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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 10th 2018
   • (edited Sep 10th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded 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](https://www.jstor.org/stable/2006940), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/carlsson.pdf)) and a bit more <a href="https://ncatlab.org/nlab/revision/diff/Segal+theorem/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Segal+theorem/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Segal+theorem">current</a>

   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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 10th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexAdjusted title <a href="https://ncatlab.org/nlab/revision/diff/Segal-Carlsson+theorem/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Segal-Carlsson+theorem/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Segal-Carlsson+theorem">current</a>

   Adjusted title

   diff, v3, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeSep 10th 2018
   • (edited Sep 10th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded 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](#Carlsson84)) characterizes the [[stable cohomotopy]] [[cohomology group|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]] [[cohomology groups|groups]] of the point, the latter also being [[isomorphism|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|equivariant]]) [[stable cohomotopy]] but ([[equivariant K-theory|equivariant]]) [[complex K-theory]] (with the role of the [[Burnside ring]] then being the [[representation ring]] of $G$). <a href="https://ncatlab.org/nlab/revision/diff/Segal-Carlsson+completion+theorem/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Segal-Carlsson+completion+theorem/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">current</a>

   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 $G$).

   diff, v4, current