# Homotopy groups of mapping cylinder

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:

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.

• 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. Commented Jul 15, 2023 at 2:44
• 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. Commented Jul 15, 2023 at 3:12
• You are right. I've just added the proof of Wall. I'd really appreciate if you could help me about my question. Commented Jul 15, 2023 at 12:55
• 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.‎ Commented Jul 18, 2023 at 15:06
• 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 $\pi_2 (X)$? Commented Jul 18, 2023 at 15:11