Homology, Homotopy and Applications

Volume 3 (2001)

Number 2

Volume of a Workshop at Stanford University

Cores of spaces, spectra, and $E_{\infty}$ ring spectra

Pages: 341 – 354

DOI: http://dx.doi.org/10.4310/HHA.2001.v3.n2.a3

Authors

P. Hu (Department of Mathematics, University of Chicago, Chicago, Ilinois, U.S.A.)

I. Kriz (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)

J. P. May (Department of Mathematics, University of Chicago, Chicago, Ilinois, U.S.A.)

Abstract

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 $BP$.

Keywords

atomic space, Brown-Peterson spectrum, localization, Einfty ring spectrum

2010 Mathematics Subject Classification

Primary 55P15, 55P42, 55P43. Secondary 55S12.

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