About: Phantom map

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 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, who constructed a stably essential phantom map from infinite-dimensional complex projective space to . The subject was analysed in the thesis of Gray, much of which was elaborated and later published in (Gray & McGibbon ). Similar constructions are defined for maps of spectra.