In this section we will prove the Whitehead and Hurewicz theorems in a rational context. Serre proved these results in \cite{serre}. In his paper he considered homology groups `modulo a class of abelian groups'. In our case of rational homotopy theory, this class will be the class of torsion groups.
Serre gave weaker axioms for his classes and proves some of the following lemmas only using these weaker axioms. However the classes we are interested in do satisfy the above (stronger) requirements. One should think of a Serre class as a class of groups we want to \emph{ignore}. We will be interested in the first two of the following examples.
\item The class $\C$ of all uniquely divisible groups. Note that these groups can be given a unique $\Q$-vector space structure (and conversely every $\Q$-vector space is uniquely divisible).
In the following arguments we will consider fibrations and need to compute homology thereof. Unfortunately there is no long exact sequence for homology of a fibration, however the following lemma expresses something similar. It is usually proven with spectral sequences, \cite[Ch. 2 Thm 1]{serre}. However in \cite{kreck} we find a more elementary proof using cellular homology.
We will first replace the fibration by a fiber bundle. This is done by going to simplicial sets and replace the induced map by a minimal fibration \cite{joyal}. The fibration $p$ induces a fibration $S(E)\tot{S(p)} S(B)$, which can be factored as $S(E)\we M \fib S(B)$, where the map $M \fib S(B)$ is minimal (and hence a fiber bundle). By realizing we obtain the following diagram:
\begin{displaymath}
\xymatrix{
{|M|}\arfib[d]&\arwe[l]{|S(E)|}\arwe[r]\arfib[d]& E \arfib[d]\\
{|S(B)|}&\ar[l]^{\id}{|S(B)|}\arwe[r]& B
}
\end{displaymath}
The fibers of all fibrations are weakly equivalent by the long exact sequence, so the assumptions of the lemma also hold for the fiber bundle. To prove the lemma, it is enough to do so for the fiber bundle $|M| \fib |S(B)|$.
So we can assume $E$ and $B$ to be a CW complexes and $E \fib B$ to be a fiber bundle. We will do induction on the skeleton $B^k$. By connectedness we can assume $B^0=\{ b_0\}$. Restrict $E$ to $B^k$ and note $E^0= F$. Now the base case is clear: $H_i(E^0, F)\to H_i(B^0, b_0)$ is a $\C$-iso.
The morphism in the middle is a $\C$-iso by induction. We will prove that the left morphism is a $\C$-iso which implies by the five lemma that the right morphism is one as well.
As we are working with relative homology $H_{i+1}(E^{k+1}, E^k)$, we only have to consider the interiors of the $k+1$-cells (by excision). Each interior of a $k+1$-cell is a product, as $p$ was a fiber bundle. So we note that we have an isomorphism:
Note that this is the graded tensor product, and that the term $H_{i+1}(B^{k+1}, B^k)\tensor H_0(F)= H_{i+1}(B^{k+1}, B^k)$ and that this identification is compatible with the induced map $p_\ast : H_{i+1}(E^{k+1}, E^k)\to H_{i+1}(B^{k+1}, B^k)$ (hence the map is surjective). To prove that the map is a $\C$-iso, we need to prove that the kernel is in $\C$. The kernel is the sum of the following terms, with $1\leq l \leq i+1$:
Now we can use the assumption that $H_l(F)\in\C$ for $1\leq l < n$ and that for $B \in\C$ we have $A \tensor B \in\C$ for all $A$ (by \LemmaRef{Serre-properties}). This concludes that the kernel $H_{i+1-l}(B^{k+1}, B^k)\tensor H_l(F)$ is indeed in $\C$. And hence the induced map is a $\C$-iso for all
If $\pi_i(X)\in C$ for all $i<n$, then $H_i(X)\in C$ for all $i<n$ and the Hurewicz map $h: \pi_i(X)\to H_i(X)$ is a $\C$-isomorphism for all $i \leq n$.
We will prove the lemma by induction on $n$. Note that the base case ($n =1$) follows from the $1$-connectedness. For the induction step assume that $H_i(X)\in\C$ for all $i<n-1$ and that $h_{n-1}: \pi_{n-1}(X)\to H_{n-1}(X)$ is a $\C$-iso. Now given is that $\pi_{n-1}(X)\in\C$ and hence $H_{n-1}(X)\in\C$. \todo{kromme zin}
It remains to show that $h_n$ is a $\C$-iso. Use the Whitehead tower from \LemmaRef{whitehead-tower} to obtain $\cdots\fib X(3)\fib X(2)= X$. Note that each $X(j)$ is also $1$-connected and that $X(2)= X(1)= X$.
Note that $X(j+1)\fib X(j)$ is a fibration with $F = K(\pi_j(X), j-1)$ as its fiber. So by \LemmaRef{homology-em-space} we know $H_i(F)\in\C$ for all $i$. Apply \LemmaRef{kreck} to obtain a $\C$-iso $H_i(X(j+1))\to H_i(X(j))$ for all $j < n$ and all $i > 0$. This proves the claim.
Considering this claim for all $j < n$ gives a chain of $\C$-isos $H_i(X(n))\to H_i(X(n-1))\to\cdot\to H_i(X(2))= H_i(X)$ for all $i \leq n$. Consider the following diagram:
where the map on the top is an isomorphism by the classical Hurewicz theorem (and $X(n)$ is $(n-1)$-connected), the map on the left is an isomorphism by the Whitehead tower and the map on the right is a $\C$-iso by the claim.
If $\pi_i(X, A)\in\C$ for all $i<n$, then $H_i(X, A)\in\C$ for all $i<n$ and the Hurewicz map $h: \pi_i(X, A)\to H_i(X, A)$ is a $\C$-isomorphism for all $i \leq n$.
Consider the mapping cylinder $B_f$ of $f$, i.e. factor the map $f$ as a cofibration followed by a trivial fibration $f: A \cof B_f \fib B$. The inclusion $A \subset B_f$ gives a long exact sequence of homotopy groups and homology groups: