Homology, Homotopy and Applications

Volume 19 (2017)

Number 1

Bousfield localization of ghost maps

Pages: 371 – 389

DOI: http://dx.doi.org/10.4310/HHA.2017.v19.n1.a18

Authors

Mark Hovey (Department of Mathematics, Wesleyan University, Middletown, Connecticut, U.S.A.)

Keir Lockridge (Department of Mathematics, Gettysburg College, Gettysburg, Pennsylvania, U.S.A.)

Abstract

In homotopy theory, a ghost map is a map that induces the zero map on all stable homotopy groups. Bousfield localization is the homotopy-theoretic analogue of localization for rings and modules. In this paper, we consider the Bousfield localization of ghost maps. In particular, we pose the question: for which localization functors is it the case that the localization of a ghost is always a ghost? On the category of $p$-local spectra, we conjecture that the only localizations satisfying this property are the zero functor, the identity functor, and localization with respect to the rational Eilenberg–Mac Lane spectrum $H\mathbb{Q}$.We significantly narrow the field of possible counter-examples (one interesting outstanding possibility is the Brown–Comenetz dual of the sphere) and we consider a weaker version of the question at hand.

Keywords

Bousfield localization, ghost map, ghost-preserving localization

2010 Mathematics Subject Classification

55P42, 55P60

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