Groups of homotopy classes of phantom maps

Kihara, Hiroshi

doi:10.2140/agt.2018.18.583

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Groups of homotopy classes of phantom maps

Hiroshi Kihara

Algebraic & Geometric Topology 18 (2018) 583–612

DOI: 10.2140/agt.2018.18.583

Abstract

We introduce a new approach to phantom maps which largely extends the rational-ization-completion approach developed by Meier and Zabrodsky. Our approach enables us to deal with the set $Ph (X, Y)$ of homotopy classes of phantom maps and the subset $SPh (X, Y)$ of homotopy classes of special phantom maps simultaneously. We give a sufficient condition for $Ph (X, Y)$ and $SPh (X, Y)$ to have natural group structures, which is much weaker than the conditions obtained by Meier and McGibbon. Previous calculations of $Ph (X, Y)$ have generally assumed that $[X, Ω Ŷ]$ is trivial, in which case generalizations of Miller’s theorem are directly applicable, and calculations of $SPh (X, Y)$ have rarely been reported. Here, we calculate not only $Ph (X, Y)$ but also $SPh (X, Y)$ in many important cases of nontrivial $[X, Ω Ŷ]$ .

Keywords

phantom maps, special phantom maps, group structure

Mathematical Subject Classification 2010

Primary: 55Q05

Secondary: 55P60

References

Publication

Received: 24 April 2017

Revised: 16 June 2017

Accepted: 5 July 2017

Published: 10 January 2018

Authors

Hiroshi Kihara
Center for Mathematical Sciences University of Aizu Aizuwakamatsu Fukushima Japan

Volume 18, issue 1 (2018)