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I am curious: is there a canonical way to decompose a finite simplicial complex into a wedge sum up to homotopy equivalence? More formally:

Let $X$ be a finite simplicial complex. Is $X$ homotopy equivalent to a wedge sum $X_1 \vee \cdots \vee X_n$ such that

  1. no $X_i$ is non-trivially a wedge sum, i.e. no $X_i$ is homotopically equivalent two a wedge sum between two non-contractible spaces;
  2. if $X$ is homotopically equivalent to a wedge sum $Y_1 \vee \cdots \vee Y_m$ such that no $Y_i$ is non-trivially a wedge sum, then $m=n$ and there is a permutation $\sigma \in \mathrm{Sym}(n)$ such that each $X_i$ is homotopically equivalent to $Y_{\sigma(i)}$?

This sounds like a natural question, but I do not find anything on the subjet in the literature. Does anyone know a relevant reference? Or, in the opposite direction, some exotic homotopy equivalence between two wedge sums?

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    $\begingroup$ $2.$ fails miserably. I gave a classical example in this answer. It's a nice exercise to show that $X,Y$ there are not (even stably) homotopy equivalent. $\endgroup$
    – Tyrone
    Commented Sep 10 at 12:15
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    $\begingroup$ Perhaps it's worth mentioning that some positive results do hold that are relevant to the low-dimensional or group-theoretic context. If $X$ is aspherical then such a decomposition does exist: this is the Grushko decomposition (en.wikipedia.org/wiki/…). Also, there is a positive result if you are willing to strengthen homotopy equivalence to Nielsen equivalence, i.e. homotopy equivalence of 1-skeleta rel. the higher cells. See, for instance, Lemma 4.6 here: arxiv.org/abs/2210.09853 . $\endgroup$
    – HJRW
    Commented Sep 10 at 14:11

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Cancellation does not hold for wedge sums of finite complexes. See for example Hilton and Roitberg, "On principal $S^3$ bundles over spheres". This refers to a paper of Hilton, "On the Grothendieck group of compact polyhedra".

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