Title: Cores of spaces, spectra, and $E_{\infty}$ ring spectra
Authors: P. Hu, I. Kriz, and J.P. May
Classification: 55P15, 55P42, 55P43, 55S12
Address: Dept. Math., University of Chicago, Chicago, IL 60637, USA.
E-mail: pohu@math.uchicago.edu
Address: Dept. Math., University of Michigan, Ann Arbor, MI 48109-1107, USA
E-mail: ikriz@math.lsa.umich.edu
Address: Dept. Math., University of Chicago, Chicago, IL 60637, USA.
E-mail: may@math.uchicago.edu

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.
{Appeared: Homology, homotopy, and applications 3(2001), 341--354}