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Fix a prime $p$. I have a sketch of a proof that if $X$ is a finite simply-connected CW complex with $\mathrm{dim}(X) < p$ then for some $t\in \mathbb{N}$, the $p$-localization $\Sigma^t X_{(p)}$ is a wedge of Moore spaces.

(Basically, the idea is that all the interesting attaching maps are Whitehead products, hence stably trivial.)

Questions:

  1. Does anyone know a reference for this?
  2. If it's not true, I'd love to know that too!

EDIT: Perhaps this is just a theorem of tame homotopy theory?

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    $\begingroup$ I think this was in Hans-Werner Henn's thesis and his first papers. $\endgroup$ – Neil Strickland Jun 15 '19 at 5:56
  • $\begingroup$ @NeilStrickland I looked at MathSciNet, and I only saw stuff about nearly rational spaces---rational spaces with fundamental group. $\endgroup$ – Jeff Strom Jun 16 '19 at 1:37
  • $\begingroup$ I think you missed writing "$p$-local" in the body of the question. $\endgroup$ – Dan Petersen Jun 16 '19 at 9:45
  • $\begingroup$ Yes, it needs the $p$-local hypothesis $\endgroup$ – Jeff Strom Jun 16 '19 at 15:10
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This result appears in the PhD thesis of Hans-Werner Henn, and in this paper:

@article {MR884630,
    AUTHOR = {Henn, Hans-Werner},
     TITLE = {Classification of {$p$}-local low-dimensional spectra},
   JOURNAL = {J. Pure Appl. Algebra},
  FJOURNAL = {Journal of Pure and Applied Algebra},
    VOLUME = {45},
      YEAR = {1987},
    NUMBER = {1},
     PAGES = {45--71},
      ISSN = {0022-4049},
   MRCLASS = {55P42 (55Q70)},
  MRNUMBER = {884630},
MRREVIEWER = {Frederick Cohen},
       DOI = {10.1016/0022-4049(87)90083-1},
       URL = {https://doi.org/10.1016/0022-4049(87)90083-1},
}

(For some reason it seems that this took several years to be published, and so is not Henn's earliest paper, contrary to my previous comment.)

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