^{1}

^{2}

^{1}

^{2}

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

Recall that a

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

For a simplicial complex

With a pure simplicial complex

For any set

If

Observe that if

Let

If

In [

Let

Then

If

There are exactly

There are exactly

The topological properties of these normal

All the simplicial complexes considered in this paper
are finite (i.e., with finite vertex-set). The vertex-set of a simplicial
complex

If

For a face

If the number of

A

For a simplicial complex

If

Consider the binary relation “

Two weak

For two simplicial complexes

Let

For two

Let

Let

Remark

In Figure

We know that if

There are exactly

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

These four neighbourly 8-vertex combinatorial
3-manifolds were found by Grünbaum and Sreedharan (in [

Observe that

Some nonneighbourly 8-vertex combinatorial
3-manifolds. It follows from Lemma

(a)

For

From the proof of Lemma

Since

Since the nonedge graphs of the members of

For

For

Since

By Lemma

Some 8-vertex neighbourly normal
3-pseudomanifolds:

It follows from the definition that

(a) The geometric carriers of

For a normal 3-pseudomanifold

Part (b) follows from the fact that

Let

Let

Observe that the singular vertices in

Some 8-vertex nonneighbourly normal
3-pseudomanifolds:

For

By Lemma

Observe that (i)

Observe that (i)

Observe that (i)

Since a member of

The 3-dimensional

All the

Let

If

A 3-dimensional

Let

Let

(a)

(b) For

Let

Let

Let

Let

If

The result now follows from Lemmas

The combinatorial

In [

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

links of singular vertices | Geometric carriers, Homology | ||||
---|---|---|---|---|---|

8 | 8 | all are | |||

2 | 2 | both are | |||

3 | 5 | ||||

1 | 1 | ||||

4 | 8 | all are | |||

,, | ,, | ,, | all are | ||

1 | 2 | both are | |||

,, | ,, | ,, | both are in | ||

,, | ,, | ,, | ,, | ||

,, | ,, | ,, | ,, | ||

,, | ,, | ,, | ,, |

[Here

For

From

The only nonedge in

Similarly, from

By the same way, one can construct a

Let

For

Let

Let

Let

Since

Consider the face

Let

Let

There exists a

First consider the case when there exists a vertex

Let

Now, assume that

By the claim, there exists a 2-simplex

Let

Let

Since

The links of all the vertices
cannot be isomorphic to

Otherwise, let

Consider the case when

If

Let

Let us assume that

There is a vertex (other than the
vertex 8), say 7, whose link is isomorphic to

In the second case,

There is a vertex whose link is a
7-vertex

There exists a vertex in

If there is vertex whose link is isomorphic to

By the claim, we can assume that

Only singular vertex in

Since

Let

Let

There exist (at least) two
vertices whose links are isomorphic to

Exactly one vertex whose link is
isomorphic to

Exactly one vertex whose link is
isomorphic to

There is no vertex whose link is
isomorphic to

There is no vertex whose link is
isomorphic to

There is no vertex whose link is
isomorphic to

Since

Let

Recall that

The removable edges in

The removable edges in

The removable edges in

By the same arguments as in the case for

Hasse diagram of the poset of the 8-vertex
combinatorial 3-manifolds (the partial order relation is as defined in Section

Let

Recall that

The removable edges in

The removable edges in

The removable edges in

The removable edges in

The removable edges of

The removable edges in

The removable edges in

The removable edges in

The removable edges in

By the same arguments as in the case for

Hasse diagram of the poset of all the 3-pseudomanifolds

Let

Let

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.