For a given map $\phi :X\longrightarrow Y$, the mapping cylinder of $\phi$ is defined by $M_{\phi}:=Y\cup_{\phi} (X \times \{ 1\})$. Denote $\pi_n (M_{\phi},X \times \{ 1\} )$ by $\pi_n (\phi)$. The map $\phi$ is called $n$-connected if $X$ and $Y$ are connected and $\pi_i (\phi)=0$ for $1\leq i\leq n$.
Let $K$ be a CW-complex, $X$ have the homotopy type of one, and suppose that $\phi :K\to X$ to be $(n-1)$-connected. If $n\geq 3$, $\phi$ induces an isomorhphism of fundamental groups, so we can regard $\pi_n (\phi)$ as a $\mathbb{Z}\pi_1 (X)$-module. Select $\mathbb{Z}\pi_1 (X)$-generators $\{ \alpha_i \}$ for $\pi_n (\phi)$. Then the $\partial \alpha_i$ belong to $\pi_{n-1} (K)$: use them to attach $n$-cells to $K$. Now use the $\alpha_i$ themeselves to extend $\phi$ over these cells (recall that an element of $\pi_n (\phi )$ is represented by a pair of maps $\alpha :S^{n-1}\longrightarrow X$ and $\beta :D^n \longrightarrow Y$ with $\beta\rvert_{S^{n-1}}=\phi \circ \alpha$). If the resulting space is $L$ and map $\psi :L\to X$, then since the map $\alpha$ in the exact sequence $$\pi_n (L,K)\overset{\alpha}{\to}\pi_n (\phi)\to \pi_n (\psi)\to \pi_{n-1}(L,K)=0$$ is onto (for, if $n\geq 3$, $\pi_n (L,K)\cong H_n (\tilde{L},\tilde{K})=C_n (\tilde{L})$, and the $\alpha_i$ were chosen as generators of $\pi_n (\phi)$), $\pi_n (\psi)$ vanishes, and so $\psi$ is $n$-connected.
The above argument belongs to C.T.C. Wall's paper “Finite conditions for CW-complexes" page 59. There are two things that I don't understand in the argument.
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Why if $\phi$ induces an isomorhphism of fundamental groups, we can regard $\pi_n (\phi)$ as a $\mathbb{Z}\pi_1 (X)$-module?
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I really appreciate if someone could explain me why such an exact sequence exists, $\pi_{n-1}(L,K)=0$, $\pi_n (L,K)\cong H_n (\tilde{L},\tilde{K})=C_n (\tilde{L})$, and $\alpha$ is onto in more detail. Clearly, $\pi_{n-1}(L,K)=0$ and $\alpha$ is onto, then by the exact sequence we have $\pi_n (\psi)=0$.
Best Answer
As to (1):
If we choose a basepoint in $K$, then $\phi$ can be viewed as a map of based spaces. Let $F$ be the homotopy fiber of $\phi$. Then there is a well-defined action $\Omega K \times F \to F$ which induces a $\Bbb Z[\pi_1(X)]$-module structure on $H_*(F)$.
As $F$ is $1$-connected (by the assumptions), the Hurewicz theorem says that $$\pi_n(\phi) = \pi_{n-1}(F) \cong H_{n-1}(F).$$ So $\pi_n(\phi)$ is indeed a $\Bbb Z[\pi_1(X)]$-module.
As for (2):
Let $F'$ be the homotopy fiber of the map $\psi$, and let $F''$ be the homotopy fiber of the map $K\to L$. Then one has a commuting diagram: $\require{AMScd}$ $$ \begin{CD} F @>>> K @>\phi >> X \\ @VVV @VVV @VVV \\ F' @>>> L @>>\psi > X \end{CD} $$ where the rows are homotopy fiber sequences. Taking homotopy groups vertically gives a long exact sequence $$ \cdots\to \pi_*(F'') \to \pi_*(F) \to \pi_*(F') \to \pi_{*-1}(F'') \to \cdots $$ i.e., $$ \cdots\to \pi_*(L,K) \to \pi_*(\phi)) \to \pi_*(\psi) \to \pi_{*-1}(L,K) \to \cdots $$ The given connectivity assumptions then say that this long exact sequence terminates on the right when $* = n$.