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# 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

#### 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.