phantom map

In homotopy theory, phantom maps are continuous maps $f: X \to Y$ of CW-complexes for which the restriction of $f$ to any finite subcomplex $Z \subset X$ is inessential (i.e., nullhomotopic). {{harvs|txt|last1 = Adams|first1=J. Frank|authorlink1=Frank Adams|last2=Walker|first2=Grant|year=1964}} 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 {{harvs|txt|last1 = Gray|first1=Brayton|year=1966}}, who constructed a stably essential phantom map from infinite-dimensional complex projective space to $S^3$ .{{Cite web |last=Mathew |first=Akhil |date=2012-06-13 |title=An example of a phantom map |url=https://amathew.wordpress.com/2012/06/13/an-example-of-a-phantom-map/ |url-status=live |archive-url=https://web.archive.org/web/20210731084253/https://amathew.wordpress.com/2012/06/13/an-example-of-a-phantom-map/ |archive-date=2021-07-31 |access-date= |website=Climbing Mount Bourbaki |language=en}} The subject was analysed in the thesis of Gray, much of which was elaborated and later published in {{harvs|last1=Gray | last2=McGibbon |year=1993}}. Similar constructions are defined for maps of spectra.{{Cite web |last=Lurie |first=Jacob |date=2010-04-27 |title=Phantom Maps (Lecture 17) |url=https://www.math.ias.edu/~lurie/252xnotes/Lecture17.pdf |url-status=live |archive-url=https://web.archive.org/web/20220130234317/https://www.math.ias.edu/~lurie/252xnotes/Lecture17.pdf |archive-date=2022-01-30}}

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.{{Cite book |last=Neeman |first=Amnon |title=Triangulated Categories |publisher=Princeton University Press |year=2010}}

References

{{Citation | last1=Adams | first1=J. Frank |authorlink1=Frank Adams| last2=Walker | first2=G. | title=An example in homotopy theory | doi=10.1017/S0305004100077422 | mr=0166786 | year=1964 | journal=Mathematical Proceedings of the Cambridge Philosophical Society | volume=60 | issue=3 | pages=699–700| bibcode=1964PCPS...60..699A }}
{{Citation | last=Gray | first=Brayton I. | title=Spaces of the same $n$ -type, for all $n$ | mr=0196743 | year=1966 | journal=Topology | volume=5 | issue=3 | pages=241–243| doi=10.1016/0040-9383(66)90008-5 | doi-access=free}}
{{Citation | last1=Gray |first1=Brayton | last2=McGibbon | first2=C.A. | title= Universal phantom maps | year=1993 | journal=Topology | volume=32 |issue=2 | pages=371–294|doi=10.1016/0040-9383(93)90027-S | doi-access= }}

Category:Homotopy theory