Simplicial Structures in Topology

190 V Triangulable Manifolds
The definition of orientability as given here is obviously incomplete and not very
useful: in fact, a good definition must be independent from the triangulation and
based on an invariant by homeomorphisms. The next result is meant to correct this
flaw.
(V.3.2) Theorem. Let |K| be a triangulable n-manifold; 6 then
Hn(|K|;Z) ∼ = Z ⇐⇒ |K| is orientable.
Proof. Let us suppose that |K| is orientable and let z = ∑σ be the formal sum of all
n-simplexes of K. The definition of orientability implies that z and all its integral
multiples are n-cycles. On the other hand, if z ′ ∈ Zn(K) has a term which is a
multiple rτ of an n-simplex τ, then the terms of z ′ are of the form rτ ′ ,whereτ ′ runs
over the set of all n-simplexes of K intersecting τ in (n − 1)-simplexes (otherwise,
z ′ would not be a cycle); but then, by part 3 of Theorem (V.1.5), z ′ would equal the
formal sum r ∑σ = rz,whereσ runs over the set of all n-simplexes of K. Therefore,
Hn(|K|;Z) ∼ = Z.
With a similar argument we may prove that, conversely, given any orientation of
the n-simplexes of K,then-cycles of Cn(K) are of the type rz where z = ∑±σ and σ
runs over the set of all n-simplexes. Since ∂n(z)=0, we conclude that it is possible
to give an orientation to the n-simplexes, according to the Definition (V.3.1). �
Because of this last result, the definition of orientability of a triangulable
n-manifold does not depend on the triangulation; furthermore, two triangulable
connected n-manifolds of the same homotopy type are either both orientable or both
nonorientable.
Let V be a triangulable n-manifold and let us suppose that V ∼ = |K|, whereK =
(X,Φ). We consider the barycentric subdivision K (1) =(Φ,Φ (1) ) and associate
with every simplex σ ∈ Φ (that is to say, a vertex of K (1) ) the subset Bσ of Φ (1)
defined by
eσ = {{σ,σ 1 ,...,σ r }|σ ⊂ σ 1 ⊂ ...⊂ σ r } .
Note that, if dimσ = p, the dimension of the simplexes of eσ is ≤ n − p; in particular,
if dimσ = n, eσ coincides with the vertex σ (namely, the barycenter b(σ) )
of K (1) .
We now consider the closure and the boundary of eσ , that is to say, the simplicial
subcomplexes eσ and • eσ of K (1) (see Sect. III.5). The reader is asked to note that • eσ
is the set of all simplexes of eσ not having b(σ) as a vertex; hence,
eσ = • eσ ∗ b(σ)
and, therefore, |eσ | is contractible.
The reader is also asked to review Definitions (III.5.3) and (III.5.5) before turning
to the next result.
6 We remember that our definition of triangulable manifold requires K to be connected by
1-simplexes.