# Splitting low-dimensional $p$-local CW complexes for large $p$

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?

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

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.)