Phantom map

In homotopy theory, phantom maps are continuous maps [math]\displaystyle{ f: X \to Y }[/math] of CW-complexes for which the restriction of [math]\displaystyle{ f }[/math] to any finite subcomplex [math]\displaystyle{ Z \subset X }[/math] is inessential (i.e., nullhomotopic). J. Frank Adams and Grant Walker (1964) produced the first known nontrivial example of such a map with [math]\displaystyle{ Y }[/math] 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 [math]\displaystyle{ S^3 }[/math]. 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 [math]\displaystyle{ \alpha }[/math] be a regular cardinal. A morphism [math]\displaystyle{ f: x \longrightarrow y }[/math] in the homotopy category of spectra is called an [math]\displaystyle{ \alpha }[/math]-phantom map if, for any spectrum s with fewer than [math]\displaystyle{ \alpha }[/math] cells, any composite [math]\displaystyle{ s \longrightarrow x \xrightarrow{f} y }[/math] vanishes.

References

• Adams, J. Frank; Walker, G. (1964), "An example in homotopy theory", Proc. Cambridge Philos. Soc. 60 (3): 699–700, doi:10.1017/S0305004100077422, Bibcode: 1964PCPS...60..699A 
• Gray, Brayton I. (1966), "SPACES OF THE SAME n-TYPE, FOR ALL n", Topology 5 (3): 241–243, doi:10.1016/0040-9383(66)90008-5 
• Gray, Brayton; McGibbon, C.A. (1993), "Universal phantom maps", Topology 32 (2): 371–294, doi:10.1016/0040-9383(93)90027-S