I've read one proof, rather long, in Allen Hatcher's book. There the key is Lemma 4.70, which uses the relative Hurewicz Theorem. But there is another, shorter proof in J.P.May's book "A concise course in algebraic topology", chapter 22.4. My problem with that proof is that I can't find any way that it uses simplicity of the space. We discussed this argument as proof for general CW-complexes on a seminar, and could not find an error.

I can write that part of the proof, which is omitted in the text, here: Let B $\subseteq C\left(\alpha_n\right)$ be the image of $X \times [0,2/3)$, A is the image of $X \times (1/3,1] \cup X_n$. Here $A$ and $B$ are as in the definition of an excisive triad.

Then if $C=A \cap B$, there are homotopy equivalences of pairs $(A,C) \sim \left(M\left(\alpha_n\right),X\right)$ and $\left(B,C\right) \sim \left(CX,X\right)$. Also, if we identify $\pi_k\left(A,C\right) \simeq \pi_k\left(M\left(\alpha_n\right),X\right) \simeq \pi_{k-1}\left(F\left(X \to M\left(\alpha_n \right) \right)\right) \simeq \pi_{k-1}\left(F\left(X \to X_n\right) \right)$. The only nontrivial identification here is by definition of relative homotopy groups.

Moreover, this identification commutes with morphisms to $\pi_{k-1}\left(C\left(\alpha_n\right)\right)$. The first of these is $\pi_{k-1}(\eta) : \pi_{k-1}\left(F\left(X \to X_n\right) \right) \to \pi_k\left(C\left(\alpha_n\right)\right)$ and the homomorphism induced by the morphism of pairs $(A,C) \to \left(C\left(\alpha_n\right),B\right)$ : $\pi_k\left(A,C\right) \to \pi_k\left(C\left(\alpha_n\right),B\right) \simeq \pi_k\left(C\left(\alpha_n\right),\ast \right)$.

This commutativity is evident if we choose the projection as the homotopy equivalence $M(\alpha_n) \to X_n$ and retraction along $I$ as the deformation retraction of $B$ to $\ast$. So, we only need to check that the morphisms induced by $(A,C) \to \left(C\left(\alpha_n\right),B\right)$ are isomorphisms for k up to n+2.

This is true by the Homotopy excision theorem, applied to $A,B$ with $n_{ex}=2, m_{ex}=n+2$. The theorem applies because of homotopy long exact sequences of pairs which say $\pi_l(CX,X) \simeq \pi_{l-1}(X)$, so $(B,C)$ is $1$-connected, which is $n_{ex}-1$-connected, so $n_{ex}=2$. The other exact sequence is $$ \pi_l(X) \to \pi_l(X_n) \to \pi_l(A,C) \to \pi_{l-1}(X) \to \pi_{l-1}(X_n).$$ Here the first arrow is an epimorphism for all $l$ and the last is a monomorphism for $l-1 \leq n$. So, $\pi_l(A,C)=0$ if $l \leq n+1$. This means that $(A,C)$ is $(n+1)$ - connected, so $m_{ex}=n+2$. Thus, the theorem applies.

  • $\begingroup$ The application of homotopy excision in May's proof does not give the claimed range for the map $\eta$ to be an isomorphism. $\endgroup$ – Oscar Randal-Williams Apr 27 '14 at 19:02
  • $\begingroup$ @OscarRandal-Williams I added the version of the proof of the part you mention which we discussed at the seminar, to my post. $\endgroup$ – Alexei Apr 27 '14 at 21:10
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    $\begingroup$ @Alexi: yes, the homotopy excision theorem applies, but the statement of that theorem shows that $(A,C) \to (C(\alpha_n), B)$ is a $(n+2 = m_{ex} + n_{ex}-2)$-equivalence, which on $\pi_{n+2}$ only means it is a surjection: May's proof uses that it is an isomorphism in this degree. $\endgroup$ – Oscar Randal-Williams Apr 27 '14 at 22:10
  • $\begingroup$ @OscarRandal-Williams Thank you, I misunderstood the notion of $n$-equivalence. Is there any way to fix this proof, maybe by using a modified version of the homotopy excision theorem? $\endgroup$ – Alexei Apr 28 '14 at 8:26

Mea culpe! Kate Ponto and I corrected the sloppy ``proof'' in Concise in Chapter 3 of More Concise Algebraic Topology, which gives a pedantically careful treatment of a more general result needed in our treatment of localizations and completions of nilpotent spaces. There we construct Postnikov towers of nilpotent spaces (which are restricted to have homotopy groups in some preassigned collection of abelian groups) rather than just simple spaces. The treatment can no longer claim to be brief, since the cited Chapter is 23 pages, but I hope people may find it instructive. It is based on a wholesale dualization of cellular theory to cocellular theory, in particular showing that two familiar theorems of Whitehead admit line by line dual proofs, and it gives a kind of cocellular functoriality of the construction of Postnikov towers: see Theorem 3.5.4. The role of the fundamental group is pinpointed in Lemma 3.4.2. That gives several characterizations of exactly when, given a based map $X \to Y$ between connected spaces with fiber a $K(A,n)$, there is a "$k$-invariant'' $k\colon Y \to K(A,n+1)$ whose fiber is equivalent to $X$ over $Y$. One characterization is that $\pi_1(Y)$ must act trivially $A$.

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