### 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 <pohu@math.uchicago.edu>

I. Kriz <ikriz@math.lsa.umich.edu>

J.P. May <may@math.uchicago.edu>