I am reading Hatcher Chapter V on spectral sequence. This is a paragraph after Theorem 5.3:
The Kunneth formula and the universal coefficient theorem then combine to give an isomorphism $$H_n(B\times F;G)\cong \bigoplus_p H_p(B;H_{n-p}(F;G))$$
where $B$ is a simply connected space, $F$ is any topological space, and $G$ is an arbitrary abelian group.
I guess the logic would be
$$H_n(B\times F;G)\cong \bigoplus_{p} H_p(B;G)\otimes H_{n-p}(F;G)\cong \bigoplus_p H_p(B;H_{n-p}(F;G))$$
But for the first isomorphism, as far as I know, it is only true when $G$ is a field then we can use Kunneth formula.
Moreover, for the second isomorphism, by the universal coefficient theorem, there should be $\mathrm{Tor}$ part:
$$0 \to H_i(X; \mathbf{Z})\otimes A \, \overset{\mu}\to \, H_i(X;A) \to \operatorname{Tor}(H_{i-1}(X; \mathbf{Z}),A)\to 0.$$
The last term is zero if one of them is free, which is not generally true.
So I want to know what real argument we use here. I know both of theorems are pure homological algebra results, then probably we can adjust the condition properly so that we can get the desired result?
Best Answer
It is convenient for an answer to this question to generalise the usual homology $H_*(C,A)$ of a chain complex $C$ (of abelian groups, and with $C_n= 0$ for $n < 0$) from coefficients in an abelian group $A$, to the case where $A$ is also a similar chain complex. We work with the definition $H_*(C;A)=H_*(C \otimes A)$.
So we are thinking of the questioner's $H_n(B \times F;G)$ as $H_n(B ; C(F;G))$, where $C(F;G)$ is of course $C(F) \otimes G$, the chains of $F$ with coefficients in $G$.
Then we use three basic and well known properties of such chain complexes:
1) for any chain complexes $F,A $ such that $F$ is free, and morphism $\phi: H_*(F) \to H_*(A)$ of graded groups, there is a morphism $f: F \to A$ of chain complexes such that $H_*(f)=\phi$; in particular, if $F$ is free. there is a morphism $f: F \to H_*(F)$ such that $H_*(f)$ is the identity.
2) if $F$ is a free chain complex and $g: A \to B$ is a morphism of chain complexes such that $H_*(g): H_*(A) \to H_*(B) $ is an isomorphism, then $1\otimes g: F \otimes A \to F \otimes B$ induces an isomorphism of homology;
3) if $A$ is a chain complex there is a free chain complex $L$ such that there is a morphism $a: L \to A$ inducing an isomorphism in homology.
From this we deduce that if $A$ is a chain complex and $F$ is a free chain complex then there is an isomorphism $$\kappa_F:H_*(F;A) \to H_*(F;H_*(A))$$ which can be chosen to be natural with respect to maps of $F$. To get $\kappa_F$ we choose a free chain complex $L$ and a morphism $a: L \to A$ inducing an isomorphism in homology. Then we choose a morphism $b: L \to H_*(L) $ inducing an isomorphism in homology.
This can lead to specific calculations of $\kappa_F$.
This is the dual of arguments for cohomology in this paper Chains as coeficients, (Proc. LMS (3) 14 (1964) 545-65) and examples are given there of non naturality. The original problem as suggested by M.G. Barratt was to get some results on Postnikov invariants of function spaces $X^Y$ by induction on the Postnikov system of $X$.