Phantom map

In homotopy theory, phantom maps are continuous maps defined on a direct limit of spaces in which each restriction is inessential. The first known example involved the filtration of a finite dimensional CW complex by finite sub-complexes ([AW]). The first example in which the filtration was by the skeleta led to the name [G]. In this case, a stably essential map was constructed from infinite dimensional complex projective space to S^3. The subject was analysed in the thesis of Gray, much of which was further developed and later published in [GM]. Similar constructions are defined for maps of spectra [L]

References[]

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

External links[]

Lurie, J., Phantom maps (PDF)

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