Simplicial Structures in Topology

82 II Simplicial Complexes
obtained by removing the n-face opposite to the vertex x0 from the simplicial complex
•
σ, n-simplicial cone with vertex {x0} . Clearly
H0(C(σ),Z) ∼ = Z
because C(σ) is connected. A similar proof to the one used for σ shows that
Hj(C(σ),Z) ∼ = 0forevery0< j < n. We note that, when j = n, thevertexx0
belongs to every n-simplex of C(σ) and so
(∀c ∈ Cn(C(σ)))c = kn−1∂n(c),
allowing us to conclude that the trivial cycle 0 is the only n-cycle of C(σ); inother
words, Hn(C(σ);Z) ∼ = 0.
In the next example we refer to the construction of an acyclic carrier.
Homology of the (abstract) cone –LetvK = v ∗ K be the join of a simplicial
complex K =(X,Φ) and of a simplicial complex with a single vertex (and
simplex) v.
(II.4.6) Lemma. The cones vK are acyclic simplicial complexes.
Proof. Let v be the simplicial complex with the single vertex v andnoothersimplex;
it is clear that v (considered as a simplicial complex) is an acyclic simplicial
complex. The chain complex C(v) is a subcomplex of C(vK); letι : C(v) → C(vK)
be the inclusion. Consider the simplicial function
It is readily seen that the chain morphism
c: vK → v , y ∈ vΦ ↦→{v} .
C(c)ι : C(v) −→ C(v)
coincides with the identity homomorphism of C(v); then, for every n ∈ Z,thecomposite
Hn(c)Hn(ι) equals the identity. Let us prove that ιC(c) and the identity homomorphism
1 C(vK) of C(vK) are homotopic. We define
sn : Cn(vK) → Cn+1(vK)
on the oriented n-simplexes σ ∈ vΦ (understood as a chain) by the formula
�
0 if v ∈ σ,
sn(σ)=
vσ if v �∈ σ;
sn may be linearly extended to a homomorphism of Cn(vK). Let us take a look into
the properties of these functions.
Case 1: n = 0–Letxbe any vertex of vK.
�
x − v if x �= v,
(1C0(vK) − ιC0(c))(x)=
0 if x = v.