Volume 18, issue 1 (2018)

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Groups of homotopy classes of phantom maps

Hiroshi Kihara

Algebraic & Geometric Topology 18 (2018) 583–612

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,ΩŶ] 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,ΩŶ].

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phantom maps, special phantom maps, group structure
Mathematical Subject Classification 2010
Primary: 55Q05
Secondary: 55P60
Received: 24 April 2017
Revised: 16 June 2017
Accepted: 5 July 2017
Published: 10 January 2018
Hiroshi Kihara
Center for Mathematical Sciences
University of Aizu