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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: enter image description here

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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.

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  • $\begingroup$ Thank you so much for your explanation. Could you please explain me what is the use of free $\mathbb{Z}\pi_1 (X)$-module $\pi_2 (f)$ here? You are using only this hypothesis that $f:K\to X$ induces an isomorphism of fundamental groups. $\endgroup$
    – Mahtab
    Commented Jul 15, 2023 at 2:44
  • $\begingroup$ What you are reading is part of an argument, and I do not have it in front of me. The idea of using a relative homotopy class to attach a cell and extend a map is often a useful one, but I do not know exactly what Wall is using it for here. $\endgroup$ Commented Jul 15, 2023 at 3:12
  • $\begingroup$ You are right. I've just added the proof of Wall. I'd really appreciate if you could help me about my question. $\endgroup$
    – Mahtab
    Commented Jul 15, 2023 at 12:55
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    $\begingroup$ Dear @TomGoodwillie , I really appreciate your help. I got it now. As you told, there is an argument in the Wall's paper which is briefly as following: Assume that ‎‎$‎‎n\geq 3$, ‎‎$‎‎‎\phi ‎:K^{n-1}\to X‎$ ‎is ‎‎$‎‎(n-1)$-connected, $‎‎\pi_n (‎\phi‎)$ ‎is ‎free, and ‎‎$‎‎X$ ‎satisfie‎s ‎$‎‎D_n$‎. ‎Perform ‎the ‎above ‎construction ‎with ‎the ‎‎$‎‎‎\alpha‎_i$ ‎as ‎free ‎generators ‎of ‎‎$‎‎\pi_n (\phi)$. ‎Consider ‎the ‎resulting ‎‎$‎‎L$ ‎as a‎ ‎subcomple‎x of ‎$‎‎X$. ‎By ‎the ‎argument ‎presented ‎in‎ Wall p. 63, the inclusion of ‎$‎‎L$ ‎in ‎‎$‎‎X$ ‎is a‎ ‎homotopy ‎equivalence.‎ $\endgroup$
    – Mahtab
    Commented Jul 18, 2023 at 15:06
  • $\begingroup$ Sorry I have another related question: Whats is relationship between the number of free genertors of $\pi_2 (f)$ (which we attach to $K$ ) and the number of free genertors of $\pi_2 (X)$? I mean how do you know the number of 2-cell attaching to $K$? is it related to $\pi_2 (X)$? $\endgroup$
    – Mahtab
    Commented Jul 18, 2023 at 15:11

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