‎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?

‎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 $i\geq 3$?

‎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 $\phi$ is a homotopy equivalence?

‎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?