ar X iv : m at h / 02 06 06 7 v 1 [ m at h . A T ] 7 J un 2 00 2 MINIMAL ATOMIC COMPLEXES

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 M U considered as a commutative S-algebra, although it is a core of M U 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 homo-morphism 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 Σ ∞ CP ∞ and Σ ∞ HP ∞ at any prime p are minimal atomic.