Three-Dimensional Pseudomanifolds on Eight Vertices

Abstract

A normal pseudomanifold is a pseudomanifold in which the links of simplices are also pseudomanifolds. So, a normal 2-pseudomanifold triangulates a connected closed 2-manifold. But, normal 𝑑-pseudomanifolds form a broader class than triangulations of connected closed 𝑑-manifolds for 𝑑≥3. Here, we classify all the 8-vertex neighbourly normal 3-pseudomanifolds. This gives a classification of all the 8-vertex normal 3-pseudomanifolds. There are 74 such 3-pseudomanifolds, 39 of which triangulate the 3-sphere and other 35 are not combinatorial 3-manifolds. These 35 triangulate six distinct topological spaces. As a preliminary result, we show that any 8-vertex 3-pseudomanifold is equivalent by proper bistellar moves to an 8-vertex neighbourly 3-pseudomanifold. This result is the best possible since there exists a 9-vertex nonneighbourly 3-pseudomanifold which does not allow any proper bistellar moves.

1. Introduction

Recall that a simplicial complex is a collection of nonempty finite sets (sets of vertices) such that every nonempty subset of an element is also an element. For 𝑖≥0, the elements of size 𝑖+1 are called the 𝑖-simplices (or 𝑖-faces) of the complex.

A simplicial complex is usually thought of as a prescription for construction of a topological space by pasting geometric simplices. The space thus obtained from a simplicial complex 𝐾 is called the geometric carrier of 𝐾 and is denoted by |𝐾|. We also say that 𝐾triangulates|𝐾|. A combinatorial 2-manifold (resp., combinatorial2-sphere) is a simplicial complex which triangulates a closed surface (resp., the 2-sphere 𝑆²).

For a simplicial complex 𝐾, the maximum of 𝑘 such that 𝐾 has a 𝑘-simplex, is called the dimension of 𝐾. A 𝑑-dimensional simplicial complex 𝐾 is called pure if each simplex of 𝐾 is contained in a 𝑑-simplex of 𝐾. A 𝑑-simplex in a pure 𝑑-dimensional simplicial complex is called a facet. A 𝑑-dimensional pure simplicial complex 𝐾 is called a weak pseudomanifold if each (𝑑−1)-simplex of 𝐾 is contained in exactly two facets of 𝐾.

With a pure simplicial complex 𝐾 of dimension 𝑑≥1, we associate a graph Λ(𝐾) as follows. The vertices of Λ(𝐾) are the facets of 𝐾 and two vertices of Λ(𝐾) are adjacent if the corresponding facets intersect in a (𝑑−1)-simplex of 𝐾. If Λ(𝐾) is connected, then 𝐾 is called strongly connected. A strongly connected weak pseudomanifold is called a pseudomanifold. Thus, for a 𝑑-pseudomanifold 𝐾,Λ(𝐾) is a connected (𝑑+1)-regular graph. This implies that 𝐾 has no proper subcomplex which is also a 𝑑-pseudomanifold; (or else, the facets of such a subcomplex would provide a disconnection of Λ(𝑋)).

For any set 𝑉 with #(𝑉)=𝑑+2 (𝑑≥0), let 𝐾 be the simplicial complex whose simplexes are all the nonempty proper subsets of 𝑉. Then 𝐾 is a 𝑑-pseudomanifold and triangulates the 𝑑-sphere 𝑆^𝑑. This 𝑑-pseudomanifold 𝐾 is called the standard 𝑑-sphere and is denoted by 𝑆^𝑑_𝑑+2(𝑉) (or 𝑆^𝑑_𝑑+2). By convention, 𝑆⁰₂ is the only 0-pseudomanifold.

If 𝜎 is a face of a simplicial complex 𝐾, then the link of 𝜎 in 𝐾, denoted by lk_𝐾(𝜎) (or lk(𝜎)), is by definition the simplicial complex whose faces are the faces 𝜏 of 𝐾 such that 𝜏 is disjoint from 𝜎 and 𝜎∪𝜏 is a face of 𝐾. Clearly, the link of an 𝑖-face in a weak 𝑑-pseudomanifold is a weak (𝑑−𝑖−1)-pseudomanifold. For 𝑑≥1, a connected weak 𝑑-pseudomanifold is said to be a normal 𝑑-pseudomanifold if the links of all the simplices of dimension ≤𝑑−2 are connected. Thus, any connected triangulated 𝑑-manifold (triangulation of a closed 𝑑-manifold) is a normal 𝑑-pseudomanifold. Clearly, the normal 2-pseudomanifolds are just the connected combinatorial 2-manifolds; but, normal 𝑑-pseudomanifolds form a broader class than connected triangulated 𝑑-manifolds for 𝑑≥3.

Observe that if 𝑋 is a normal pseudomanifold, then 𝑋 is a pseudomanifold. (If Λ(𝑋) is not connected, then, since 𝑋 is connected, Λ(𝑋) has two components 𝐺₁ and 𝐺₂ and two intersecting facets 𝜎₁, 𝜎₂ such that 𝜎_𝑖∈𝐺_𝑖,𝑖=1,2. Choose 𝜎₁,𝜎₂ among all such pairs such that dim(𝜎₁∩𝜎₂) is maximum. Then dim(𝜎₁∩𝜎₂)≤𝑑−2 and lk_𝑋(𝜎₁∩𝜎₂) is not connected, a contradiction.) Notice that all the links of positive dimensions (i.e., the links of simplices of dimension ≤𝑑−2) in a normal 𝑑-pseudomanifold are normal pseudomanifolds. Thus, if 𝐾 is a normal 3-pseudomanifold, then the link of a vertex in 𝐾 is a combinatorial 2-manifold. A vertex 𝑣 of a normal 3-pseudomanifold 𝐾 is called singular if the link of 𝑣 in 𝐾 is not a 2-sphere. The set of singular vertices is denoted by SV(𝐾). Clearly, the space |𝐾|⧵SV(𝐾) is a pl 3-manifold. If SV(𝐾)=∅ (i.e., the link of each vertex is a 2-sphere), then 𝐾 is called a combinatorial 3-manifold. A combinatorial 3-sphere is a combinatorial 3-manifold which triangulates the topological 3-sphere 𝑆³.

Let 𝑀 be a weak 𝑑-pseudomanifold. If 𝛼 is a (𝑑−𝑖)-face of 𝑀,0<𝑖≤𝑑, such that lk_𝑀(𝛼)=𝑆^𝑖−1_𝑖+1(𝛽) and 𝛽 is not a face of 𝑀 (such a face 𝛼 is said to be a removable face of 𝑀), then consider the weak 𝑑-pseudomanifold (denoted by 𝜅_𝛼(𝑀)) whose facet-set is {𝜎∶𝜎afacetof𝑀,𝛼⊈𝜎}∪{𝛽∪𝛼⧵{𝑣}∶𝑣∈𝛼}. The operation 𝜅_𝛼∶𝑀↦𝜅_𝛼(𝑀) is called a bistellar 𝑖-move. For 0<𝑖<𝑑, a bistellar 𝑖-move is called a proper bistellar move. If 𝜅_𝛼 is a proper bistellar 𝑖-move and lk_𝑀(𝛼)=𝑆^𝑖−1_𝑖+1(𝛽), then 𝛽 is a removable 𝑖-face of 𝜅_𝛼(𝑀) (with lk_{𝜅𝛼(𝑀)}(𝛽)=𝑆^{𝑑−𝑖−1}_{𝑑−𝑖+1}(𝛼)) and 𝜅_𝛽∶𝜅_𝛼(𝑀)↦𝑀 is an bistellar (𝑑−𝑖)-move. For a vertex 𝑢, if lk_𝑀(𝑢)=𝑆^𝑑−1_𝑑+1(𝛽), then the bistellar 𝑑-move 𝜅_{𝑢}∶𝑀↦𝜅_{𝑢}(𝑀)=𝑁 deletes the vertex 𝑢 (we also say that 𝑁 is obtained from 𝑀 by collapsing the vertex 𝑢). The operation 𝜅_𝛽∶𝑁↦𝑀 is called a bistellar 0-move (we also say that 𝑀 is obtained from 𝑁 by starring the vertex 𝑢 in the facet 𝛽 of 𝑁). The 10-vertex combinatorial 3-manifold 𝐴³₁₀ in Example 3.15 is not neighbourly and does not allow any bistellar 1-move. In [1], Bagchi and Datta have shown that if the number of vertices in a nonneighbourly combinatorial 3-manifold is at most 9, then the 3-manifold admits a bistellar 1-move. Existence of the 9-vertex 3-pseudomanifold 𝐵³₉ in Example 3.16 shows that Bagchi and Datta's result is not true for 9-vertex 3-pseudomanifolds. Here we prove the following theorem.

Theorem 1.1. If 𝑀 is an 8-vertex 3-pseudomanifold, then there exists a sequence of bistellar 1-moves 𝜅_𝐴1,…,𝜅_𝐴𝑚, for some 𝑚≥0, such that 𝜅_𝐴𝑚(⋯(𝜅_𝐴1(𝑀))) is a neighbourly 3-pseudomanifold.

In [2], Altshuler has shown that every combinatorial 3-manifold with at most 8 vertices is a combinatorial 3-sphere. In [3], Grünbaum and Sreedharan have shown that there are exactly 37 polytopal 3-spheres on 8 vertices (namely, 𝑆³_8,1,…,𝑆³_8,37 in Examples 3.1 and 3.3). They have also constructed the nonpolytopal sphere 𝑆³_8,38. In [4], Barnette proved that there is only one more nonpolytopal 8-vertex 3-sphere (namely, 𝑆³_8,39). In [5], Emch constructed an 8-vertex normal 3-pseudomanifold (namely, 𝑁₁ in Example 3.5) as a block design. This is not a combinatorial 3-manifold and its automorphism group is PGL(2,7) (cf. [6]). In [7], Altshuler has constructed another 8-vertex normal 3-pseudomanifold (namely, 𝑁₅ in Example 3.5). In [8], Lutz has shown that there exist exactly three 8-vertex normal 3-pseudomanifolds which are not combinatorial 3-manifolds (namely, 𝑁₁,𝑁₅ and 𝑁₆ in Example 3.5) with vertex-transitive automorphism groups. Here we prove the following theorem.

Theorem 1.2. Let 𝑆³_8,35,…,𝑆³_8,38,𝑁₁,…,𝑁₁₅ be as in Examples 3.1 and 3.5.
(i)Then 𝑆³_8,𝑖≇𝑆³_8,𝑗,𝑁_𝑘≇𝑁_𝑙, and 𝑆³_8,𝑚≇𝑁_𝑛 for 35≤𝑖<𝑗≤38,1≤𝑘<𝑙≤15,35≤𝑚≤38, and 1≤𝑛≤15.(ii)If 𝑀 is an 8-vertex neighbourly normal 3-pseudomanifold, then 𝑀 is isomorphic to one of 𝑆³_8,35,…,𝑆³_8,38,𝑁₁,…,𝑁₁₅.

Corollary 1.3. There are exactly 39 combinatorial 3-manifolds on 8 vertices, all of which are combinatorial 3-spheres.

Corollary 1.4. There are exactly 35 normal 3-pseudomanifolds on 8 vertices which are not combinatorial 3-manifolds. These are 𝑁₁,…,𝑁₃₅ defined in Examples 3.5 and 3.8.

The topological properties of these normal 3-pseudomanifolds are given in Section 3.

2. Preliminaries

All the simplicial complexes considered in this paper are finite (i.e., with finite vertex-set). The vertex-set of a simplicial complex 𝐾 is denoted by 𝑉(𝐾). We identify the 0-faces of a complex with the vertices. The 1-faces of a complex 𝐾 are also called the edges of 𝐾.

If 𝐾,𝐿 are two simplicial complexes, then an isomorphism from 𝐾 to 𝐿 is a bijection 𝜋∶𝑉(𝐾)→𝑉(𝐿) such that for 𝜎⊆𝑉(𝐾),𝜎 is a face of 𝐾 if and only if 𝜋(𝜎) is a face of 𝐿. Two complexes 𝐾, 𝐿 are called isomorphic when such an isomorphism exists. We identify two complexes if they are isomorphic. An isomorphism from a complex 𝐾 to itself is called an automorphism of 𝐾. All the automorphisms of 𝐾 form a group under composition, which is denoted by Aut(𝐾).

For a face 𝜎 in a simplicial complex 𝐾, the number of vertices in lk_𝐾(𝜎) is called the degree of 𝜎 in 𝐾 and is denoted by deg_𝐾(𝜎) (or by deg(𝜎)). If every pair of vertices of a simplicial complex 𝐾 form an edge, then 𝐾 is called neighbourly. For a simplicial complex 𝐾, if 𝑈⊆𝑉(𝐾), then 𝐾[𝑈] denotes the induced complex of 𝐾 on the vertex-set 𝑈.

If the number of 𝑖-faces of a 𝑑-dimensional simplicial complex 𝐾 is 𝑓_𝑖(𝐾)(0≤𝑖≤𝑑), then the number 𝜒(𝐾)∶=∑^𝑑_𝑖=0(−1)^𝑖𝑓_𝑖(𝐾) is called the Euler characteristic of 𝐾.

A graph is a simplicial complex of dimension ≤1. A finite 1-pseudomanifold is called a cycle. An 𝑛-cycle is a cycle on 𝑛 vertices and is denoted by 𝐶_𝑛 (or by 𝐶_𝑛(𝑎₁,…,𝑎_𝑛) if the edges are 𝑎₁𝑎₂,…,𝑎_𝑛−1𝑎_𝑛,𝑎_𝑛𝑎₁).

For a simplicial complex 𝐾, the graph consisting of the edges and vertices of 𝐾 is called the edge-graph of 𝐾 and is denoted by EG(𝐾). The complement of EG(𝐾) is called the nonedge graph of 𝐾 and is denoted by NEG(𝐾). For a weak 3-pseudomanifold 𝑀 and an integer 𝑛≥3, we define the graph 𝐺_𝑛(𝑀) as follows. The vertices of 𝐺_𝑛(𝑀) are the vertices of 𝑀. Two vertices 𝑢 and 𝑣 form an edge in 𝐺_𝑛(𝑀) if 𝑢𝑣 is an edge of degree 𝑛 in 𝑀. Clearly, if 𝑀 and 𝑁 are isomorphic, then 𝐺_𝑛(𝑀) and 𝐺_𝑛(𝑁) are isomorphic for each 𝑛.

If 𝑀 is a weak 3-pseudomanifold and 𝜅_𝛼∶𝑀↦𝜅_𝛼(𝑀)=𝑁 is a bistellar 1-move, then, from the definition, (𝑓₀(𝑁),𝑓₁(𝑁),𝑓₂(𝑁),𝑓₃(𝑁))=(𝑓₀(𝑀),𝑓₁(𝑀)+1,𝑓₂(𝑀)+2,𝑓₃(𝑀)+1) and deg_𝑁(𝑣)≥deg_𝑀(𝑣) for any vertex 𝑣. If 𝜅_𝛼∶𝑀↦𝜅_𝛼(𝑀)=𝐿 is a bistellar 3-move, then (𝑓₀(𝐿),𝑓₁(𝐿),𝑓₂(𝐿),𝑓₃(𝐿))=(𝑓₀(𝑀)−1,𝑓₁(𝑀)−4,𝑓₂(𝑀)−6,𝑓₃(𝑀)−3).

Consider the binary relation “≤” on the set of weak 3-pseudomanifolds as 𝑀≤𝑁 if there exists a finite sequence of bistellar 1-moves 𝜅_𝛼1,…,𝜅_𝛼𝑚, for some 𝑚≥0, such that 𝑁=𝜅_𝛼𝑚(⋯𝜅_𝛼1(𝑀)). Clearly, this ≤ is a partial order relation.

Two weak 𝑑-pseudomanifolds 𝑀 and 𝑁 are bistellar equivalent (denoted by 𝑀∼𝑁) if there exists a finite sequence of bistellar operations leading from 𝑀 to 𝑁. If there exists a finite sequence of proper bistellar operations leading from 𝑀 to 𝑁, then we say 𝑀 and 𝑁 are properly bistellar equivalent and we denote this by 𝑀≈𝑁. Clearly, “∼" and “≈" are equivalence relations on the set of pseudomanifolds. It is easy to see that 𝑀∼𝑁 implies that |𝑀| and |𝑁| are pl homeomorphic.

For two simplicial complexes 𝑋 and 𝑌 with disjoint vertex sets, the simplicial complex 𝑋∗𝑌∶=𝑋∪𝑌∪{𝜎∪𝜏∶𝜎∈𝑋,𝜏∈𝑌} is called the join of 𝑋 and 𝑌.

Let 𝐾 be an 𝑛-vertex (weak) 𝑑-pseudomanifold. If 𝑢 is a vertex of 𝐾 and 𝑣 is not a vertex of 𝐾, then consider the simplicial complex Σ_𝑢𝑣𝐾 on the vertex set 𝑉(𝐾)∪{𝑣} whose set of facets is {𝜎∪{𝑢}∶𝜎 is a facet of 𝐾 and 𝑢∉𝜎}∪{𝜏∪{𝑣}∶𝜏 is a facet of 𝐾}. Then Σ_𝑢𝑣𝐾 is a (weak) (𝑑+1)-pseudomanifold and |Σ_𝑢𝑣𝐾| is the topological suspension 𝑆|𝐾| of |𝐾| (cf. [9]). It is easy to see that the links of 𝑢 and 𝑣 in Σ_𝑢𝑣𝐾 are isomorphic to 𝐾. This Σ_𝑢𝑣𝐾 is called the one-point suspension of 𝐾.

For two 𝑑-pseudomanifolds 𝑋 and 𝑌, a simplicial map 𝑓∶𝑋→𝑌 is called a 𝑘-fold branched covering (with discrete branch locus) if |𝑓||_{|𝑋|⧵𝑓}⁻¹_(𝑈)∶|𝑋|⧵𝑓⁻¹(𝑈)→|𝑌|⧵𝑈 is a 𝑘-fold covering for some 𝑈⊆𝑉(𝑌). (We say that 𝑋 is a branched cover of 𝑌 and 𝑌 is a branched quotient of 𝑋.) The smallest such 𝑈 (so that |𝑓||_{|𝑋|⧵𝑓}⁻¹_(𝑈)∶|𝑋|⧵𝑓⁻¹(𝑈)→|𝑌|⧵𝑈 is a covering) is called the branch locus. If 𝑁 is a 𝑘-fold branched quotient of 𝑀 and 𝑁 is obtained from 𝑁 by collapsing a vertex (resp., starring a vertex in a facet), then 𝑁 is the branched quotient of 𝑀, where 𝑀 can be obtained from 𝑀 by collapsing 𝑘 vertices (resp., starring 𝑘 vertices in 𝑘 facets). For proper bistellar moves we have the following lemma.

Lemma 2.1. Let 𝑀 and 𝑁 be two 𝑑-pseudomanifolds and 𝑓∶𝑀→𝑁 be a 𝑘-fold branched covering. For 1≤𝑙<𝑑−1, if 𝛼 is a removable 𝑙-face, then 𝑓⁻¹(𝛼) consists of 𝑘 removable 𝑙-faces 𝛼₁,…,𝛼_𝑘(𝑠𝑎𝑦) and 𝜅_𝛼𝑘(⋯(𝜅_𝛼1(𝑀))) is a 𝑘-fold branched cover of 𝜅_𝛼(𝑁).

Proof. Let lk_𝑁(𝛼)=𝑆^{𝑑−𝑙−1}_{𝑑−𝑙+1}(𝛽). Since the dimension of 𝛼 is >0,𝑓⁻¹(𝛼) consists of 𝑘𝑙-faces, 𝛼₁,…,𝛼_𝑘 (say) of 𝑀. Let lk_𝑀(𝛼_𝑖)=𝑆^{𝑑−𝑙−1}_{𝑑−𝑙+1}(𝛽_𝑖) and 𝑀_𝑖∶=𝑀[𝛼_𝑖∪𝛽_𝑖] for 1≤𝑖≤𝑘. Since 𝑓 is simplicial, 𝛽_𝑖 is not a face of 𝑀 and hence 𝛼_𝑖 is removable for each 𝑖. Since 0<𝑙<𝑑−1, it follows that 𝑀_𝑖 is neighbourly. For 𝑖≠𝑗, if 𝑥≠𝑦∈𝑉(𝑀_𝑖)∩𝑉(𝑀_𝑗), then 𝑥𝑦 is an edge in 𝑀_𝑖∩𝑀_𝑗 and hence the number of edges in 𝑓⁻¹(𝑓(𝑥)𝑓(𝑦)) is less than 𝑘, a contradiction. So, #(𝑉(𝑀_𝑖)∩𝑉(𝑀_𝑗))≤1 for 𝑖≠𝑗. This implies that 𝛽_𝑖 is not a face in 𝜅_𝛼𝑗(𝑀) and hence 𝛼_𝑖 is removable in 𝜅_𝛼𝑗(𝑀) for 𝑖≠𝑗. The result now follows.

Remark 3.14 shows that Lemma 2.1 is not true for 𝑙=𝑑−1 (i.e., for bistellar 1-moves) in general.

Example 2.2. In Figure 2, we present some weak 2-pseudomanifolds on at most seven vertices. The degree sequences are presented parenthetically below the figures. Each of 𝑆₁,…,𝑆₉ triangulates the 2-sphere, each of 𝑅₁,…,𝑅₄ triangulates the real projective plane and 𝑇 triangulates the torus. Observe that 𝑃₁,𝑃₂ are not pseudomanifolds.
We know that if 𝐾 is a weak 2-pseudomanifold with at most six vertices, then 𝐾 is isomorphic to 𝑆₁,…,𝑆₄ or 𝑅₁ (cf. [9]). In [10], we have seen the following.

Figure 1 

(a)

(b)

(c)

(d)

Figure 002 

Proposition 2.3. There are exactly 13 distinct 2-dimensional weak pseudomanifolds on 7 vertices, namely, 𝑆₅,…,𝑆₉,𝑅₂,…,𝑅₄,𝑇,𝑃₁,…,𝑃₃, and 𝑃₄.

3. Examples

We identify a weak pseudomanifold with the set of facets in it.

Example 3.1. These four neighbourly 8-vertex combinatorial 3-manifolds were found by Grünbaum and Sreedharan (in [3], these are denoted by 𝑃⁸₃₅,𝑃⁸₃₆,𝑃⁸₃₇ and ℳ, resp.). It follows from Lemma 3.4 that these are combinatorial 3-spheres. It was shown in [3] that the first three of these are polytopal 3-spheres and the last one is a nonpolytopal sphere: 𝑆³_8,35𝑆={1234,1267,1256,1245,2345,2356,2367,3467,3456,4567,1238,1278,2378,1348,3478,1458,4578,1568,1678,5678},_38,36𝑆={1234,1256,1245,1567,2345,2356,2367,3467,3456,4567,1268,1678,2678,1238,2378,1348,3478,1458,1578,4578},_38,37𝑆={1234,1256,1245,1457,2345,2356,2367,3467,3456,4567,1568,1578,5678,1268,2678,1238,2378,1348,1478,3478},_38,38={1234,1237,1267,1347,1567,2345,2367,3467,3456,4567,2358,2368,3568,1268,1568,1248,2458,1478,1578,4578}. (3.1)

Lemma 3.2. 𝑆³_8,𝑖≇𝑆³_8,𝑗 for 35≤𝑖<𝑗≤38.

Proof. Observe that 𝐺₆(𝑆³_8,35)=𝐶₈(1,2,…,8),𝐺₆(𝑆³_8,36)=(𝑉,{23,34,45,67,78,81}), 𝐺₆(𝑆³_8,37)=(𝑉,{23,34,56,78,81}), and 𝐺₆(𝑆³_8,38)=(𝑉,{17,23,58}), where 𝑉={1,…,8}. Since 𝐾≅𝐿 implies 𝐺₆(𝐾)≅𝐺₆(𝐿),𝑆³_8,𝑖≇𝑆³_8,𝑗, for 35≤𝑖<𝑗≤38.

Example 3.3. Some nonneighbourly 8-vertex combinatorial 3-manifolds. It follows from Lemma 3.4 that these are combinatorial 3-spheres. For 1≤𝑖≤34, the sphere 𝑆³_8,𝑖 is isomorphic to the polytopal sphere 𝑃⁸_𝑖 in [3] and the sphere 𝑆³_8,39 is isomorphic to the nonpolytopal sphere found by Barnette in [4]. We consecutively define 𝑆³_8,39=𝜅₄₆𝑆³_8,38,𝑆³_8,33=𝜅₂₇𝑆³_8,37,𝑆³_8,32=𝜅₄₈𝑆³_8,37,𝑆³_8,31=𝜅₅₈𝑆³_8,37,𝑆³_8,30=𝜅₂₄𝑆³_8,37,𝑆³_8,29=𝜅₂₇𝑆³_8,31,𝑆³_8,28=𝜅₂₄𝑆³_8,31,𝑆³_8,27=𝜅₁₃𝑆³_8,31,𝑆³_8,25=𝜅₅₇𝑆³_8,31,𝑆³_8,24=𝜅₄₈𝑆³_8,31,𝑆³_8,23=𝜅₃₅𝑆³_8,31,𝑆³_8,26=𝜅₄₆𝑆³_8,27,𝑆³_8,22=𝜅₂₄𝑆³_8,25,𝑆³_8,21=𝜅₆₈𝑆³_8,25,𝑆³_8,20=𝜅₄₈𝑆³_8,25,𝑆³_8,19=𝜅₁₇𝑆³_8,25,𝑆³_8,18=𝜅₂₇𝑆³_8,25,𝑆³_8,12=𝜅₁₅𝑆³_8,25,𝑆³_8,11=𝜅₃₅𝑆³_8,25,𝑆³_8,17=𝜅₂₄𝑆³_8,19,𝑆³_8,34=𝜅₂₇𝑆³_8,26=𝑆⁰₃(1,3)∗𝑆⁰₃(2,7)∗𝑆⁰₃(4,6)∗𝑆⁰₃(5,8),𝑆³_8,16=𝜅₁₃𝑆³_8,19,𝑆³_8,15=𝜅₂₈𝑆³_8,18,𝑆³_8,14=𝜅₄₇𝑆³_8,20,𝑆³_8,10=𝜅₁₅𝑆³_8,19,𝑆³_8,9=𝜅₃₅𝑆³_8,19,𝑆³_8,8=𝜅₄₇𝑆³_8,19,𝑆³_8,13=𝜅₃₈𝑆³_8,16,𝑆³_8,7=𝜅₂₄𝑆³_8,8,𝑆³_8,6=𝜅₃₅𝑆³_8,8,𝑆³_8,5=𝜅₄₈𝑆³_8,8,𝑆³_8,4=𝜅₁₅𝑆³_8,8,𝑆³_8,3=𝜅₄₈𝑆³_8,4,𝑆³_8,2=𝜅₄₈𝑆³_8,6,𝑆³_8,1=𝜅₁₆𝑆³_8,4. (3.2)

Lemma 3.4. (a) 𝑆³_8,𝑖≈𝑆³_8,𝑗, for 1≤𝑖,𝑗≤39, (𝑏)𝑆³_8,𝑚 is a combinatorial 3-sphere for 1≤𝑚≤39, and (𝑐)𝑆³_8,𝑘≇𝑆³_8,𝑙 for 1≤𝑘<𝑙≤39.

Proof. For 0≤𝑖≤6, let 𝒮_𝑖 denote the set of 𝑆³_8,𝑗's with 𝑖 nonedges. Then 𝒮₀={𝑆³_8,35,𝑆³_8,36,𝑆³_8,37,𝑆³_8,38},𝒮₁={𝑆³_8,30,𝑆³_8,31,𝑆³_8,32,𝑆³_8,33,𝑆³_8,39},𝒮₂={𝑆³_8,23,𝑆³_8,24,𝑆³_8,25,𝑆³_8,27,𝑆³_8,28,𝑆³_8,29},𝒮₃={𝑆³_8,11,𝑆³_8,12,𝑆³_8,18,𝑆³_8,19,𝑆³_8,20,𝑆³_8,21,𝑆³_8,22,𝑆³_8,26},𝒮₄={𝑆³_8,8,𝑆³_8,9,𝑆³_8,10,𝑆³_8,14,𝑆³_8,15,𝑆³_8,16,𝑆³_8,17,𝑆³_8,34},𝒮₅={𝑆³_8,4,𝑆³_8,5,𝑆³_8,6,𝑆³_8,7,𝑆³_8,13}, and 𝒮₆={𝑆³_8,1,𝑆³_8,2,𝑆³_8,3}.
From the proof of Lemma 4.7, 𝑆³_8,35≈𝑆³_8,30≈𝑆³_8,36≈𝑆³_8,30≈𝑆³_8,37≈𝑆³_8,32≈𝑆³_8,38. Thus, 𝑆³_8,𝑖≈𝑆³_8,𝑗 for 35≤𝑖,𝑗≤38. Now, if 𝑆³_8,𝑖∈𝒮₂∪𝒮₃∪𝒮₄∪𝒮₅∪𝒮₆, then, from the definition of 𝑆³_8,𝑖,𝑆³_8,𝑖≈𝑆³_8,31≈𝑆³_8,37. This proves part (a).
Since 𝑆³_8,34 is a join of spheres, 𝑆³_8,34 is a combinatorial 3-sphere. Clearly, if 𝑀≈𝑁 and 𝑀 is a combinatorial 3-sphere, then 𝑁 is so. Part (b) now follows from part (a).
Since the nonedge graphs of the members of 𝒮₆ (resp., 𝒮₅) are pairwise nonisomorphic, the members of 𝒮₆ (resp., 𝒮₅) are pairwise nonisomorphic.
For 𝑆³_8,𝑖,𝑆³_8,𝑗∈𝒮₄(𝑖<𝑗) and NEG(𝑆³_8,𝑖)≅NEG(𝑆³_8,𝑗) imply (𝑖,𝑗)=(8,9) or (14,15). Since 𝑀≅𝑁 implies 𝐺₆(𝑀)≅𝐺₆(𝑁) and 𝐺₆(𝑆³_8,8)≇𝐺₆(𝑆³_8,9),𝐺₆(𝑆³_8,14)≇𝐺₆(𝑆³_8,15), the members of 𝒮₄ are pairwise nonisomorphic.
For 𝑆³_8,𝑖≠𝑆³_8,𝑗∈𝒮₃ and NEG(𝑆³_8,𝑖)≅NEG(𝑆³_8,𝑗) imply {𝑖,𝑗}={11,12} or 18≤𝑖≠𝑗≤21. Let ∑₁={𝑆³_8,11,𝑆³_8,12},∑₂={𝑆³_8,18,𝑆³_8,19,𝑆³_8,20,𝑆³_8,21},∑₃={𝑆³_8,22}and∑₄={𝑆³_8,26}. Since the nonedge graph of a member in Σ_𝑖 is nonisomorphic to the nonedge graph of a member of Σ_𝑗 for 𝑖≠𝑗, a member of Σ_𝑖 is nonisomorphic to a member of Σ_𝑗. Observe that 𝐺₆(𝑆³_8,11)≇𝐺₆(𝑆³_8,12) and for 18≤𝑖<𝑗≤21,𝐺₆(𝑆³_8,𝑖)≅𝐺₆(𝑆³_8,𝑗) implies (𝑖,𝑗)=(18,19). Since 𝐺₃(𝑆³_8,18)≇𝐺₃(𝑆³_8,19), the members of 𝒮₃ are pairwise nonisomorphic.
Since 𝐺₃(𝑆³_8,𝑖)≇𝐺₃(𝑆³_8,𝑗) for 𝑆³_8,𝑖≠𝑆³_8,𝑗∈𝒮₂, the members of 𝒮₂ are pairwise nonisomorphic. By the same reasoning, the members of 𝒮₁ are pairwise nonisomorphic.
By Lemma 3.2, the members of 𝒮₀ are pairwise nonisomorphic. Since a member of 𝒮_𝑖 is nonisomorphic to a member of 𝒮_𝑗 for 𝑖≠𝑗, the above imply part (c).

Example 3.5. Some 8-vertex neighbourly normal 3-pseudomanifolds: 𝑁₁𝑁={1248,1268,1348,1378,1568,1578,2358,2378,2458,2678,3468,3568,4578,4678,1247,1257,1367,1467,2347,2567,3457,3567,1236,2346,1345,1235,1456,2456},₂={1248,2458,2358,3568,3468,4678,4578,1578,1568,1268,2678,2378,1378,1348,1247,2457,2357,3567,3467,1567,1267,1347}=Σ₇₈𝑁𝑇,₃𝑁={1248,1268,1348,1378,1568,1578,2358,2378,2458,2678,3468,3568,4578,4678,1234,2347,2456,2467,3456,3457,1235,1256,1357},₄𝑁={1248,1268,1348,1378,1568,1578,2358,2378,2458,2678,3468,3568,4578,4678,1245,1256,2356,2367,3467,1347,1457},₅𝑁={1258,1268,1358,1378,1468,1478,2368,2378,2458,2478,3458,3468,1257,1267,1367,1457,2357,2467,3457,3467,2356,2456,1356,1456},₆𝑁={1358,1378,1468,1478,1568,2368,2378,2458,2478,2568,3458,3468,1235,1245,1457,1567,2357,2567,3457,1236,1246,1367,2467,3467},₇𝑁={1268,1258,1358,1378,1478,1468,2378,2368,2458,2478,3468,3458,1356,1367,2357,2356,3467,3457,1256,1467,2457},₈=𝜅₃₄₈𝜅₂₃₈𝜅₅₆𝜅₆₇𝑁₇,𝑁₉=𝜅₂₃₅𝜅₆₇𝑁₇,𝑁₁₀=𝜅₁₄₈𝜅₆₇𝑁₇,𝑁₁₁=𝜅₃₄₈𝜅₅₆𝑁₁₀,𝑁₁₂=𝜅₄₅₇𝜅₂₃𝑁₉,𝑁₁₃=𝜅₅₆₇𝜅₂₃𝑁₉,𝑁₁₄=𝜅₁₃₈𝜅₅₇𝑁₈≅Σ₇₈𝑅₂,𝑁₁₅=𝜅₁₅₈𝜅₂₃𝑁₉. (3.3) All the vertices of 𝑁₁ are singular and their links are isomorphic to the 7-vertex torus 𝑇. There are two singular vertices in 𝑁₂ and their links are isomorphic to 𝑇. The singular vertices in 𝑁₃ are 8, 3, 4, 2, 5 and their links are isomorphic to 𝑇,𝑅₂,𝑅₂,𝑅₃, and 𝑅₃, respectively. There is only one singular vertex in 𝑁₄ whose link is isomorphic to 𝑇. All the vertices of 𝑁₅ (resp., 𝑁₆) are singular and their links are isomorphic to 𝑅₄ (resp., 𝑅₃). Each of 𝑁₇,…,𝑁₁₅ has exactly two singular vertices and their links are 7-vertex ℝ𝑃²'s. Thus, each 𝑁_𝑖 is a normal 3-pseudomanifold.
It follows from the definition that 𝑁_𝑖≈𝑁_𝑗 for 7≤𝑖,𝑗≤15. Here we prove the following lemmas.

Lemma 3.6. (a) The geometric carriers of 𝑁₁,𝑁₂,𝑁₃,𝑁₄,𝑁₅, and 𝑁₇ are distinct (non-homeomorphic), (𝑏)𝑁_𝑖≉𝑁_𝑗 for 1≤𝑖<𝑗≤7, (𝑐)𝑁₅∼𝑁₆.

Proof. For a normal 3-pseudomanifold 𝑋, let 𝑛_𝑠(𝑋) denote the number of singular vertices. Clearly, if 𝑀 and 𝑁 are two normal 3-pseudomanifolds with homeomorphic geometric carriers, then (𝑛_𝑠(𝑀),𝜒(𝑀))=(𝑛_𝑠(𝑁),𝜒(𝑁)). Now, (𝑛_𝑠(𝑁₁),𝜒(𝑁₁))=(8,8),(𝑛_𝑠(𝑁₂),𝜒(𝑁₂))=(2,2),(𝑛_𝑠(𝑁₃),𝜒(𝑁₃))=(5,3), (𝑛_𝑠(𝑁₄), 𝜒(𝑁₄))=(1,1), (𝑛_𝑠(𝑁₅),𝜒(𝑁₅))=(8,4),(𝑛_𝑠(𝑁₇),𝜒(𝑁₇))=(2,1). This proves part (𝑎).
Part (b) follows from the fact that 𝑁_𝑖 is neighbourly and has no removable edge and, hence, there is no proper bistellar move from 𝑁_𝑖 for 1≤𝑖≤6.
Let 𝑁^′₅ be obtained from 𝑁₅ by starring a new vertex 0 in the facet 1358. Let 𝑁₅^′′=𝜅_{0}(𝜅₀₈(𝜅₁₅₆(𝜅₀₇(𝜅₀₃(𝜅₀₃₅(𝜅₆₈(𝜅₀₂(𝜅₂₆₈(𝜅₁₃(𝜅₁₃₅(𝜅₁₃₈(𝜅₁₅₈(𝑁^′₅))))))))))))), then 𝑁₅^′′ is isomorphic to 𝑁₆ via the map (2,3)(5,8). This proves part (c).

Lemma 3.7. 𝑁_𝑘≇𝑁_𝑙 for 1≤𝑘<𝑙≤15.

Proof. Let 𝑛_𝑠 be as above. Clearly, if 𝑀 and 𝑁 are two isomorphic 3-pseudomanifolds, then (𝑛_𝑠(𝑀),𝑓₃(𝑀))=(𝑛_𝑠(𝑁),𝑓₃(𝑁)). Now, (𝑛_𝑠(𝑁₁),𝑓₃(𝑁₁))=(8,28),(𝑛_𝑠(𝑁₂),𝑓₃(𝑁₂))=(2,22),(𝑛_𝑠(𝑁₃),𝑓₃(𝑁₃))=(5,23),(𝑛_𝑠(𝑁₄),𝑓₃(𝑁₄))=(1,21),(𝑛_𝑠(𝑁₅),𝑓₃(𝑁₅))=(𝑛_𝑠(𝑁₆),𝑓₃(𝑁₆))=(8,24), and (𝑛_𝑠(𝑁_𝑖),𝑓₃(𝑁_𝑖))=(2,21) for 7≤𝑖≤15. Since the links of each vertex in 𝑁₅ is isomorphic to 𝑅₄ and the links of each vertex in 𝑁₆ is isomorphic to 𝑅₃, it follows that 𝑁₅≇𝑁₆. Thus, 𝑁_𝑖≇𝑁_𝑗 for 1≤𝑖≤6, 1≤𝑗≤15, 𝑖≠𝑗.
Observe that the singular vertices in 𝑁_𝑖 are 3 and 8 for 7≤𝑖≤15. Moreover, (i) lk_𝑁7(3)≅lk_𝑁7(8)≅𝑅₄, (ii) lk_𝑁8(3)≅𝑅₄ and lk_𝑁8(8)≅𝑅₃, (iii) lk_𝑁9(3)≅𝑅₂ and lk_𝑁9(8)≅𝑅₄, (iv) lk_𝑁10(3)≅lk_𝑁10(8)≅𝑅₃ and deg_𝑁10(38)=6, (v) lk_𝑁11(3)≅lk_𝑁11(8)≅𝑅₃ and deg_𝑁11(38)=5, (vi) lk_𝑁12(3)≅𝑅₂,lk_𝑁12(8)≅𝑅₃ and 𝐺₃(𝑁₁₂)=(𝑉,{32,21,17,75,54,46}), (vii) lk_𝑁13(3)≅𝑅₂,lk_𝑁13(8)≅𝑅₃ and 𝐺₃(𝑁₁₃)=(𝑉,{32,21, 17,75,56,67,64,42}), (viii) lk_𝑁14(3)≅lk_𝑁14(8)≅𝑅₂ and deg_𝑁14(38)=3. (xi) lk_𝑁15(3)≅lk_𝑁15(8)≅𝑅₂ and deg_𝑁15(38)=6. These imply that there is no isomorphism between 𝑁_𝑖 and 𝑁_𝑗 for 7≤𝑖<𝑗≤15. This completes the proof.

Example 3.8. Some 8-vertex nonneighbourly normal 3-pseudomanifolds: 𝑁₁₆=𝜅₆₇𝑁₇,𝑁₁₇=𝜅₂₄𝑁₈,𝑁₁₈=𝜅₂₃₈𝜅₅₆𝜅₆₇𝑁₇,𝑁₁₉=𝜅₅₇𝑁₈,𝑁₂₀=𝜅₅₆𝑁₁₀,𝑁₂₁=𝜅₁₂𝑁₉,𝑁₂₂=𝜅₁₄𝑁₁₁,𝑁₂₃=𝜅₂₃𝑁₉,𝑁₂₄=𝜅₃₈𝑁₁₄,𝑁₂₅=𝜅₅₆𝑁₁₆,𝑁₂₆=𝜅₁₂𝑁₁₆,𝑁₂₇=𝜅₅₆𝑁₁₇,𝑁₂₈=𝜅₅₇𝑁₁₈,𝑁₂₉=𝜅₁₅𝑁₁₈,𝑁₃₀=𝜅₁₂𝑁₂₃,𝑁₃₁=𝜅₂₄𝑁₂₂,𝑁₃₂=𝜅₂₄𝑁₂₆,𝑁₃₃=𝜅₅₇𝑁₂₅,𝑁₃₄=𝜅₄₅𝑁₂₈,𝑁₃₅=𝜅₅₈𝑁₂₉. (3.4)

Lemma 3.9. (𝑎)𝑁_𝑖≇𝑁_𝑗 for 1≤𝑖<𝑗≤35 and (𝑏)𝑁_𝑘≈𝑁_𝑙 for 7≤𝑘,𝑙≤35.

Proof. For 0≤𝑖≤3, let 𝒩_𝑖 denote the set of 3-pseudomanifolds defined in Examples 3.5 and 3.8 with 𝑖 nonedges. Then 𝒩₀={𝑁₁,…,𝑁₁₅},𝒩₁={𝑁₁₆,…,𝑁₂₄},𝒩₂={𝑁₂₅,…,𝑁₃₁}, and 𝒩₃={𝑁₃₂,…,𝑁₃₅}. The singular vertices in 𝑁_𝑖 are 3 and 8 for 7≤𝑖≤35.
By Lemma 3.7, the members of 𝒩₀ are pairwise nonisomorphic.
Observe that (i) lk_𝑁16(3)≅𝑅₄ and lk_𝑁16(8)≅𝑅₃, (ii) lk_𝑁17(3)≅lk_𝑁17(8)≅𝑅₄, (iii) lk_𝑁18(3)≅lk_𝑁18(8)≅𝑅₃ and 𝐺₆(𝑁₁₈)=(𝑉,{73,31,18,84}), (iv) lk_𝑁19(3)≅lk_𝑁19(8)≅𝑅₃ and 𝐺₆(𝑁₁₉)=(𝑉,{63,31,18,86}), (v) lk_𝑁20(3)≅lk_𝑁20(8)≅𝑅₃ and 𝐺₆(𝑁₂₀)=(𝑉,{74,28,83,31}), (vi) lk_𝑁21(3)≅𝑅₂,lk_𝑁21(8)≅𝑅₃ and 𝐺₆(𝑁₂₁)=(𝑉,{48,83,37,36}), (vii) lk_𝑁22(3)≅𝑅₂,lk_𝑁22(8)≅𝑅₃ and 𝐺₆(𝑁₂₂)=(𝑉,{28,86,63,37,38}), (viii) lk_𝑁23(3)≅𝑅₁ and lk_𝑁23(8)≅𝑅₃, (ix) lk_𝑁24(3)≅lk_𝑁24(8)≅𝑅₁. These imply that there is no isomorphism between any two members of 𝒩₁.
Observe that (i) lk_𝑁25(3)≅𝑅₃ and lk_𝑁25(8)≅𝑅₄, (ii) lk_𝑁26(3)≅lk_𝑁26(8)≅𝑅₃ and 𝐺₆(𝑁₂₆)=(𝑉,{53,38,84}), (iii) lk_𝑁27(3)≅lk_𝑁27(8)≅𝑅₃,𝐺₆(𝑁₂₇)=(𝑉,{78,81,13,37}) and NEG(𝑁₂₇)={24,56}, (iv) lk_𝑁28(3)≅lk_𝑁28(8)≅𝑅₃,𝐺₆(𝑁₂₈)=(𝑉,{18,84,43,31}) and NEG(𝑁₂₈)={75,56}, (v) lk_𝑁29(3)≅𝑅₃ and lk_𝑁29(8)≅𝑅₂, (vi) lk_𝑁30(3)≅𝑅₁ and lk_𝑁30(8)≅𝑅₃, (vii) lk_𝑁31(3)≅lk_𝑁31(8)≅𝑅₂. These imply that there is no isomorphism between any two members of 𝒩₂.
Observe that (i) lk_𝑁32(3)≅lk_𝑁32(8)≅𝑅₃, (ii) lk_𝑁33(3)≅lk_𝑁33(8)≅𝑅₄, (iii) lk_𝑁34(3)≅lk_𝑁34(8)≅𝑅₂, (iv) lk_𝑁35(3)≅𝑅₂ and lk_𝑁35(8)≅𝑅₁. These imply that there is no isomorphism between any two members of 𝒩₃.
Since a member of 𝒩_𝑖 is nonisomorphic to a member of 𝒩_𝑗 for 𝑖≠𝑗, the above imply part (a). Part (b) follows from the definition of 𝑁_𝑘 for 8≤𝑘≤35.

The 3-dimensional Kummer variety 𝐾³ is the torus 𝑆¹×𝑆¹×𝑆¹ modulo the involution 𝜎∶𝑥↦−𝑥. It has 8 singular points corresponding to 8 elements of order 2 in the abelian group 𝑆¹×𝑆¹×𝑆¹. In [11], Kühnel showed that 𝑁₅ triangulates 𝐾³. For a topological space 𝑋,𝐶(𝑋) denotes a cone with base 𝑋. Let 𝐻=𝐷²×𝑆¹ denote the solid torus. As a consequence of the above lemmas we get.

Corollary 3.10. All the 8-vertex normal 3-pseudomanifolds triangulate seven distinct topological spaces, namely, |𝑆³_8,𝑗|=𝑆³ for 1≤𝑗≤38,|𝑁₁|,|𝑁₂|=𝑆(𝑆¹×𝑆¹),|𝑁₃|,|𝑁₄|=𝐻∪(𝐶(𝜕𝐻)),|𝑁₅|=|𝑁₆|=𝐾³,and |𝑁_𝑖|=𝑆(ℝ𝑃²) for 7≤𝑖≤35.

Proof. Let 𝐾 be an 8-vertex normal 3-pseudomanifold. If 𝐾 is a combinatorial 3-sphere, then it triangulates the 3-sphere 𝑆³.
If 𝐾 is not a combinatorial 3-sphere, then, by Lemma 3.9(b), |𝐾| is (pl) homeomorphic to |𝑁₁|,…,|𝑁₆|, or |𝑁₇|. Since 𝑁₂=Σ₇₈𝑇,|𝑁₂| is homeomorphic to the suspension 𝑆(𝑆¹×𝑆¹). In 𝑁₄, the facets not containing the vertex 8 form a solid torus whose boundary is the link of 8. This implies that |𝑁₄|=𝐻∪(𝐶(𝜕𝐻)). It follows from Lemma 3.6(c) that |𝑁₆| is (pl) homeomorphic to |𝑁₅|=𝐾³. Since 𝑁₂₄ is isomorphic to the suspension 𝑆⁰₂∗𝑅₁, |𝑁₂₄|=𝑆(ℝ𝑃²). Therefore, by Lemma 3.9(b), |𝑁_𝑖| is (pl) homeomorphic to |𝑁₂₄|=𝑆(ℝ𝑃²) for 7≤𝑖≤35. The result now follows from Lemma 3.6(a).

A 3-dimensional pseudocomplex𝐾 is an ordered pair (Δ,Φ), where Δ is a finite collection of disjoint tetrahedra and Φ is a family of affine isomorphisms between pairs of 2-faces of the tetrahedra in Δ. Let |𝐾| denote the quotient space obtained from the disjoint union ⊔_𝜎∈Δ𝜎 by setting 𝑥=𝜑(𝑥) for 𝜑∈Φ. The quotient of a tetrahedron 𝜎∈Δ in |𝐾| is called a 3-simplex in |𝐾| and is denoted by |𝜎|. Similarly, the quotient of 2-faces, edges, and vertices of tetrahedra are called 2-simplices, edges, and vertices in |𝐾|, respectively. If |𝐾| is homeomorphic to a topological space 𝑋, then 𝐾 is called a pseudotriangulation of 𝑋. A 3-dimensional pseudocomplex 𝐾=(Δ,Φ) is said to be regular if the following hold: (i) each 3-simplex in |𝐾| has four distinct vertices, and (ii) for 2≤𝑖≤3, no two distinct 𝑖-simplices in |𝐾| have the same set of vertices. So, for 2≤𝑖≤3, an 𝑖-simplex 𝛼 in |𝐾| is uniquely determined by its vertices and denoted by 𝑢₁⋯𝑢_𝑖+1, where 𝑢₁,…,𝑢_𝑖+1 are vertices of 𝛼. (But, the edges in |𝐾| may not form a simple graph.) So, we can identify a regular pseudocomplex 𝐾=(Δ,Φ) with 𝒦∶={|𝜎|∶𝜎∈Δ}. Simplices and edges in |𝐾| are said to be simplices and edges of 𝒦. Clearly, a pure 3-dimensional simplicial complex is a regular pseudocomplex.

Let ℳ be a regular pseudotriangulation of 𝑋 and 𝑎𝑏𝑐𝑑,𝑎𝑏𝑐𝑒 be two 3-simplices in ℳ. If 𝑎𝑑𝑒,𝑏𝑑𝑒,𝑐𝑑𝑒 are not 2-simplices in ℳ, then 𝒩∶=(ℳ⧵{𝑎𝑏𝑐𝑑,𝑎𝑏𝑐𝑒})∪{𝑎𝑏𝑑𝑒,𝑎𝑐𝑑𝑒,𝑏𝑐𝑑𝑒} is also a regular pseudotriangulation of 𝑋. We say that 𝒩 is obtained from ℳ by the generalized bistellar 1-move𝜅_𝑎𝑏𝑐. If there is no edge between 𝑑 and 𝑒 in ℳ, then 𝜅_𝐹 is called a bistellar 1-move. If there exist 3-simplices of the form 𝑥𝑦𝑢𝑣,𝑥𝑧𝑢𝑣,𝑦𝑧𝑢𝑣 in a regular pseudotriangulation 𝒫 of 𝑌 and 𝑥𝑦𝑧 is not a 2-simplex, then 𝒬∶=(𝒫⧵{𝑥𝑦𝑢𝑣,𝑥𝑧𝑢𝑣,𝑦𝑧𝑢𝑣})∪{𝑥𝑦𝑧𝑢,𝑥𝑦𝑧𝑣} is also a regular pseudotriangulation of 𝑌. We say that 𝒬 is obtained from 𝒫 by the generalized bistellar 2-move𝜅_𝐸, where 𝐸 is the common edge in 𝑥𝑦𝑢𝑣,𝑥𝑧𝑢𝑣, and 𝑦𝑧𝑢𝑣. If 𝐸 is the only edge between 𝑢 and 𝑣 in 𝒫, then 𝜅_𝐸 is called a bistellar 2-move.

Let 𝑀 be a pseudotriangulation of a closed 3-manifold and 𝑁 a 3-pseudomanifold. A simplicial map 𝑓∶𝑀→𝑁 is said to be a 𝑘-fold branched covering (with discrete branch locus) if there exists 𝑈⊆𝑉(𝑁) such that |𝑓||_{|𝑀|⧵𝑓}⁻¹_(𝑈)∶|𝑀|⧵𝑓⁻¹(𝑈)→|𝑁|⧵𝑈 is a 𝑘-fold covering. The smallest such 𝑈 (so that |𝑓||_{|𝑀|⧵𝑓}⁻¹_(𝑈)∶|𝑀|⧵𝑓⁻¹(𝑈)→|𝑁|⧵𝑈 is a covering) is called the branch locus. It is known that 𝑁₁ can be regarded as a branched quotient of a regular hyperbolic tessellation (cf. [6]). In [11], Kühnel has shown that 𝑁₅ is a 2-fold branched quotient of a pseudotriangulation of the 3-dimensional torus. Here we prove the following theorem.

Theorem 3.11. (a) 𝑁₂₄ is a 2-fold branched quotient of a 14-vertex combinatorial 3-sphere.
(b) For 7≤𝑖≤35,𝑁_𝑖 is a 2-fold branched quotient of a 14-vertex regular pseudotriangulation of the 3-sphere.

Lemma 3.12. Let 𝑀 be a regular pseudotriangulation of a 3-manifold and 𝑁 be a normal 3-pseudomanifold. Let 𝑓∶𝑀→𝑁 be a 𝑘-fold branched covering with at most two vertices in the branch locus. If 𝜅_𝑒∶𝑁↦𝑁 is a bistellar 2-move, then there exist 𝑘 generalized bistellar 2-moves 𝜅_𝑒1,…,𝜅_𝑒𝑘 such that 𝜅_𝑒𝑘(⋯(𝜅_𝑒1(𝑀))) is a 𝑘-fold branched cover of 𝑁.

Proof. Let lk_𝑁(𝑒)=𝑆¹₃({𝑥,𝑦,𝑧}). Let 𝑓⁻¹(𝑒) consist of the edges 𝑒₁,…,𝑒_𝑘. Let the end points of 𝑒_𝑖 be 𝑢_𝑖, 𝑣_𝑖, the 3-simplices containing 𝑒_𝑖 be 𝑢_𝑖𝑣_𝑖𝑥_𝑖𝑦_𝑖,𝑢_𝑖𝑣_𝑖𝑥_𝑖𝑧_𝑖,𝑢_𝑖𝑣_𝑖𝑦_𝑖𝑧_𝑖, and 𝑓(𝑥_𝑖)=𝑥,𝑓(𝑦_𝑖)=𝑦,𝑓(𝑧_𝑖)=𝑧 for 1≤𝑖≤𝑘. Since 𝑥𝑦𝑧 is not a simplex in 𝑁, it follows that 𝑥_𝑖𝑦_𝑖𝑧_𝑖 is not a 2-simplex in 𝑀. Let 𝑀_𝑖 be the pseudocomplex consists of 𝑢_𝑖𝑣_𝑖𝑥_𝑖𝑦_𝑖,𝑢_𝑖𝑣_𝑖𝑥_𝑖𝑧_𝑖, and 𝑢_𝑖𝑣_𝑖𝑦_𝑖𝑧_𝑖. Since the number of vertices in the branched locus is at most 2, it follows that the number of vertices common in 𝑀_𝑖 and 𝑀_𝑗 is at most 2 for 𝑖≠𝑗. In particular, #({𝑥_𝑖,𝑦_𝑖,𝑧_𝑖}∩{𝑥_𝑗,𝑦_𝑗,𝑧_𝑗})≤2. Therefore, 𝑥_𝑗𝑦_𝑗𝑧_𝑗 is not a 2-simplex in 𝜅_𝑒𝑖(𝑀). So, we can perform generalized bistellar 2-move 𝜅_𝑒𝑗 on 𝜅_𝑒𝑖(𝑀)=(𝑀⧵𝑀_𝑖)∪{𝑥_𝑖𝑦_𝑖𝑧_𝑖𝑢_𝑖,𝑥_𝑖𝑦_𝑖𝑧_𝑖𝑣_𝑖} for 𝑖≠𝑗. Clearly, 𝑀∶=𝜅_𝑒𝑘(⋯𝜅_𝑒1(𝑀)) is a 𝑘-fold branched cover of 𝑁 (via the map 𝑓, where 𝑓(𝑤)=𝑓(𝑤) for 𝑤∈𝑉(𝑀)=𝑉(𝑀) and 𝑓(𝑥_𝑖𝑦_𝑖𝑧_𝑖𝑢_𝑖)=𝑥𝑦𝑧𝑢 and 𝑓(𝑥_𝑖𝑦_𝑖𝑧_𝑖𝑣_𝑖)=𝑥𝑦𝑧𝑣).

Lemma 3.13. Let 𝑀 be a regular pseudotriangulation of a 3-manifold and 𝑁 be a normal 3-pseudomanifold. Let 𝑓∶𝑀→𝑁 be a 𝑘-fold branched covering with at most two vertices in the branch locus. If 𝜅_𝐹∶𝑁↦𝑁 is a bistellar 1-move, then there exist 𝑘 generalized bistellar 1-moves 𝜅_𝐹1,…,𝜅_𝐹𝑘 such that 𝜅_𝐹𝑘(⋯(𝜅_𝐹1(𝑀))) is a 𝑘-fold branched cover of 𝑁.

Proof. Let 𝐹=𝑥𝑦𝑧 and lk_𝑁(𝐹)={𝑢,𝑣}. Let 𝑓⁻¹(𝐹) consist of the 2-simplices 𝐹₁,…,𝐹_𝑘. Let 𝐹_𝑖=𝑥_𝑖𝑦_𝑖𝑧_𝑖 and the 3-simplices containing 𝐹_𝑖 be 𝑥_𝑖𝑦_𝑖𝑧_𝑖𝑢_𝑖 and 𝑥_𝑖𝑦_𝑖𝑧_𝑖𝑣_𝑖 and 𝑓(𝑥_𝑖,𝑦_𝑖,𝑧_𝑖,𝑢_𝑖,𝑣_𝑖)=(𝑥,𝑦,𝑧,𝑢,𝑣) for 1≤𝑖≤𝑘. Since 𝑓 is simplicial, it follows that 𝑥_𝑖𝑢_𝑖𝑣_𝑖,𝑦_𝑖𝑢_𝑖𝑣_𝑖, and 𝑧_𝑖𝑢_𝑖𝑣_𝑖 are not 2-simplices in 𝑀. Let 𝑀_𝑖 be pseudocomplex {𝑥_𝑖𝑦_𝑖𝑧_𝑖𝑢_𝑖,𝑥_𝑖𝑦_𝑖𝑧_𝑖𝑣_𝑖}. Since the number of vertices in the branched locus is at most 2, it follows that 𝑥_𝑗𝑢_𝑗𝑣_𝑗,𝑦_𝑗𝑢_𝑗𝑣_𝑗, and 𝑧_𝑗𝑢_𝑗𝑣_𝑗 are not 2-simplices in 𝜅_𝐹𝑖(𝑀) for 𝑖≠𝑗. Then (by the similar arguments as in the proof of Lemma 3.12) 𝜅_𝐹𝑘(⋯𝜅_𝐹1(𝑀)) is a 𝑘-fold branched cover of 𝑁.

Proof of Theorem 3.11. If ℐ denotes the boundary of the icosahedron, then there exists a simplicial 2-fold covering 𝑓∶ℐ→𝑅₁. Consider the simplicial map 𝑓∶𝑆⁰₂({𝑎,𝑏})∗ℐ→𝑆⁰₂({𝑐,𝑑})∗𝑅₁ given by 𝑓(𝑎)=𝑐,𝑓(𝑏)=𝑑 and 𝑓(𝑢)=𝑓(𝑢) for 𝑢∈𝑉(ℐ). Then 𝑓 is a 2-fold branched covering with branch locus {𝑐,𝑑}. Since 𝑁₂₄ is isomorphic to the suspension 𝑆⁰₂∗𝑅₁, it follows that 𝑁₂₄ is a 2-fold branched quotient of the 14-vertex combinatorial 3-sphere 𝑆⁰₂({𝑎,𝑏})∗ℐ (with branch locus {3,8}). This proves part (a).
The result now follows from Lemmas 3.9(a), 3.12, and 3.13. (In fact, to obtain a 2-fold branched cover 𝑁₁₄ of 𝑁₁₄ from 𝑅₁∗𝑆⁰₂, one needs one bistellar 1-move and then one generalized bistellar 1-move; and all other moves required in the proof are bistellar moves on regular pseudotriangulations of 𝑆³.)

Remark 3.14. The combinatorial 3-sphere 𝑅₁∗𝑆⁰₂ is a 2-fold branched cover of 𝑁₂₄ and 𝑁₁₄ can be obtained from 𝑁₂₄ by a bistellar 1-move. Now, if 𝑓∶𝑀→𝑁₁₄ is a 2-fold branched covering and 𝑀 is a combinatorial 3-manifold, then (since lk_𝑁14(8) is a 7-vertex triangulated ℝ𝑃²) the link of any vertex in 𝑓⁻¹(8) is a 14-vertex triangulated 𝑆² and hence 𝑓₀(𝑀)>14. (Similarly, for 𝑖≠24, if 𝑁_𝑖 is a branched quotient of a combinatorial 3-manifold 𝑀, then 𝑓₀(𝑀)>14.) So, there does not exist a combinatorial 3-sphere 𝑀 which is a branched cover of 𝑁₁₄ and which can be obtained from 𝑅₁∗𝑆⁰₂ by proper bistellar moves.

In [7], Altshuler observed that 𝑁₁ is orientable and |𝑁₁| is simply connected. In [8], Lutz showed that (𝐻₁(𝑁₁),𝐻₂(𝑁₁),𝐻₃(𝑁₁))=(0,ℤ⁸,ℤ). The normal 3-pseudomanifold 𝑁₃ is the only among all the 35 which has singular vertices of different types, namely, one singular vertex whose link is a triangulated torus and four singular vertices whose links are triangulated real projective planes. Using polymake [12], we find that (𝐻₁(𝑁₃),𝐻₂(𝑁₃),𝐻₃(𝑁₃))=(0,ℤ²⊕ℤ₂,0). We summarized all the findings about 𝑁₁,…,𝑁₃₅ in Table 1.

Table 1 

8-vertex normal 3-pseudomanifolds which are not combinatorial 3-manifolds.

Example 3.15. For 𝑑≥2, let 𝐾^𝑑_2𝑑+3=𝑣_𝑖⋯𝑣_𝑗−1𝑣_𝑗+1⋯𝑣_𝑖+𝑑+1∶𝑖+1≤𝑗≤𝑖+𝑑