‎For a given map $\phi‎ :‎X\longrightarrow Y$‎, ‎the mapping cylinder of $\phi$ is defined by $M\_{\phi}:=Y\bigcup\_{\phi} (X \times \{ 1\})$‎. ‎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 $X$ be a CW-complex‎. ‎Then conditions $\mathcal{F}\_i$ and $\mathcal{D}\_i$ on $X$ are defined inductively as follows‎:


‎$\mathcal{F}\_1$‎: ‎the group $\pi\_1 (X)$ is finitely generated‎.


‎$\mathcal{F}\_2$‎: ‎the group $\pi\_1 (X)$ is finitely presented‎, ‎and for any 1-dimensional finite CW-complex $K$ and any map $\phi‎ :‎K\longrightarrow X$ inducing an isomorphism of fundamental groups‎, ‎$\pi\_2 (\phi )$ is a finitely generated module over $\mathbb{Z}\pi\_1 (X)$‎.


‎$\mathcal{F}\_n$‎: ‎the condition $\mathcal{F}\_{n-1}$ holds‎, ‎and for any $(n-1)$-dimensional finite CW-complex $K$ and any $(n-1)$-connected map $\phi‎ :‎K\longrightarrow X$‎, ‎$\pi\_n (\phi )$ is a finitely generated $\mathbb{Z}\pi\_1 (X)$-module‎.


‎$\mathcal{D}\_n$‎: ‎$H\_i(\tilde{X})=0$ for $i>n$‎, ‎and $H^{n+1}(X;\mathcal{B})=0$ for all coefficient bundles $\mathcal{B}$.


Proposition 3.3 in th paper "Finiteness conditions for CW-complexes" of C.T.C. Wall states that : If CW-complex $X$ satisfies the conditions $\mathcal{D}\_2$ and $\mathcal{F}\_2$, and $\pi\_1 (X)$ is free, then $X$ has the homotopy of a finite bouquet of 1-spheres and 2-spheres. 


Clearly, there exist a finite bouquet of circles $K$ and a map $\phi :K\longrightarrow X$ inducing an isomorphism of fundamental groups. 


Wall proved that $\pi\_2 (\phi)$ is a free $\mathbb{Z}\pi\_1 (X)$-module. So we can attach a finite 2-cells to $K$, necessarily with trivial attaching map, to make a new complex $L$. Since $\pi\_2 (\phi)$ is free, we can extend the map $\phi$ to a $2$-connected map $\psi :L\longrightarrow X$. 


My question is that:


**Why $\psi$ is a homotopy equivalence? Equivalently, why does $\psi$ is i-connected for all $i\geq 3$?**



A question about Wall's construction for CW-complexes
The given conditions imply that the map $\psi:L\rightarrow X$ induces an isomorphism of fundamental groups, and so it lifts to a map $\tilde\psi: \widetilde L\rightarrow \widetilde X$. The conditions also imply that $\tilde\psi$ is an isomorphism on homology. Since these spaces are simply-connected, their homology groups being isomorphic implies that their homotopy groups are isomorphic. But for $i>1$, $\pi\_i(X)=\pi\_i(\widetilde X)$ and similarly for $L$. Hence $\psi$ is $i$-connected for all $i$. 


