Minimal Atomic Complexes

We define minimal atomic complexes and irreducible complexes, and we prove that they are the same. The irreducible complexes admit homological characterizations that make them easy to recognize. These concepts apply both to spaces and to spectra. On the spectrum level, our characterizations allow us to show that such familiar spectra as ko, eo2, 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 irreducible and therefore minimal atomic. Up to equivalence, the minimal atomic complexes admit descriptions as CW complexes with restricted attaching maps, called nuclear complexes, and this description can be refined further to nuclear minimal complexes, which are nuclear and have zero differential on their mod p chains. As an illustrative example, we construct BoP as a nuclear complex.