#### 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 -pseudomanifolds form a broader class than triangulations of connected closed -manifolds for . 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 ,
the elements of size are called the -*simplices* (or -*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 is called the *geometric carrier* of and is denoted by .
We also say that *triangulates*.
A *combinatorial -manifold* (resp., *combinatorial*-*sphere*) is a simplicial complex which
triangulates a closed surface (resp., the 2-sphere ).

For a simplicial complex ,
the maximum of such that has a -simplex, is called the *dimension* of .
A -dimensional simplicial complex is called *pure* if each simplex of is contained in a -simplex of .
A -simplex in a pure -dimensional simplicial complex is called a *facet*. A -dimensional pure simplicial complex is called a *weak pseudomanifold* if each -simplex of is contained in exactly two facets of .

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

For any set with (), let be the simplicial complex whose simplexes are
all the nonempty proper subsets of .
Then is a -pseudomanifold and triangulates the -sphere .
This -pseudomanifold is called the *standard -sphere* and is denoted by (or ). By convention, is the only -pseudomanifold.

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

Observe that if is a normal pseudomanifold, then is a pseudomanifold. (If is not connected, then, since is connected, has two components and and two intersecting facets , such that .
Choose among all such pairs such that is maximum. Then and is not connected, a contradiction.) Notice
that all the links of positive dimensions (i.e., the links of simplices of
dimension ) in a normal -pseudomanifold are normal pseudomanifolds.
Thus, if is a normal 3-pseudomanifold, then the link of
a vertex in is a combinatorial 2-manifold. A vertex of a normal 3-pseudomanifold is called *singular* if the link of in is not a 2-sphere. The set of singular
vertices is denoted by
Clearly, the space is a pl 3-manifold. If (i.e., the link of each vertex is a 2-sphere),
then is called a *combinatorial -manifold*. A *combinatorial -sphere* is a combinatorial 3-manifold
which triangulates the topological 3-sphere .

Let be a weak -pseudomanifold. If is a -face of such that and is not a face of (such a face is said to be a *removable* face of ), then consider the weak -pseudomanifold (denoted by ) whose facet-set is .
The operation is called a *bistellar -move*. For ,
a bistellar -move is called a *proper bistellar move*. If is a proper bistellar -move and ,
then is a removable -face of (with ) and is an bistellar -move. For a vertex ,
if ,
then the bistellar -move deletes the vertex (we also say that is obtained from by *collapsing* the vertex ). The operation is called a bistellar -*move* (we also say that is obtained from by *starring* the vertex in the facet of ). The 10-vertex combinatorial 3-manifold 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 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 is an -vertex -pseudomanifold, then there exists a sequence
of bistellar -moves ,
for some ,
such that is a neighbourly -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, in Examples 3.1 and 3.3). They have also constructed the nonpolytopal sphere . In [4], Barnette proved that there is only one more nonpolytopal 8-vertex 3-sphere (namely, ). In [5], Emch constructed an 8-vertex normal 3-pseudomanifold (namely, in Example 3.5) as a block design. This is not a combinatorial 3-manifold and its automorphism group is (cf. [6]). In [7], Altshuler has constructed another 8-vertex normal 3-pseudomanifold (namely, 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, and in Example 3.5) with vertex-transitive automorphism groups. Here we prove the following theorem.

Theorem 1.2. *Let be as in Examples 3.1 and 3.5.*

(i)*Then and for and .*(ii)*If is an -vertex neighbourly normal -pseudomanifold, then is isomorphic to one of .*

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

Corollary 1.4. *There are exactly normal -pseudomanifolds on vertices which are not combinatorial -manifolds. These are defined in Examples 3.5 and 3.8.*

The topological properties of these normal -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 is denoted by .
We identify the 0-faces of a complex with the vertices. The -faces of a complex are also called the *edges* of .

If are two simplicial complexes, then an *isomorphism* from to is a bijection such that for is a face of if and only if is a face of .
Two complexes , are called *isomorphic* when such an isomorphism
exists. We identify two complexes if they are isomorphic. An isomorphism from a
complex to itself is called an *automorphism* of .
All the automorphisms of form a group under composition, which is
denoted by .

For a face in a simplicial complex ,
the number of vertices in is called the *degree* of in and is denoted by (or by ). If every pair of vertices of a simplicial
complex form an edge, then is called *neighbourly*. For a simplicial complex ,
if ,
then denotes the induced complex of on the vertex-set .

If the number of -faces of a -dimensional simplicial complex is then the number is called the *Euler characteristic* of .

A *graph* is a simplicial complex of dimension .
A finite 1-pseudomanifold is called a *cycle*. An -cycle is a cycle on vertices and is denoted by (or by if the edges are ).

For a simplicial complex ,
the graph consisting of the edges and vertices of is called the *edge-graph* of and is denoted by .
The complement of is called the *nonedge graph* of and is denoted by .
For a weak 3-pseudomanifold and an integer ,
we define the graph as follows. The vertices of are the vertices of .
Two vertices and form an edge in if is an edge of degree in .
Clearly, if and are isomorphic, then and are isomorphic for each .

If is a weak 3-pseudomanifold and is a bistellar 1-move, then, from the definition, and for any vertex . If is a bistellar -move, then .

Consider the binary relation “” on the set of weak 3-pseudomanifolds as if there exists a finite sequence of bistellar -moves , for some , such that . Clearly, this is a partial order relation.

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

For two simplicial complexes and with disjoint vertex sets, the simplicial
complex is called the *join* of and .

Let be an -vertex (weak) -pseudomanifold. If is a vertex of and is not a vertex of ,
then consider the simplicial complex on the vertex set whose set of facets is is a facet of and is a facet of .
Then is a (weak) -pseudomanifold and is the topological suspension of (cf. [9]). It is easy to see that the links of and in are isomorphic to .
This is called the *one-point suspension* of .

For two -pseudomanifolds and ,
a simplicial map is called a -*fold branched covering* (with discrete
branch locus) if is a -fold covering for some .
(We say that is a *branched cover* of and is a *branched quotient* of .)
The smallest such (so that is a covering) is called the *branch locus*. If is a -fold branched quotient of and is obtained from by collapsing a vertex (resp., starring
a vertex in a facet), then is the branched quotient of ,
where can be obtained from by collapsing vertices (resp., starring vertices in facets). For proper bistellar moves we have
the following lemma.

Lemma 2.1. *Let and be two -pseudomanifolds and be a -fold branched covering. For ,
if is a removable -face, then consists of removable -faces and is a -fold branched cover of .*

*Proof. *Let .
Since the dimension of is consists of -faces, (say) of .
Let and for .
Since is simplicial, is not a face of and hence is removable for each .
Since ,
it follows that is neighbourly. For ,
if ,
then is an edge in and hence the number of edges in is less than ,
a contradiction. So, for .
This implies that is not a face in and hence is removable in for .
The result now follows.

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

*Example 2.2. *In Figure 2, we present some weak -pseudomanifolds on at most seven
vertices. The degree sequences are
presented parenthetically below the figures. Each of triangulates the 2-sphere, each of triangulates the real projective plane and triangulates the torus. Observe that are not pseudomanifolds.

We know that if is a weak 2-pseudomanifold with at most six
vertices, then is isomorphic to or (cf. [9]). In [10], we have seen the following.

**(a)**

**(b)**

**(c)**

**(d)**

Proposition 2.3. *There are exactly distinct -dimensional weak pseudomanifolds on vertices, namely, ,
and .*

#### 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 and ,
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:

Lemma 3.2. * for .*

*Proof. *Observe that , ,
and ,
where .
Since implies ,
for .

*Example 3.3. *Some nonneighbourly 8-vertex combinatorial
3-manifolds. It follows from Lemma 3.4 that these are combinatorial 3-spheres.
For ,
the sphere is isomorphic to the polytopal sphere in [3] and the sphere is isomorphic to the nonpolytopal sphere found
by Barnette in [4]. We
consecutively define

Lemma 3.4. *(a) ,
for , is a combinatorial -sphere for ,
and for .*

*Proof. *For ,
let denote the set of 's with nonedges. Then and .

From the proof of Lemma 4.7, .
Thus, for .
Now, if ,
then, from the definition of .
This proves part (a).

Since is a join of spheres, is a combinatorial 3-sphere. Clearly, if and is a combinatorial -sphere, then is so. Part (b) now follows from part (a).

Since the nonedge graphs of the members of (resp., ) are pairwise nonisomorphic, the members of (resp., ) are pairwise nonisomorphic.

For and imply or .
Since implies and the members of are pairwise nonisomorphic.

For and imply or . Let
Since the nonedge graph of a member in is nonisomorphic to the nonedge graph of a
member of for ,
a member of is nonisomorphic to a member of .
Observe that and for implies .
Since ,
the members of are pairwise nonisomorphic.

Since for ,
the members of are pairwise nonisomorphic. By the same
reasoning, the members of are pairwise nonisomorphic.

By Lemma 3.2, the members of are pairwise nonisomorphic. Since a member of is nonisomorphic to a member of for ,
the above imply part (c).

*Example 3.5. *Some 8-vertex neighbourly normal
3-pseudomanifolds: All the vertices of are singular and their links are isomorphic to
the 7-vertex torus .
There are two singular vertices in and their links are isomorphic to .
The singular vertices in are 8, 3, 4, 2, 5 and their links are
isomorphic to and ,
respectively. There is only one singular vertex in whose link is isomorphic to .
All the vertices of (resp., ) are singular and their links are isomorphic
to (resp., ). Each of has exactly two singular vertices and their
links are 7-vertex 's. Thus, each is a normal 3-pseudomanifold.

It follows from the definition that for .
Here we prove the following lemmas.

Lemma 3.6. *(a) The geometric carriers of ,
and are distinct (non-homeomorphic), for , .*

*Proof. *For a normal 3-pseudomanifold ,
let denote the number of singular vertices.
Clearly, if and are two normal 3-pseudomanifolds with
homeomorphic geometric carriers, then .
Now, , , , .
This proves part .

Part (b) follows from the fact that is neighbourly and has no removable edge and,
hence, there is no proper bistellar move from for .

Let be obtained from by starring a new vertex 0 in the facet .
Let ,
then is isomorphic to via the map .
This proves part (c).

Lemma 3.7. * for .*

*Proof. *Let be as above. Clearly, if and are two isomorphic 3-pseudomanifolds, then .
Now, ,
and for .
Since the links of each vertex in is isomorphic to and the links of each vertex in is isomorphic to ,
it follows that .
Thus, for , , .

Observe that the singular vertices in are 3 and 8 for .
Moreover, (i) ,
(ii) and ,
(iii) and ,
(iv) and ,
(v) and ,
(vi) and ,
(vii) and , ,
(viii) and .
(xi) and .
These imply that there is no isomorphism between and for .
This completes the proof.

*Example 3.8. *Some 8-vertex nonneighbourly normal
3-pseudomanifolds:

Lemma 3.9. * for and for .*

*Proof. *For ,
let denote the set of 3-pseudomanifolds defined in
Examples 3.5 and 3.8 with nonedges. Then ,
and .
The singular vertices in are 3 and 8 for .

By Lemma 3.7, the members of are pairwise nonisomorphic.

Observe that (i) and ,
(ii) ,
(iii) and ,
(iv) and ,
(v) and ,
(vi) and ,
(vii) and ,
(viii) and ,
(ix) .
These imply that there is no isomorphism between any two members of .

Observe that (i) and ,
(ii) and ,
(iii) and ,
(iv) and ,
(v) and ,
(vi) and ,
(vii) .
These imply that there is no isomorphism between any two members of .

Observe that (i) ,
(ii) ,
(iii) ,
(iv) and .
These imply that there is no isomorphism between any two members of .

Since a member of is nonisomorphic to a member of for ,
the above imply part (a).
Part (b) follows from the definition of for .

The 3-dimensional *Kummer variety * is the torus modulo the involution .
It has 8 singular points corresponding to 8 elements of order 2 in the abelian
group .
In [11], Kühnel showed
that triangulates .
For a topological space denotes a cone with base .
Let denote the solid torus. As a consequence of
the above lemmas we get.

Corollary 3.10. *All the -vertex normal -pseudomanifolds triangulate seven distinct
topological spaces, namely, for and for .*

*Proof. *Let be an -vertex normal -pseudomanifold. If is a combinatorial -sphere, then it triangulates the 3-sphere .

If is not a combinatorial -sphere, then, by Lemma 3.9(b), is (pl) homeomorphic to ,
or .
Since is homeomorphic to the suspension .
In ,
the facets not containing the vertex 8 form a solid torus whose boundary is the
link of 8. This implies that .
It follows from Lemma 3.6(c) that is (pl) homeomorphic to .
Since is isomorphic to the suspension , .
Therefore, by Lemma 3.9(b), is (pl) homeomorphic to for .
The result now follows from Lemma 3.6(a).

A 3-dimensional *pseudocomplex* is an ordered pair ,
where is a finite collection of disjoint tetrahedra
and is a family of affine isomorphisms between
pairs of 2-faces of the tetrahedra in .
Let denote the quotient space obtained from the
disjoint union by setting for .
The quotient of a tetrahedron in is called a -*simplex* in and is denoted by .
Similarly, the quotient of 2-faces, edges, and vertices of tetrahedra are
called -*simplices*, *edges*, and *vertices* in ,
respectively. If is homeomorphic to a topological space ,
then is called a *pseudotriangulation* of .
A 3-dimensional pseudocomplex is said to be *regular* if the following hold: (i) each
3-simplex in has four distinct vertices, and (ii) for ,
no two distinct -simplices in have the same set of vertices. So, for ,
an -simplex in is uniquely determined by its vertices and
denoted by ,
where are vertices of .
(But, the edges in may not form a simple graph.) So, we can
identify a regular pseudocomplex with .
Simplices and edges in are said to be simplices and edges of .
Clearly, a pure 3-dimensional simplicial complex is a regular pseudocomplex.

Let be a regular pseudotriangulation of and be two 3-simplices in .
If are not 2-simplices in ,
then is also a regular pseudotriangulation of .
We say that is obtained from by the *generalized bistellar -move*.
If there is no edge between and in ,
then is called a *bistellar -move*. If there exist 3-simplices of the
form in a regular pseudotriangulation of and is not a 2-simplex, then is also a regular pseudotriangulation of .
We say that is obtained from by the *generalized bistellar -move*,
where is the common edge in and .
If is the only edge between and in , then is called a *bistellar -move*.

Let be a pseudotriangulation of a closed
3-manifold and a 3-pseudomanifold. A simplicial map is said to be a -*fold branched covering* (with discrete
branch locus) if there exists such that is a -fold covering. The smallest such (so that is a covering) is called the *branch locus*. It is known that can be regarded as a branched quotient of a
regular hyperbolic tessellation (cf. [6]). In [11], Kühnel has shown that is a 2-fold branched quotient of a
pseudotriangulation of the 3-dimensional torus. Here we prove the following
theorem.

Theorem 3.11. *(a) is a -fold branched quotient of a -vertex combinatorial -sphere.**(b) For is a -fold branched quotient of a -vertex regular pseudotriangulation of the -sphere.*

Lemma 3.12. *Let be a regular pseudotriangulation of a -manifold and be a normal -pseudomanifold. Let be a -fold branched covering with at most two
vertices in the branch locus. If is a bistellar -move, then there exist generalized bistellar -moves such that is a -fold branched cover of .*

*Proof. *Let .
Let consist of the edges .
Let the end points of be , ,
the 3-simplices containing be ,
and for .
Since is not a simplex in ,
it follows that is not a 2-simplex in .
Let be the pseudocomplex consists of and .
Since the number of vertices in the branched locus is at most 2, it follows
that the number of vertices common in and is at most 2 for .
In particular, .
Therefore, is not a 2-simplex in .
So, we can perform generalized bistellar 2-move on for .
Clearly, is a -fold branched cover of (via the map ,
where for and and ).

Lemma 3.13. *Let be a regular pseudotriangulation of a -manifold and be a normal -pseudomanifold. Let be a -fold branched covering with at most two
vertices in the branch locus. If is a bistellar -move, then there exist generalized bistellar -moves such that is a -fold branched cover of .*

*Proof. *Let and .
Let consist of the 2-simplices .
Let and the 3-simplices containing be and and for .
Since is simplicial, it follows that , and are not 2-simplices in .
Let be pseudocomplex .
Since the number of vertices in the branched locus is at most 2, it follows
that and are not 2-simplices in for .
Then (by the similar arguments as in the proof of Lemma 3.12) is a -fold branched cover of .

*Proof of Theorem 3.11. *If denotes the boundary of the icosahedron, then
there exists a simplicial 2-fold covering .
Consider the simplicial map given by and for .
Then is a 2-fold branched covering with branch
locus .
Since is isomorphic to the suspension ,
it follows that is a 2-fold branched quotient of the 14-vertex
combinatorial 3-sphere (with branch locus ). 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 of from ,
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 .)

*Remark 3.14. *The combinatorial -sphere is a -fold branched cover of and can be obtained from by a bistellar 1-move. Now, if is a -fold branched covering and is a combinatorial -manifold, then (since is a -vertex triangulated ) the link of any vertex in is a 14-vertex triangulated and hence .
(Similarly, for ,
if is a branched quotient of a combinatorial
3-manifold ,
then .)
So, there does not exist a combinatorial -sphere which is a branched cover of and which can be obtained from by proper bistellar moves.

In [7], Altshuler observed that is orientable and is simply connected. In [8], Lutz showed that . The normal -pseudomanifold 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 . We summarized all the findings about in Table 1.

*Example 3.15. *For ,
let