Phantom map explained

In homotopy theory, phantom maps are continuous maps $f:\; X\; \backslash to\; Y$ of CW-complexes for which the restriction of $f$ to any finite subcomplex $Z\; \backslash subset\; X$ is inessential (i.e., nullhomotopic). produced the first known nontrivial example of such a map with $Y$ finite-dimensional (answering a question of Paul Olum). Shortly thereafter, the terminology of "phantom map" was coined by, who constructed a stably essential phantom map from infinite-dimensional complex projective space to $S^3$.^[1] The subject was analysed in the thesis of Gray, much of which was elaborated and later published in . Similar constructions are defined for maps of spectra.^[2]

Definition

Let

\alpha be a regular cardinal. A morphism f: x \longrightarrow 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 \longrightarrow x \xrightarrow{f} y vanishes.^[3]

Notes and References

1. Web site: Mathew . Akhil . 2012-06-13 . An example of a phantom map . live . https://web.archive.org/web/20210731084253/https://amathew.wordpress.com/2012/06/13/an-example-of-a-phantom-map/ . 2021-07-31 . Climbing Mount Bourbaki . en.
2. Web site: Lurie . Jacob . 2010-04-27 . Phantom Maps (Lecture 17) . live . https://web.archive.org/web/20220130234317/https://www.math.ias.edu/~lurie/252xnotes/Lecture17.pdf . 2022-01-30.
3. Book: Neeman, Amnon . Triangulated Categories . . 2010.