In homotopy theory, phantom maps are continuous maps f : X → Y {\textstyle f:X\to Y} of CW-complexes for which the restriction of f {\textstyle f} to any finite subcomplex Z ⊂ X {\textstyle Z\subset X} is inessential (i.e., nullhomotopic). J. Frank Adams and Grant Walker (1964) produced the first known nontrivial example of such a map with Y {\textstyle Y} 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 S 3 {\textstyle S^{3}} . 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.
Definition Let α {\displaystyle \alpha } be a regular cardinal. A morphism f : x → y {\displaystyle f:x\to y} in the homotopy category of spectra is called an α {\displaystyle \alpha } -phantom map if, for any spectrum s with fewer than α {\displaystyle \alpha } cells, any composite s → x → f y {\displaystyle s\to x\xrightarrow {f} y} vanishes.
References
Adams, J. Frank; Walker, G. (1964), "An example in homotopy theory", Mathematical Proceedings of the Cambridge Philosophical Society, 60 (3): 699–700, Bibcode:1964PCPS...60..699A, doi:10.1017/S0305004100077422, MR 0166786 Gray, Brayton I. (1966), "Spaces of the same n {\displaystyle n} -type, for all n {\displaystyle n} ", Topology, 5 (3): 241–243, doi:10.1016/0040-9383(66)90008-5, MR 0196743 Gray, Brayton; McGibbon, C.A. (1993), "Universal phantom maps", Topology, 32 (2): 371–294, doi:10.1016/0040-9383(93)90027-S