Phantom map

In homotopy theory, phantom maps are continuous maps f : X → Y {\textstyle f:X\to Y} of CW-complexes for which the restriction of f {\textstyle f} to any finite subcomplex Z ⊂ X {\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 Y {\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 S 3 {\textstyle S^{3}} .[1] 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.[2]

Definition

Let α {\displaystyle \alpha } be a regular cardinal. A morphism f : x ⟶ y {\displaystyle f:x\longrightarrow y} in the homotopy category of spectra is called an α {\displaystyle \alpha } -phantom map if, for any spectrum s with fewer than α {\displaystyle \alpha } cells, any composite s ⟶ x → f y {\displaystyle s\longrightarrow x\xrightarrow {f} y} vanishes.[3]

References

  1. ^ Mathew, Akhil (2012-06-13). "An example of a phantom map". Climbing Mount Bourbaki. Archived from the original on 2021-07-31.
  2. ^ Lurie, Jacob (2010-04-27). "Phantom Maps (Lecture 17)" (PDF). Archived (PDF) from the original on 2022-01-30.
  3. ^ Neeman, Amnon (2010). Triangulated Categories. Princeton University Press.
  • Adams, J. Frank; Walker, G. (1964), "An example in homotopy theory", Proc. Cambridge Philos. Soc., 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
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