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.

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 (additions in the suffixes are modulo ). It was shown in [13] the following : (i) is a triangulated -manifold for all , (ii) triangulates for even, and triangulates the twisted product (the twisted -bundle over ) for odd. For , is the unique nonsimply connected -vertex triangulated -manifold (cf. [14]). The combinatorial 3-manifolds was first constructed by Walkup in [15].
From , we construct the following 10-vertex combinatorial 3-manifold: [Geometrically, first we remove a pl 3-ball consisting of five 3-simplices from . This gives a pl 3-manifold with boundary and the boundary is a 2-sphere. Then we add a cone with base this boundary and vertex . So, the new polyhedron is pl homeomorphic to . This implies that the simplicial complex is a combinatorial 3-manifold.]
The only nonedge in is and there is no common -face in the links of and in . So, does not allow any bistellar -move. So, is a 10-vertex nonneighbourly combinatorial 3-manifold which does not admit any bistellar 1-move.
Similarly, from , we construct the following 12-vertex triangulated 4-manifold: The only nonedge in is and there is no common -face in the links of and in . So, does not allow any bistellar -move. So, is a 12-vertex nonneighbourly triangulated 4-manifold which does not admit any bistellar 1-move.
By the same way, one can construct a -vertex nonneighbourly triangulated -manifold (from ) which does not admit any bistellar 1-move for all .

Example 3.16. Let be as in Example 3.5. Let be obtained from by starring two vertices and in the facets and , respectively, that is, . Then is a 10-vertex normal 3-pseudomanifold. Let be obtained from by identifying the vertices and . Let the new vertex be 9. Then The degree 3 edges in are , , and ; but none of these edges is removable. So, no bistellar -moves are possible from . The only nonedge in is . Since there is no common -face in the links of and , no bistellar 1-move is possible. So, is a -vertex nonneighbourly 3-pseudomanifold which does not admit any proper bistellar move.

4. Proofs

For , by an we mean a combinatorial 2-sphere on vertices. If is a bistellar 1-move, then for . Here we prove the following.

Lemma 4.1. Let be an -vertex -pseudomanifold and be a vertex of degree . If , then there exists a bistellar -move such that .

Proof. Let and . Let . If , then the induced complex is a 3-pseudomanifold. Since , is a proper subcomplex of . This is not possible. So, and hence is a nonedge in . Then is a bistellar 1-move. Since is an edge in , is a required bistellar 1-move.

Lemma 4.2. Let be an -vertex -pseudomanifold and be a vertex of degree . If , then there exists a bistellar -move such that .

Proof. Since , the link of in is of the form for some vertices of . If both and are facets, then the induced subcomplex is a 3-pseudomanifold. This is not possible since . So, without loss of generality, assume that is not a facet. Again, if , and all are facets, then the induced subcomplex is a 3-pseudomanifold, which is not possible. So, assume that is not a facet.
Consider the face . Suppose . From the above, . So, is a nonedge and hence is a required bistellar 1-move.

Lemma 4.3. Let be a nonneighbourly -vertex -pseudomanifold and be a vertex of degree . If the degree of each vertex is at least , then there exists a bistellar -move such that .

Proof. Let be a vertex with and be a nonedge. Let .
Claim 1. There exists a -face such that and are facets.
First consider the case when there exists a vertex such that . Let .
Let . Since , is an edge. Thus contains 7 vertices. If one of is a -face, say , then and are facets. In this case, serves the purpose. So, assume that are nonfaces in . Then there are at least three -faces (not containing ) containing the edges in . Also, there are at least three -faces containing in . So, the number of -faces in is at least . This implies that or 4 and is a 7-vertex or . Since , it follows that is isomorphic to or (defined in Section 2). In each case, (since or 4, and is a nonedge) there exists an edge in such that and are -faces in and hence serves the purpose.
Now, assume that has no vertex of degree 5. Then must be of the form . If possible, let is not a facet for . Consider the -face . There exists a vertex such that is a facet. Assume, without loss of generality, that is a facet. Since (resp., ), (resp., ) is not a facet. So, the facet (other than ) containing must be . Similarly, the facet (other than ) containing must be . Then and are three facets containing , a contradiction. This proves the claim.

By the claim, there exists a 2-simplex such that . Since is a nonedge of is a bistellar 1-move. Since is an edge in , it follows that .

Proof of Theorem 1.1. Let be an -vertex -pseudomanifold. Then, by Lemma 4.1, there exist bistellar 1-moves , for some , such that the degree of each vertex in is at least 5. Therefore, by Lemma 4.2, there exist bistellar 1-moves , for some , such that the degree of each vertex in is at least 6. Then, by Lemma 4.3, there exist bistellar 1-moves , for some , such that the degree of each vertex in is 7. This proves the theorem.

Lemma 4.4. Let be an -vertex combinatorial -manifold. If is neighbourly, then is isomorphic to , or .

Proof. Since is a neighbourly combinatorial 3-manifold, by Proposition 2.3, the link of any vertex is isomorphic to , or .
Claim 1. The links of all the vertices cannot be isomorphic to ().
Otherwise, let . Consider the vertex 2. Since the degree of 2 is 7, or is not a facet. Assume, without loss of generality, that is not a facet. Again, if is a facet, then and hence . So, is not a facet. Similarly, is not a facet. Then the facet other than containing must be . Similarly, is a facet. This implies that . Thus . Similarly, . Then, the 7-vertex 2-sphere contains five vertices of degree 5. This is not possible. This proves the claim.
Case 1. Consider the case when has a vertex, (say ) whose link is isomorphic to . Assume, without loss of generality, that the facets containing the vertex are , , , , , , , , , and . Since . Hence the facet other than containing the face is one of , , or .
If , then, clearly, or . If , then on completing , we see that , , thereby showing that , an impossibility. Hence, and, therefore, . There are two possibilities for the completion of . If , , , from the links of and , we see that . Here, . If then . Thus, .
Subcase 1.1. . Since , either or is a facet. In the first case, . Here, , a contradiction. So, and hence . From the links of the vertices and , we see that . Here, by the map .Subcase 1.2. . By the same argument as in Case 1.1 (replace the vertex 1 by vertex 2), we get . From and . Here, by the map .Case 2. has no vertex whose link is isomorphic to but has a vertex whose link is isomorphic to . Using the same method as in Case 1.1, we find that .Case 3. has no vertex whose link is isomorphic to or but has a vertex whose link is isomorphic to . Using the same method as in Case 1.1, we find that .Case 4. has no vertex whose link is isomorphic to , , or but has a vertex (say 8) whose link is isomorphic to . The facets through can be assumed to be , , , , , , , , , and . Clearly, , . If , then from and , we see that , , , thereby showing that . Hence . Now, we can assume, without loss of generality, that the facets required to complete are , , and . Now, consider . If , then after completing the links of 2 and 7, we observe that . Hence . The links of , , and show that , , , , , and . Here, by the map . This completes the proof.

Lemma 4.5. Let be an -vertex neighbourly normal -pseudomanifold. If has one vertex whose link is the -vertex torus , then is isomorphic to , or .

Proof. Let us assume that and the link of the vertex 8 is the 7-vertex torus . So, the facets containing 8 are , , , , , , , , , , , , , and . We have the following cases. Case 1. There is a vertex (other than the vertex 8), say 7, whose link is isomorphic to . Then has no vertex of degree 3 and hence . This implies that the facet (other than ) containing is or . In the first case, . Thus, . Then, from the links of and , we get . Now, from . Then, from the links of , and , we get . Here, .
In the second case, . Thus, . Now, from the links of and , we get . Here, .
Case 2. There is a vertex whose link is a 7-vertex .
Claim 1. There exists a vertex in whose link is isomorphic to .
If there is vertex whose link is isomorphic to , then we are done. Otherwise, since acts transitively on , assume that (resp., ). Since , we may assume that the degree 4 vertex (resp., vertices) in is 1 (resp., are , , ). Then, from . This implies that is a nonsphere and . Hence . This proves the claim.

By the claim, we can assume that . Again, we may assume that the vertex 1 is of degree 3 in . Then, from . Considering the links of the edges , , , , and , we get . Here, .
Case 3. Only singular vertex in is . So, the link of each vertex (other than vertex 8) is an (a 7-vertex 2-sphere). Since is a degree 6 vertex in , it follows that is isomorphic to one of or (defined in Example 2.2) for any vertex . If , then (since ), we may assume that the other degree 6 vertex in is 3. Then, from the links of and , , , are facets containing , a contradiction. If , then (since ) we may assume that the degree 5 vertices in are 2, 3, and 5. Then cannot be an , a contradiction. So, . Since acts transitively on , it follows that the link of each vertex is isomorphic to .
Since and , we may assume that the degree 5 vertices in are 4 and 5. Since , it follows that . Then, from . Now, from the links of and , we get . Then, from . Here . This completes the proof.

Lemma 4.6. Let be an -vertex neighbourly normal -pseudomanifold. If is not a combinatorial -manifold and has no vertex whose link is isomorphic to the -vertex torus then is isomorphic to or . Proof. Let be the number of singular vertices in . Since is neighbourly, by Proposition 2.3, the link of any vertex is either a 7-vertex or a 7-vertex . So, the number of facets through a singular (resp., nonsingular) vertex is 12 (resp., 10). Let be the number of facets of . Consider the set is a facet of and is a vertex . Then . This implies is even. Since is not a combinatorial 3-manifold, it follows that and hence . So, has at least two vertices whose links are isomorphic to , or . Case 1. There exist (at least) two vertices whose links are isomorphic to . Assume that . Then . Since we may assume that or .Subcase 1.1. . Since , it follows that 1, 2, 3, 4 are degree 5 vertices in . Since , assume without loss that . Then, from , we get . This shows that is an . Since , it follows that . Then, from , . Then, from the links of and , . Here .Subcase 1.2. . So, . Since , the degree 4 vertices in are either , or . In the first case, on completion of , we observe that , , remain nonedges in . So, the degree 4 vertices in are , and . Then , , , , , and are facets. Since and , either or is an . In the former case, is a facet. This is not possible from . So, is an . Then, from . Now, from . Here, .Case 2. Exactly one vertex whose link is isomorphic to and there exists a vertex whose link is isomorphic to . Using the same method as in Case 1, we find that .Case 3. Exactly one vertex whose link is isomorphic to , there is no vertex whose link is isomorphic to and there exists (at least) a vertex whose link is isomorphic to . Using the same method as in Case 1, we find that .Case 4. There is no vertex whose link is isomorphic to and there exist (at least) two vertices whose links are isomorphic to . Assume that , so that . Using the same method as in Case 1, we get the following: (i) if , then and (ii) if , then is isomorphic to or .Case 5. There is no vertex whose link is isomorphic to , there exists exactly one vertex whose link is isomorphic to and there exists (at least) a vertex whose link is isomorphic to . Using the same method as in Case 1, we find that is isomorphic to or .Case 6. There is no vertex whose link is isomorphic to or and there exist (at least) two vertices whose links are isomorphic to . Using the same method as in Case 1, we find that is isomorphic to or . This completes the proof.Proof of Theorem 1.2. Since 's are combinatorial 3-manifolds and 's are not combinatorial 3-manifolds, for . Part (a) now follows from Lemmas 3.2, 3.7. Part (b) follows from Lemmas 4.4, 4.5, and 4.6.

Lemma 4.7. Let be as in the proof of Lemma 3.4. If a combinatorial -manifold is obtained from a member of by a bistellar -move, then is isomorphic to a member of for . Moreover, no bistellar -move is possible from a member of .

Proof. Recall that . The removable edges in are , , , , , , , , and . Since , up to isomorphisms, it is sufficient to consider the bistellar -moves and only. Here and by the map .
The removable edges in are , , , , , , and . Since , it is sufficient to consider the bistellar 2-moves and only. Here and by the map .
The removable edges in are , . Since is an automorphism of , it is sufficient to consider the bistellar 2-moves and only. Here and by the map .
The removable edges in are , and . Since it is sufficient to consider the bistellar 2-moves only. Here by the map . This proves the result for .

By the same arguments as in the case for , one proves for the cases for . We summarize these cases in Figure 3 below. Last part follows from the fact that none of or has any removable edges.

Lemma 4.8. Let be as in the proof of Lemma 3.9. If a -pseudomanifold is obtained from a member of by a bistellar -move, then is isomorphic to a member of for . Moreover, no bistellar -move is possible from a member of .

Proof. Recall that . Since there are no degree 3 edges in , , , and , no bistellar -moves are possible from , or . The degree 3 edges in (resp., in ) are (resp., ). But, none of these edges is removable. So, bistellar -moves are not possible from or .
The removable edges in are , and . Since , , and are automorphisms of , it follows that up to isomorphisms, we only have to consider the bistellar 2-move . Here, .
The removable edges in are , and . Since we only consider the bistellar 2-moves , , and . Here, , and .
The removable edges in are , and . Since , we consider only , and . Here, and .
The removable edges in are , and . Since , we consider the bistellar 2-moves and only. Here, and .
The removable edges of are , and . Since , we only consider the bistellar 2-moves and . Here, and (by the map ).
The removable edges in are , , , and . Here, (by the map ), (by the map ), and (by the map ).
The removable edges in are , and . Since , we only consider , and . Here, (by the map ), (by the map ), and .
The removable edges in are . Since , we only consider and . Here, and .
The removable edges in are . Since , we only consider the bistellar 2-moves and . Here, and (by the map ). This proves the result for .

By the same arguments as in the case for , one proves the same for other cases (namely, for ) as well. We summarize these cases in Figure 4. Last part follows from the fact that, for has no removable edge.

Proof of Corollary 1.3. Let be as in the proof of Lemma 3.4. Let be an 8-vertex combinatorial 3-manifold. Then, by Theorem 1.1, there exist bistellar 1-moves , for some , such that is a neighbourly 8-vertex 3-pseudomanifold. Since bistellar moves send a combinatorial 3-manifold to a combinatorial 3-manifold, is a combinatorial 3-manifold. Then, by Theorem 1.2, . In other words, , where and , , for , are bistellar 2-moves. Therefore, by Lemma 4.7, . The result now follows from Lemma 3.4.

Proof of Corollary 1.4. Let be as in the proof of Lemma 3.9. Let be an 8-vertex normal 3-pseudomanifold. Then, by Theorem 1.1, there exist bistellar 1-moves , for some , such that is a neighbourly 3-pseudomanifold. Since bistellar moves send a normal 3-pseudomanifold to a normal 3-pseudomanifold, is normal. Hence, by Theorem 1.2, . In other words, , where and , , for , are bistellar 2-moves. Therefore, by Lemma 4.8, . The result now follows from Lemma 3.9.

Acknowledgments

The authors thank the anonymous referees for many useful comments which helped to improve the presentation of this paper. The first author was partially supported by DST (Grant no. SR/S4/MS-272/05) and by UGC-SAP/DSA-IV.