be a 3-ball. Let x be a vertex in ∂B. Let e 1 , e 2 be two edges in link (x, ∂B). If we identify the two triangles x * e 1 and x * e 2 , this is a legitimate Mogami gluing. Let v 1 , w 1 be the two endpoints of e 1 . Similarly, let v 2 , w 2 be the two endpoint of e 2 , labeled so that the vertex that is identified to v 1 is v 2 . Let Q be the obtained pseudomanifold (which is not a ball, this time.) Let us call v the vertex of Q resulting from the identification of the two vertices v 1 and v 2 . It is easy to see that link (v, ∂Q) is a cycle. It is obtained from C 1 = link (v 1 , ∂B) and C 2 = link (v 2 , ∂B) with an operation that is an LC gluing plus a merging. More precisely, C 1 and C 2 do not have an edge in common; they share only the vertex x. However, the cycle link (v, ∂Q) can be obtained from C 1 and C 2 by first identifying [x, w 1 ] (which is in C 1 ) and [x, w 2 ] (which is in C 2 ), and then by performing a merging at the resulting edge [x, w].
Example 30. Let P be a pseudomanifold obtained from a 3-ball by performing one Mogami gluing of 2 triangles sharing only a vertex v, and then another Mogami gluing of 2 triangles sharing only a vertex w = v, such that v and w belong to adjacent triangles in P . Then: • link (v, ∂P ) is the disjoint union of two cycles, A v and B v ; • link (w, ∂P ) is also the disjoint union of two cycles, A w and B w ; • link (v, ∂P )∩link (w, ∂P ) consists of an edge e, which (up to relabeling) belongs to A v ∩A w .
Let us identify the two triangles v * e and w * e, and let Q be the resulting pseudomanifold. With the usual abuse of notation, let us call v be the vertex of Q obtained from the identification of v and w. It is easy to see that link (v, ∂Q) is a disjoint union of three cycles, namely B v , B w , and a third cycle obtained by merging A v and A w . In particular, ∂Q is not homeomorphic to ∂P . (This pathology is due to the presence of two different singularities in P , which are identified in the gluing; on LC pseudomanifolds, fold does preserve the homeomorphism type).
Example 31. Let us start with an annulus of 4 squares, and let us subdivide each square into four triangles by inserting the two diagonals ( Figure 5). Let w be one of the four square barycenters. Let a, b, c, d be the four corners of the square containing w, labeled so that ab and cd are free edges (i.e. edges that belong to one triangle only). Let , now that v is carried onto w we have in P two distinct triangles ∆ 1 and ∆ 2 that share one edge and also the opposite vertex. Hence P (which topologically is homeomorphic to v * A, cf. Example 28) is not a simplicial complex. Remark 32. No 3-ball without interior vertices can be obtained via Mogami gluings from the pseudomanifold P of Example 31. In fact, assume by contradiction that R is a ball without interior vertices obtained with a Mogami construction from P . Because R is a simplicial complex, in R the two triangles ∆ 1 and ∆ 2 that share the vertices v, a, b must be identified at some point; so we might as well glue them immediately. Let us call Q be the pseudomanifold obtained from P by gluing ∆ 1 ≡ ∆ 2 . We may assume that R is obtained via Mogami gluings from Q. Since R is a ball without interior vertices, the link of v in R must be a disk. Since Q is not a simplicial complex the notion of "link of v in Q" is not well-defined; but we can look at the spherical link L of v in Q, which is what we would get by intersecting Q with a sphere of small radius centered at v. (In simplicial complexes, this is isomorphic to the vertex link.) Up to homeomorphism, we can think of L as a 2-dimensional simplicial complex obtained from a finely triangulated annulus by identifying (coherently, without twists) two parallel edges in different components of the boundary. Note that L is not planar, in the sense that no simplicial complex homeomorphic to L can be drawn in R 2 without self-intersections. Now, any further Mogami step performed on Q will possibly modify L only via identifications in its boundary. Topologically, these steps may transform the spherical link of v into a torus, but not into a 2-ball (or a 2-sphere). A contradiction.
In fact, the topological argument of Remark 32 above proves the following: We are now ready to prove our main result.
Theorem 34. Let B be a Mogami 3-ball without interior vertices. Let T N be the tree of tetrahedra from which B is constructed, via some sequence of Mogami gluings. Then, B can also be constructed from T N via some sequence of LC gluings. In particular, all Mogami 3-ball without interior vertices are LC.
Proof. If all Mogami gluings are LC gluings, there is nothing to prove. Otherwise, let us consider the first Mogami gluing ∆ 0 ≡ ∆ 0 that is not LC. Let v = ∆ 0 ∩ ∆ 0 . By definition there are disjoint edges δ 0 , δ 0 such that ∆ 0 = v * δ 0 and ∆ 0 = v * δ 0 . Let P be the pseudomanifold obtained after the gluing ∆ 0 ≡ ∆ 0 ; the vertex v is in the boundary of P , while the triangle ∆ 0 obtained from the identification is in the interior of P . We denote by δ 0 the edge opposite to v in ∆ 0 . As we saw in Figure 1, the gluing creates a singularity at v: namely, link (v, ∂P ) consists of two cycles. Since B is a 3-ball with all vertices on the boundary, the subsequent Mogami gluings in the construction of B from P will • keep the vertex v in the boundary, and • eventually "kill" one of the two connected components of link (v, ∂P ). Let us call C the "doomed" component, that is, the cycle of link (v, ∂P ) none of whose edges will eventually appear in link (v, ∂B). Let us denote by c 0 the vertex of δ 0 that belongs to C.
Our strategy is to consider this cycle C and rearrange the sequence of gluings according to Lemma 19, so that after the rearrangement, all gluings in the sequence are LC gluings, and the last pair of edges glued is a pair adjacent to the edge δ 0 . Before doing this, though, we need a delicate preliminary argument. In fact, while constructing B from P , all triangles of star (v, ∂P ) are going to be matched and sunk into the interior; but what we do not know for sure, is whether they are going to be matched to one another. A priori, there are other two possibilities that we should consider (both of which could occur multiple times): (a) for some edge e of C, it could happen that v * e is matched in an LC gluing with some triangle w * e outside star (v, ∂P ); (b) or it could also happen that v * e is matched in a Mogami gluing to another triangle that does not contain v, but contains exactly one of the two endpoints (let us call it x) of e. The steps above affect the boundary-link of v as follows.
I) The cycle C is "expanded" via a merging operation. For example, in case (a) the boundarylink of v gets merged with the boundary-link of w, as explained in Example 28. Case (b) is similar: The vertex v is identified with a vertex v 2 of the other triangle, and essentially the boundary-link of v gets merged with the boundary-link of v 2 (after an LC gluing; compare Example 29.) II) Possibly, the link of P might acquire further connected components. This happens when the vertex w identified with v is also a singularity, a case we saw in Example 30.
These cases, however, do not ruin our proof strategy -they just delay it. Our remedy in fact is to anticipate all matchings of the type (a) and (b) described above, in a "first round" of identifications. For example, if a single triangle v * e is later matched in a fold with some triangle w * e outside star (v, ∂P ), then we can rearrange the sequence by performing such LC gluing immediately. After all identifications of type (a) and (b) have been carried out, if P 1 is the resulting pseudomanifold, we ask ourselves again: are all triangles of star (v, ∂P 1 ) going to be matched exclusively with one another? If not, we repeat the procedure above, in a second round of identifications, and we call the obtained pseudomanifold P 2 . And so on. The effect of these rounds on the boundary-link of v is to expand it by inglobating new edges. We make a crucial claim: in these rounds of identifications, the components of the boundarylink of v remain separate. The proof of this claim relies on Lemma 33. In fact, suppose by contradiction that passing from P 1 to P 2 , say, we have included into C an edge [a, b] that belongs to another component of the boundary-link of v (which is what we have done in Example 31.) This means that in P 2 we have a singularity v, and two distinct triangles containing v and the edge [a, b]. So if we want to obtain a simplicial complex, we are forced to glue the two triangles together; and with the same proof of Remark 32, no matter how we continue this Mogami construction, we are never going to achieve a 3-ball without interior vertices. A contradiction. (This shows that the components of the boundary-link of v never have an edge in common; in analogous way, adapting Lemma 33, one proves they cannot have vertices in common, either.) Eventually, after a finite number of rounds, we will reach a pseudomanifold P such that: • link (v, ∂P ) consists of k ≥ 2 connected components, • B is obtained with a list of Mogami gluings from P , a process in which exactly k − 1 of the components of link (v, ∂P ) are going to be "killed", • if C is the connected component of link (v, ∂P ) obtained from C via merging operations, then for any edge e of C there exists an edge f of C such that, in one of the Mogami gluings that leads from P to B, the triangle v * e is identified with v * f . In fact, we can repeat the reasoning above until the last property holds for all the k−1 "doomed" connected components of link (v, ∂P ).
Note that C contains all vertices of C. This is because the merging operation does not delete any vertex. In particular, the vertex c 0 = δ 0 ∩ C of C will be present in C as well.
We are now in the position to use Lemma 19. The Mogami construction that leads from P to B yields a complete matching of the edges of C . Clearly, ordering the edges in link (v, ∂P ) is the same as ordering the triangles in star (v, ∂P ); also, two edges e, f are adjacent in the link of v if and only if v * e and v * f are adjacent in the star of v. Let us thus reorder the gluings involving triangles in star (v, ∂P ), according to Lemma 19, so that the vertex c 0 is deactivated last. In this order, the identifications "killing" the component C are all LC gluings. Furthermore, it is easy to see that all gluings mentioned above (those leading from P to P , plus all LC gluings that kill C ) can be performed before the identification ∆ 0 ≡ ∆ 0 . With this postponement the step ∆ 0 ≡ ∆ 0 becomes an LC gluing: In fact, after all other identifications have been carried out, ∆ 0 and ∆ 0 share the edge [v, c 0 ]. In conclusion, by reshuffling the Mogami sequence we got rid of the first non-LC step. By induction, we reach our claim.
Corollary 35. Let B be a 3-ball without interior vertices. The following are equivalent: (1) B is Mogami; (2) B is LC; (3) some (possibly empty) sequence of spread operations reduces B to a tree of tetrahedra; (4) B has trivial nuclei (that is, some sequence of spread and split operations reduces B to disjoint tetrahedra.) Proof. "(1) ⇔ (2)" follows from Theorem 34.
Proof. Let us fix a 3-ball A with some knotted spanning edge and with all vertices on ∂A (cf. Lemma 39). Let F A be the number of facets of A. Let us also fix a triangle ∆ A ⊂ ∂A. Now, let B be an arbitrary 3-ball with N tetrahedra, without interior vertices, and with a distinguished triangle ∆ B ⊂ ∂B. From B we can obtain a 3-ball B with N + F A tetrahedra via a unite step that consolidates the 3-balls A and B by identifying ∆ A ≡ ∆ B . (Ignore the fact that there are multiple ways to do this, according to rotation, as this amounts to an asymptotically neglectable factor.) No matter how we choose B, the union B = A ∪ B is going to contain the same knotted spanning edge of A. But since all its vertices are on the boundary, by Lemma 38 the ball B cannot be Mogami. Now note that B determines B: In fact, for any interior triangle ∆ of B with all three edges on ∂B , we could split B at ∆ and check if one of the two 3-balls obtained is combinatorially equivalent to A (if it is, the other 3-ball is B). Hence the transition from B to B yields an injective map 3-balls with N tetrahedra and with 0 interior vertices −→ non-Mogami 3-balls with N + F A tetrahedra and with 0 interior vertices If we pass to the cardinalities and let N tend to infinity, F A being constant, we conclude.
Finally, we recall the connection of knot theory with simplicial collapsibility: Proposition 41 (essentially Goodrick, cf. [Ben12, Corollary 4.25]). Let K be any knot whose group admits no presentation with 2 generators. (For example, the double trefoil). Any knot with a knotted spanning edge isotopic to K, cannot be collapsible.
Proposition 42 ([BZ11, Theorem 3.23]). For any 2-bridge knot K (for example, the trefoil), there is a collapsible 3-ball without interior vertices with a knotted spanning edge isotopic to K.
Summing up, we have the following hierarchy: Proof. Any linear subdivision of a (convex) 3-dimensional polytope (with or without interior vertices) is collapsible [Chi67] and even LC [BZ11, Theorem 3.27]. However, Rudin proved in 1958 that not all these linear subdivisions are shellable [Rud58]; her counterexample, known as "Rudin's ball", is a subdivision of a tetrahedron with all 14 vertices on the boundary. The equivalence of LC and Mogami is discussed in Corollary 35. Any knotted 3-ball described in Proposition 42 is collapsible, but cannot be Mogami by Lemma 38. Finally, 3-balls without interior vertices that are not collapsible can be produced by pairing together Lemma 39 and Proposition 41: For example, any 3-ball without interior vertices and with a knotted spanning edge isotopic to the double trefoil would do.