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(1) Every simplex of K is a face of some n-simplex in K.
(2) Every (n - 1)-simplex of K is a face of at least one, but not more than two, n-
simplices of K.
(3) Given any two n-simplices a and b in K there is a chain a = s
1
, s
2
,..., s
k
=
b of n-simplices s
i
in K so that s
i
and s
i+1
meet in an (n - 1)-simplex.
The pseudomanifold is said to be
closed
if ∂K = f.
Note that condition (3) in the definition implies that pseudomanifolds are compact
connected spaces. This is not an essential but convenient standard assumption. It is
easy to see that every (combinatorial) surface is a pseudomanifold, but not every two-
dimensional pseudomanifold is a surface. Figure 7.31(a) shows a two-dimensional
pseudomanifold with boundary that is not a surface with boundary. Figure 7.31(b)
shows a
pinched sphere
(a sphere with two points identified). The problem occurs at
the points
p
that do not have the correct neighborhood. In general, every triangula-
ble n-manifold is an n-pseudomanifold. Although the boundary of every manifold is
a manifold, Figure 7.31(a) also shows that this need not be the case for pseudoman-
ifolds. Nevertheless, pseudomanifolds have enough nice manifold-type properties, so
that they are interesting because many properties of manifolds are true simply
because they satisfy the pseudomanifold conditions. Finally, one nice fact (Theorem
7.5.2) is that it does not matter how we triangulate a pseudomanifold because every
triangulation will satisfy properties (1)-(3). One way to prove this topological invari-
ance of the combinatorial structure of a pseudomanifold is to establish the following
interesting property of the top-dimensional mod 2 homology group of a pseudoman-
ifold (and hence manifold) first.
7.5.1. Theorem.
Let
X
be an n-dimensional pseudomanifold.
(1) If ∂
X
= f, then H
n
(
X
;
Z
2
) ª
Z
2
.
(2) If ∂
X
π f, then H
n
(
X
;
Z
2
) = 0.
Proof.
See [AgoM76]. The proofs are not hard. They are similar to our computa-
tions of homology groups for
S
n
and consists in finding the obvious cycles and bound-
aries. Note that over
Z
2
the orientation of simplices does not play a role.
7.5.2. Theorem.
(Invariance of Pseudomanifolds) Let
X
be an n-dimensional
pseudomanifold and let (L,Y) be any triangulation of
X
. Then L satisfies properties
(1)-(3) in the definition of a pseudomanifold.
p
p
(a)
(b)
Figure 7.31.
Pseudomanifolds that are not manifolds.
