nForum - Discussion Feed (Segal-Carlsson completion theorem) 2022-08-18T16:04:17-04:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Urs comments on "Segal-Carlsson completion theorem" (71538) https://nforum.ncatlab.org/discussion/8962/?Focus=71538#Comment_71538 2018-09-10T06:25:28-04:00 2022-08-18T16:04:16-04:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

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 $\pi^\bullet_{st}(B G)$ of the classifying space $B G$ of a finite group $G$ as the formal completion $\widehat \pi^\bullet_S(B G)$ 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 $\pi^\bullet_{st,G}(\ast)$ of $G$-equivariant stable cohomotopy groups of the point, the latter also being isomorphic to the Burnside ring $A(G)$ of $G$:

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

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 $G$).

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Urs comments on "Segal-Carlsson completion theorem" (71537) https://nforum.ncatlab.org/discussion/8962/?Focus=71537#Comment_71537 2018-09-10T05:38:01-04:00 2022-08-18T16:04:17-04:00 Urs https://nforum.ncatlab.org/account/4/ Adjusted title diff, v3, current