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