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 ${\textstyle S^{3}}$. 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 $\alpha$ be a regular cardinal. A morphism $f:x\to y$ in the homotopy category of spectra is called an $\alpha$-phantom map if, for any spectrum s with fewer than $\alpha$ cells, any composite $s\to x\xrightarrow {f} y$ 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 $n$-type, for all $n$", 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