Title: Minimal atomic complexes
Authors: A.J. Baker and J.P. May
Classification: 55P15 55P42 (55P60)
Address: Math. Dept., University of Glasgow, Glasgow G12 8QW, Scotland.
E-mail: a.baker@maths.gla.ac.u
Address: Dept. Math., University of Chicago, Chicago, IL 60637, USA.
E-mail: may@math.uchicago.edu


Hu, Kriz and May recently reexamined ideas implicit in Priddy's elegant
homotopy theoretic construction of the Brown-Peterson spectrum at a prime p.
They discussed May's notions of nuclear complexes and of cores of spaces, 
spectra, and commutative S-algebras. Their most striking conclusions, due to 
Hu and Kriz, were negative: cores are not unique up to equivalence, and 
BP is not a core of MU considered as a commutative S-algebra, although it 
is a core of MU considered as a p-local spectrum. We investigate these ideas 
further, obtaining much more positive conclusions. We show that nuclear complexes 
have several non-obviously equivalent characterizations. Up to equivalence, they
are precisely the irreducible complexes, the minimal atomic complexes, and the 
Hurewicz complexes with trivial mod p Hurewicz homomorphism above the Hurewicz 
dimension, which we call complexes with no mod p detectable homotopy. Unlike the 
notion of a nuclear complex, these other notions are all invariant under equivalence.
This simple and conceptual criterion for a complex to be minimal atomic allows us to 
prove that many familiar spectra, such as ko, $eo_2$, and BoP at the prime 2,  all 
$BP<n>$ at any prime p, and the indecomposable wedge summands of the suspension 
spectra of $CP^\infty$ and $HP^\infty$ at any prime p are minimal atomic.