‎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.


1. Why if $\phi$ induces an isomorhphism of fundamental groups, we can regard $\pi\_n (\phi)$ as a $\mathbb{Z}\pi\_1 (X)$-module?
2. 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 $\pi\_n (\psi)=0$.



Explaining some detail in Wall's paper of CW-complexes
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 $\* = n$.


