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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2018
    • (edited Sep 10th 2018)

    added the crucial pointer to

    • Gunnar Carlsson, Equivariant Stable Homotopy and Segal’s Burnside Ring Conjecture, Annals of Mathematics Second Series, Vol. 120, No. 2 (Sep., 1984), pp. 189-224 (jstor:2006940, pdf)

    and a bit more

    diff, v3, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2018

    Adjusted title

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2018
    • (edited Sep 10th 2018)

    added the actual statement to the Idea section:

    The statement known as Segal’s conjecture (due to Graeme Segal in the 1970s, then proven by Carlsson 84) characterizes the stable cohomotopy groups π st (BG)\pi^\bullet_{st}(B G) of the classifying space BGB G of a finite group GG as the formal completion π^ S (BG)\widehat \pi^\bullet_S(B G) at the augmentation ideal (i.e. when regarded as a ring of functions: its restriction to the infinitesimal neighbourhood of the basepoint) of the ring π st,G (*)\pi^\bullet_{st,G}(\ast) of GG-equivariant stable cohomotopy groups of the point, the latter also being isomorphic to the Burnside ring A(G)A(G) of GG:

    A(G)π st,G (*)completionprojectionπ^ st,G (*)π st (BG). A(G) \simeq \pi^\bullet_{st,G}(\ast) \overset{ \text{completion} \atop \text{projection} }{\longrightarrow} \widehat \pi^\bullet_{st,G}(\ast) \simeq \pi^\bullet_{st}(B G) \,.

    This statement is the direct analogue of the Atiyah-Segal completion theorem, which makes the analogous statement for the generalized cohomology not being (equivariant) stable cohomotopy but (equivariant) complex K-theory (with the role of the Burnside ring then being the representation ring of GG).

    diff, v4, current

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