Proving the mod $\mathfrak{C}$ Whitehead theorem from the mod $\mathfrak{C}$ Hurewicz theorem

Question

I am reading Lecture Notes in Algebraic Topology by Davis and Kirk and in the book the mod $\mathfrak{C}$ Whitehead Theorem (where $\mathfrak{C}$ is a Serre class of abelian groups) is stated as:

mod $\mathfrak{C}$ Whitehead Theorem: Let $f : A \to X$ where $A$ and $X$ are simply connetced and suppose that $f : \pi_2(A) \to \pi_2(X)$ is an epimorphism. Let $\mathfrak{C}$ be a Serre class satisfying Axioms 1, 2B and 3 (stated at the very end of this question). Then the following two statements are equivalent:

1. $f_* : \pi_i(A) \to \pi_i(X)$ is a $\mathfrak{C}$-isomorphism for $i < n$ and a $\mathfrak{C}$-epimorphism for $i = n$.
2. $f_*: H_i(A) \to H_i(X)$ is a $\mathfrak{C}$-isomorphism for $i < n$ and a $\mathfrak{C}$-epimorphism for $i = n$.

Now in the book the authors say the mod $\mathfrak{C}$ Hurewicz theorem, stated below, implies the mod $\mathfrak{C}$ Whitehead Theorem.

mod $\mathfrak{C}$ relative Hurewicz Theorem: Suppose $A \subseteq X$, $A$ and $X$ are simply-connected, and $\pi_2(X, A) = 0$. Let $\mathfrak{C}$ be a Serre class satisfying Axioms 1, 2B and 3. Then

1. If $\pi_i(X, A) \in \mathfrak{C}$ for all $i < n$, then $H_i(X, A) \in \mathfrak{C}$ for all $i< n$ and the Hurewicz map $\pi_n(X, A) \to H_n(X, A)$ is a $\mathfrak{C}$-isomorphism.

2. If $H_i(X, A) \in \mathfrak{C}$ for all $i < n$, then $\pi_i(X, A) \in \mathfrak{C}$ for all $i< n$ and the Hurewicz map $\pi_n(X, A) \to H_n(X, A)$ is a $\mathfrak{C}$-isomorphism.

I, however, ran into some trouble proving the mod $\mathfrak{C}$ Whitehead Theorem from the mod $\mathfrak{C}$ Hurewicz theorem.

My guess was that the proof would follow from the mod $\mathfrak{C}$ Hurewicz theorem by induction on $n$. In the case $n=1$ it trivially holds since both $A$ and $X$ are simply connected. Suppose now that by induction that for $i< n-1$ the two statements in the mod $\mathfrak{C}$ Whitehead Theorem are equivalent. I would need to show that $f_* : H_{n-1}(A) \to H_{n-1}(X)$ is a $\mathfrak{C}$-isomorphism and $f_* : H_{n}(A) \to H_{n}(X)$ is a $\mathfrak{C}$-epimorphism.

Now to do this I assume I would need to make use of the mod $\mathfrak{C}$ relative Hurewicz Theorem, noting that since $f : \pi_2(A) \to \pi_2(X)$ is an epimorphism we have $\pi_2(X, A) = 0$ (look at the long exact sequence of homotopy groups). The problem is that I don't see how I can use the mod $\mathfrak{C}$ relative Hurewicz Theorem because I don't have any knowledge if $\pi_i(X, A) \in \mathfrak{C}$ for all $i < n$ or if $H_i(X, A) \in \mathfrak{C}$.

Furthermore, I would presume that we'd need to make use of either the long exact homotopy sequence of groups or the long exact sequence of homology groups to apply our inductive hypothesis, but I don't see how we could use either of these in a meaningful way since we are dealing with the induced map $f_*$ in either homotopy or homology. My guess is that there will exist a commutative diagram involving both these long exact sequences and the induced map $f_*$ on $H_{n-1}$ and $\pi_{n-1}$, but I don't see what that is yet.

Could someone give me a hint (and not a full solution) as to how I would go about proving the mod $\mathfrak{C}$ Whitehead Theorem from the mod $\mathfrak{C}$ Hurewicz theorem?

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Definition: A Serre class of abelian groups is a non-empty collection $\mathfrak{C}$ of abelian groups satisfying the following mandatory axiom:

1. If $0 \to A \to B \to C \to 0$ is a short exact sequence, then $B \in >\mathfrak{C}$ if and only if both $A$ and $C \in \mathfrak{C}$

as well as the following additional axioms that can optionally be met:

• (2A) If $A, B \in \mathfrak{C}$, then $A \otimes B \in \mathfrak{C}$ and $\operatorname{Tor}(A, B) \in \mathfrak{C}$.
• (2B) If $A \in \mathfrak{C}$, then $A \otimes B \in \mathfrak{C}$ for any abelian group $B$.
• (3) If $A \in \mathfrak{C}$, then $H_n(A; \mathbb{Z}) = H_n(K(A, 1); \mathbb{Z}) \in \mathfrak{C}$ for every $n >0$.

Answer 1

I've written up a full proof for this, but it is too long to post here, so instead I will give a few hints on how to prove this.

The way in which I proved this was essentially via proving $\text{mod } \mathfrak{C}$ versions of the steps of the proof of the standard Whitehead Theorem as proved in Spanier's Algebraic Topology.

Basically what one does is consider the mapping cylinder $M_f$ of $f$.

This then reduces the problem to showing that the induced inclusion map of $i : (X, x_0) \hookrightarrow (M_f, x_0)$ which is $i_∗ : H_q(X, x_0) \to H_q(M_f, x_0)$ is a $\mathfrak{C}$-isomorphism for $i < n$ and a $\mathfrak{C}$-epimorphism for $i = n$ if and only if $i_\# : \pi_q(X, x_0) \to \pi_q(M_f, x_0)$ is a $\mathfrak{C}$-epimorphism for $i < n$ and a $\mathfrak{C}$-epimorphism for $i = n$.

Now from the long exact homotopy sequence of the triple $(M_f, X, x_0)$ one can prove that $i_\#$ is a $\mathfrak{C}$-isomorphism for $q < n$ and a $\mathfrak{C}$-epimorphism for $q = n$ if and only if $\pi_q(M_f, X, x_0) \in \mathfrak{C}$ for $q ≤ n$. Similarly from the long exact homology sequence of the triple, $(M_f, X, x_0)$ one needs to prove that $i_∗$ is a $\mathfrak{C}$-isomorphism for $q < n$ and a $\mathfrak{C}$-epimorphism for $q = n$ if and only if $H_q(M_f, X, x_0) \in \mathfrak{C}$ for $q \leq n$.

Then at this point we can apply the $\text{mod } \mathfrak{C}$ relative Hurewicz theorem to finish up the proof.