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?



Splitting low-dimensional $p$-local CW complexes for large $p$
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.)


