Simplicial Structures in Topology

142 III Homology of Polyhedra
Note that d ⊗ = {d ⊗ n |n ≥ 1} is really a differential:
d ⊗ i+ j−1 d⊗ i+ j (xi ⊗ y j)=∂i−1∂i(x) ⊗ y j +(−1) i−1 ∂i(xi) ⊗ ∂ ′ j (y j)+
(−1) i ∂i(xi) ⊗ ∂ ′ j(y j)+xi ⊗ ∂ ′ j−1∂ ′ j(y j)=0.
It is sometimes convenient to leave the chain morphisms indexes out, to simplify
the definition and, therefore, also the proofs; for instance, we define the boundary
morphism
d ⊗ : C ⊗C ′ → C ⊗C ′
on generators x ⊗ y ∈ C ⊗C ′ by the formula
d ⊗ (x ⊗ y)=d(x) ⊗ y +(−1) |x| x ⊗ d ′ (y)
(remember that |x| indicates the degree of x, thatistosay,|x| = n ⇐⇒ x ∈ Cn ).
As an exercise, we leave to the reader the task of proving that the tensor product
of two free chain complexes with an augmentation homomorphism is a free chain
complex with augmentation. The reader is advised to do the exercises on tensor
products of chain complexes, at the end of this section. Let K and L be two simplicial
complexes. In Sect. III.1, we have proved that the product of two polyhedra
is a polyhedron. Actually, we have proved that, given the polyhedra |K| and |L|,
there exists a simplicial complex K × L such that |K × L| ∼ = |K|×|L| (see Theorem
(III.1.1)); the reader is also advised to review the construction of the simplicial
complex K × L in the proof of Theorem (III.1.1).
Therefore, given two simplicial complexes K and L, we can define two new functors
from the product category Csim×Csim (see examples of categories in Sect. I.2)
to the category Clp:
C× : Csim × Csim → Clp , (K,L) ↦→ C(K × L) ,
C ⊗C: Csim × Csim → Clp , (K,L) ↦→ C(K) ⊗C(L) .
The functors C× and C ⊗C are examples of free functors with models and universal
elements. In fact, as we did for the functor C : Csim → Clp, wefixatean
n-simplex σn for each n ≥ 0 and we take as models the pairs (σ j,σn− j) for C×, and
(σn,σn) for C ⊗C. The universal elements are easily described.
We note that on each model σn the functor C : Csim → Clp produces a positive,
acyclic free chain complex, that is to say,
Hi(σn,Z) ∼ �
Z if i = 0
=
0 if i > 0
(see the definition of acyclic chain complexes in Sect. II.3 and how to compute the
homology of σn in Sect. II.4). For this reason, we say that the models σn are acyclic
for the functor C.
(III.6.1) Lemma. The models (σ j,σn− j) are acyclic for the functor C× and the
models (σn,σn) are acyclic for C ⊗C.