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Phantom map

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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.

01Definition

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.

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