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Phantom map - Wikipedia

  • ️Wed Jun 13 2012

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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}}.[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]

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

  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.