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Proof.
See [AgoM76]. The proof is also not hard but too long to give here. It does
make use of Theorem 7.5.1.
To define the orientability of pseudomanifolds, we start with a combinatorial def-
inition and then show how one can use a homology group to detect this property.
Definition.
Let
X
be an n-dimensional pseudomanifold and let (K,j) be a triangu-
lation of
X
. We shall say that
X
is
orientable
if the n-simplices of K can be oriented
coherently, that is, the n-simplices of K can be oriented simultaneously in such a way
that any two n-simplices that meet in a common (n - 1)-dimensional face induce oppo-
site orientations on that face. Such a choice of orientations of n-simplices, if it exists,
is called an
orientation
of
X
. If
X
is not orientable, one calls
X
nonorientable
.
The next theorem shows that the definition is well defined and independent of
the particular triangulation that is chosen. It also shows that orientability is easily
determined.
7.5.3. Theorem.
A closed n-dimensional pseudomanifold
X
is orientable if and only
if H
n
(
X
) ª
Z
.
Proof.
See [AgoM76]. The proof again relies on finding the right cycles like in the past.
Using the results in Table 7.2.1.1 for surfaces, we see that our new rigorous defi-
nition of orientable agrees with our previous intuitive definition. More importantly,
we now have an algorithm for determining the orientability of a surface. It is also
clear that choosing an orientation of a closed n-dimensional pseudomanifold
X
is
equivalent to choosing a generator of H
n
(
X
). Theorem 7.5.1 shows that the mod 2
homology groups tell us nothing about the orientability of
X
.
Before we state another useful criterion for when a pseudomanifold is orientable
we need to discuss a few more concepts associated to pseudomanifolds.
Let K be a simplicial complex that triangulates an n-dimensional pseudomanifold
M
n
. Recall the definition of the barycentric subdivision sd(K) of the simplicial complex
K given in Section 7.2.2. Its vertices are the barycenters of the simplices in K. If b(s)
again denotes the barycenter of the simplex s, then the k-simplices of sd(K), k > 0,
are all the k-simplices of the form b(s
0
)b(s
1
)...b(s
k
) where the s
i
are distinct sim-
plices of K and s
0
s
1
...
s
k
.
k
be a k-simplex of K. Define the
dual
(
n
-
k
)
-cell
s
n-k
by
Definition.
Let s
s
n-k
=
{b(s)b(s
1
)b(s
2
)...b(s
n-k
) Ω s
i
is a simplex in K and s
s
1
s
2
...
s
n-k
}.
Call b(s) the
barycenter
of s
n-k
.
7.5.4. Example.
Consider the two-dimensional simplicial complex K shown in
Figure 7.32 whose vertices are labeled with uppercase letters and whose edges are
drawn with thick lines. The barycentric subdivision of K is drawn with thin lines and
its additional vertices are labeled with lowercase letter. The dual cell of the 0-simplex
A
is the union of the 2-cells
Aab
,
Abc
,
Acd
,
Ade
,
Aef
, and
Afa
in the barycentric
