Phantom map

In homotopy theory, phantom maps are continuous maps ${\textstyle f:X\to Y}$ of CW-complexes for which the restriction of ${\textstyle f}$ to any finite subcomplex ${\textstyle Z\subset X}$ is inessential (i.e., nullhomotopic). J. Frank Adams and Grant Walker (1964) produced the first known nontrivial example of such a map with ${\textstyle Y}$ 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 ${\textstyle S^{3}}$.^[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]

Definition

Let ${\displaystyle \alpha }$ be a regular cardinal. A morphism ${\displaystyle f:x\longrightarrow y}$ in the homotopy category of spectra is called an ${\displaystyle \alpha }$-phantom map if, for any spectrum s with fewer than ${\displaystyle \alpha }$ cells, any composite ${\displaystyle s\longrightarrow x\xrightarrow {f} y}$ vanishes.^[3]