Simplicial Structures in Topology

II.5 Homology with Coefficients 95
We now focus our attention on functor Tor(−,G). For any H ∈ Ab, wedefine
the group Tor(H,G) as follows. Let F(H) be the free group generated by all the
elements of H; the function
q: F(H) → H , h ↦→ h
is an epimorphism of F(H) onto H. Leti: kerq → F(H) be the inclusion homomorphism;
we then have a representation of H by free groups
We define
kerq ��
i ��
F(H)
Tor(H,G) := coker(i ⊗ 1G).
q
��
��
H.
By the first part of the theorem, Tor(H,G) does not depend on the presentation
of H. As for the morphisms, for any ¯f ∈ Ab(H,H ′ ), we choose the presentations
R ↣ F ↠ H and R ′ ↣ F ′ ↠ H ′ ; by Theorem (II.3.6), we obtain a chain morphism f
between the complexes C and C ′ (determined by R ↣ F and R ′ ↣ F ′ , respectively)
that extends ¯f and is unique up to chain homotopy. By taking their tensor product
by G and computing the homology groups, we obtain
H1( f ⊗ 1G): Tor(H,G) → Tor(H ′ ,G)
which is, by definition, the result of applying the torsion product on ¯f . �
When (C,∂) is the chain complex (C(K),∂) of an oriented simplicial complex
K, the previous results prove the Universal Coefficients Theorem in Homology:
(II.5.5) Theorem. The homology of a simplicial complex K with coefficients in an
Abelian group G is determined by the following short exact sequences:
Hn(K;Z) ⊗ G ��
��
Hn(K;G)
What is more, if G is free,
��
��
Tor(Hn−1(K;Z),G).
Hn(K;G) ∼ = Hn(K;Z) ⊗ G ⊕ Tor(Hn−1(K;Z),G).
Let us now see what happens when G = Q, the additive group of rational numbers.
This group is not free, but it is locally free: we say that an Abelian group G is
locally free if every finitely generated subgroup of G is free; in particular, due to the
Finitely Generated Abelian Groups Decomposition Theorem (see p. 75), a finitely
generated Abelian group is locally free if and only if it is torsion free. We now state
the following
(II.5.6) Lemma. If i: A → A ′ is a monomorphism and G is locally free, then
is a monomorphism.
i ⊗ 1G : A ⊗ G −→ A ′ ⊗ G