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In homotopy theory, phantom maps are continuous maps of CW-complexes for which the restriction of to any finite subcomplex is inessential (i.e., nullhomotopic). J. Frank Adams and Grant Walker (1964) produced the first known nontrivial example of such a map with finite-dimensional (answering a question of Paul Olum). Shortly thereafter, the terminology of "phantom map" was coined by Brayton Gray (1966), who constructed a stably essential phantom map from infinite-dimensional complex projective space to .[1] The subject was analysed in the thesis of Gray, much of which was elaborated and later published in (Gray & McGibbon 1993). Similar constructions are defined for maps of spectra.[2]
Let be a regular cardinal. A morphism in the homotopy category of spectra is called an -phantom map if, for any spectrum s with fewer than cells, any composite vanishes.[3]