Cores of spaces, spectra, and E_infty ring spectra, by P. Hu, I. Kriz, and J.P. May

In a paper that has attracted little notice, Priddy showed that the Brown-Peterson spectrum at a prime p can be constructed from the p-local sphere spectrum S by successively killing its odd dimensional homotopy groups. This seems to be an isolated curiosity, but it is not. For any space or spectrum Y that is p-local and (n_0-1)-connected and has pi_{n_0}(Y) cyclic, there is a p-local, (n_0-1)-connected ``nuclear'' CW complex or CW spectrum X and a map f : X to Y that induces an isomorphism on pi_{n_0} and a monomorphism on all homotopy groups. Nuclear complexes are atomic: a self-map that induces an isomorphism on pi_{n_0} must be an equivalence. The construction of X from Y is neither functorial nor even unique up to equivalence, but it is there. Applied to the localization of MU at p, the construction yields BP. The paper has appeared in: Homology, homotopy, and applications 3(2001), 341-354.

P. Hu <>
I. Kriz <>
J.P. May <>