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