Let $f:K\to X$ be a map, with mapping cyliner $M=X\cup_{f}(K\times \{ 1\})$. We define $\pi_n (f)$ as $\pi_n (M,K\times \{ 1\})$. An element of $\pi_n (f)$ is represented by a pair of maps $\alpha :S^{n-1}\to K$ and $\beta :D^n \to X$ with $\beta |_{S^{n-1}}=f\circ \alpha$.
C.T.C. Wall in his paper "Finiteness conditions for CW-complexes" says that if $f:K\to X$ induces an isomorphism of fundamental groups (where $K$ is a finite bouquet of 1-spheres) and $\pi_2 (f)$ is a free $\mathbb{Z}\pi_1 (X)$-module, then we can attach a finite set of 2-cells to $K$, necessarily with trivial attaching maps. Is there someone who explains me why this happens? How if $\pi_2 (f)$ is a free $\mathbb{Z}\pi_1 (X)$-module, we can attach 2-cells to $K$, necessarily with trivial attaching maps?
The proof of Wall:
Best Answer
In general, if $\lbrace \alpha_i\rbrace $ is a collection of elements of $\pi_n(f)$ and if $(\beta_i,\gamma_i)$ represents $g_i$, then $\beta_i:S^{n-1}\to K$ can be used as an attaching map to attach an $n$-cell $e_i$ to $K$. Let $K'=K\cup \lbrace e_i\rbrace_i$ be the resulting space. Then $\gamma_i$ can then be used to extend the given map $f:K\to X$ over the cell $e_i$, and we obtain a factorization $K\to K'\to X$ of $f$.
The element of $\pi_{n-1}(K)$ represented by $\beta_i$ is of course in the kernel of the map $f_\ast:\pi_{n-1}(K)\to \pi_{n-1}(X)$.
In the case you are reading about, $n$ is $2$ and $f_\ast:\pi_1(K)\to \pi_1(X)$ is assumed to be an isomorphism, so an element of the kernel must be trivial. In other words, the attaching map of each of these $2$-cells $e_i$ is trivial, so that $K'$ is a wedge (bouquet) of $K$ and some $2$-spheres.
If the group $\pi_2(f)$ is free as a module for $\mathbb Z\pi_1(X)$, and if the elements $g_i$ are chosen to form a basis, then the resulting map $\pi_2(K',K)\to \pi_2(f)$ will be an isomorphism, and presumably along with Wall's other assumptions this gives the conclusion.