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Recall the homotopy excision theorem, as stated in Hatcher (Theorem 4.23): Let $X$ be a CW complex decomposed as the union of subcomplexes $A$ and $B$ with nonempty connected intersection $C = A \cap B$. If $(A,C)$ is $m$-connected and $(B,C)$ is $n$-connected for some $n,m \geq 0$, then the map $\pi_i(A,C) \rightarrow \pi_i(X,B)$ is an isomorphism for $i<m+n$ and a surjection for $i=m+n$.

The proof of this theorem in Hatcher is fairly complicated (but elementary). The other sources I've looked at (e.g. May's book) give similar proofs. Are there other proofs? As an example, the first application Hatcher gives is to prove the Freudenthal suspension theorem, and my favorite proof of this uses the Serre spectral sequence. More generally, I often find proofs of basic homotopy theoretic results clearer if they use the Serre spectral sequence or other such things rather than being overly "elementary". But I'd also be interested in alternate elementary proofs.

The only other proof I know of is Rezk's proof using homotopy type theory, but I can't make heads or tails of it.

There is an earlier MO question here that is sort of along the same lines, but it includes desiderata like the proof being "ideologically profound" that I certainly am not looking for (in fact, I don't even know what this means, to be honest).

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    $\begingroup$ Have you looked in Tom Dieck's book? I don't have it with me, but if I was looking for an alternative proof, that's where I would start, as he has quite a few original arguments in his homotopy theory text. All the proofs I've seen have been of the "general position / transversality" nature, as in Hatcher's text. $\endgroup$
    – Ryan Budney
    Commented Dec 17, 2022 at 5:12
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    $\begingroup$ In the linked MO question, there is a link to Rezk's paper, and there is a generalization in this paper. $\endgroup$
    – Z. M
    Commented Dec 17, 2022 at 8:22
  • $\begingroup$ @RyanBudney tom Dieck quotes a proof by Puppe. It is elementary and a bit complicated, but geometrically intuitive. It does not rely on general position arguments, but on a clever choice of subsets of cubes. $\endgroup$
    – Sebastian Goette
    Commented Dec 20, 2022 at 9:46

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The proof of Theorem 9.3.5 (especially the part on page 486) in Spanier's "Algebraic Topology" may be more to your liking. It presumes you have already established the relative Hurewicz theorem, e.g. by Serre spectral sequence methods.

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  • $\begingroup$ Thanks, this is exactly the kind of proof I was looking for! It's too bad that it requires everything to be simply-connected. Do you know if it is possible to avoid this, e.g. by using local coefficients or something? $\endgroup$
    – Brenda
    Commented Dec 18, 2022 at 2:50
  • $\begingroup$ @Brenda Spanier's theorem only assumes that the relevant inclusions induce isomorphisms of \pi_1, not that all spaces are 1-connected. The first part of his proof uses universal covering spaces (i.e., local coefficients) to reduce to the 1-connected situation. In the case where you only have a surjection on \pi_1, you can look at the 1952 paper of Blakers-Massey that he cites in footnote 1, but their argument uses general position and such, which I think you wanted to avoid. $\endgroup$
    – John Rognes
    Commented Dec 18, 2022 at 10:48

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