Is there a “higher Segal conjecture”? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Convergence of spectral sequences of cohomological typeFormal-group interpretation for Lin's theorem?Hopf algebras as cohomology of $mathbbCP^infty$, $Omega S^3$ and related $H$-spacesIs every ''group-completion'' map an acyclic map?The cell structure of Thom spectraFailure of “equivariant triangulation” for finite complexes equipped with a $G$-action$RO(G)$-graded homotopy groups vs. Mackey functors(Pre)orientation vs. formal completionmaking the group completion in homology sense unique via the plus constructionIntuition - difference between Moore spectrum and Eilenberg-Mac Lane spectrum

Is there a “higher Segal conjecture”?



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Convergence of spectral sequences of cohomological typeFormal-group interpretation for Lin's theorem?Hopf algebras as cohomology of $mathbbCP^infty$, $Omega S^3$ and related $H$-spacesIs every ''group-completion'' map an acyclic map?The cell structure of Thom spectraFailure of “equivariant triangulation” for finite complexes equipped with a $G$-action$RO(G)$-graded homotopy groups vs. Mackey functors(Pre)orientation vs. formal completionmaking the group completion in homology sense unique via the plus constructionIntuition - difference between Moore spectrum and Eilenberg-Mac Lane spectrum










3












$begingroup$


The Segal conjecture describes the Spanier-Whitehead dual $D Sigma^infty_+ BG$ for certain $G$. Is there a similar description of $DSigma^infty_+ K(G,n)$ when $n geq 2$ when $G$ is finite (and abelian)?



Notes:



  • I'd be happy to understand the case of cyclic groups $G = C_p$.


  • $K(G,n)$ can be modeled by an abelian topological group, but I'm not sure it falls under the umbrella of other known generalizations of the Segal conjecture, although when $G = mathbb Z$ and $n=2$ there is a known decomposition (see Ravenel). For $G = mathbb Z^n$ and $n=2$ there is also this.



  • Let me recall that the Segal conjecture (proved by Carlsson) says that when $G$ is finite, the Spanier-Whitehead dual $DSigma^infty_+ BG$ is a certain completion of $vee_(H) subseteq G Sigma^infty_+ BW_G(H)$ where $(H) subseteq G$ ranges over conjugacy classes of subgroups and $W_G(H) = N_G(H) / H$ is the Weyl group of $H$ in $G$. In particular, when $G = C_p$ it says that



    $$DSigma^infty_+ BC_p = mathbb S vee(Sigma^infty_+ BC_p )^wedge_p$$



    where $mathbb S$ is the sphere spectrum (corresponding to the subgroup $C_p subseteq C_p$; the other term corresponds to the trivial subgroup $0 subseteq C_p$) and $(-)^wedge_p$ is $p$-completion.



  • Lin showed that $D H G = 0$ when $G$ is a finite abelian group, where $H$ indicates taking Eilenberg-MacLane spectra. Since $HG = varinjlim_n Sigma^infty-n K(G,n)$, we have $0 = DHG = varprojlim_n Sigma^n DSigma^infty K(G,n)$, and from the Milnor exact sequence we conclude that $varprojlim_n pi_ast-n DSigma^infty K(G,n) = varprojlim^1_n pi_ast-n D Sigma^infty K(G,n) = 0$. But I'm not sure how much information that is, really.


  • If we work in the $K(h)$-local or the $T(h)$-local category then by ambidexterity we have $F(Sigma^infty_+ K(G,n), Lmathbb S) = L Sigma^infty_+ K(G,n)$ where $L$ is the relevant localization. But it seems that the relevant limit does not commute with localization here.










share|cite|improve this question











$endgroup$
















    3












    $begingroup$


    The Segal conjecture describes the Spanier-Whitehead dual $D Sigma^infty_+ BG$ for certain $G$. Is there a similar description of $DSigma^infty_+ K(G,n)$ when $n geq 2$ when $G$ is finite (and abelian)?



    Notes:



    • I'd be happy to understand the case of cyclic groups $G = C_p$.


    • $K(G,n)$ can be modeled by an abelian topological group, but I'm not sure it falls under the umbrella of other known generalizations of the Segal conjecture, although when $G = mathbb Z$ and $n=2$ there is a known decomposition (see Ravenel). For $G = mathbb Z^n$ and $n=2$ there is also this.



    • Let me recall that the Segal conjecture (proved by Carlsson) says that when $G$ is finite, the Spanier-Whitehead dual $DSigma^infty_+ BG$ is a certain completion of $vee_(H) subseteq G Sigma^infty_+ BW_G(H)$ where $(H) subseteq G$ ranges over conjugacy classes of subgroups and $W_G(H) = N_G(H) / H$ is the Weyl group of $H$ in $G$. In particular, when $G = C_p$ it says that



      $$DSigma^infty_+ BC_p = mathbb S vee(Sigma^infty_+ BC_p )^wedge_p$$



      where $mathbb S$ is the sphere spectrum (corresponding to the subgroup $C_p subseteq C_p$; the other term corresponds to the trivial subgroup $0 subseteq C_p$) and $(-)^wedge_p$ is $p$-completion.



    • Lin showed that $D H G = 0$ when $G$ is a finite abelian group, where $H$ indicates taking Eilenberg-MacLane spectra. Since $HG = varinjlim_n Sigma^infty-n K(G,n)$, we have $0 = DHG = varprojlim_n Sigma^n DSigma^infty K(G,n)$, and from the Milnor exact sequence we conclude that $varprojlim_n pi_ast-n DSigma^infty K(G,n) = varprojlim^1_n pi_ast-n D Sigma^infty K(G,n) = 0$. But I'm not sure how much information that is, really.


    • If we work in the $K(h)$-local or the $T(h)$-local category then by ambidexterity we have $F(Sigma^infty_+ K(G,n), Lmathbb S) = L Sigma^infty_+ K(G,n)$ where $L$ is the relevant localization. But it seems that the relevant limit does not commute with localization here.










    share|cite|improve this question











    $endgroup$














      3












      3








      3





      $begingroup$


      The Segal conjecture describes the Spanier-Whitehead dual $D Sigma^infty_+ BG$ for certain $G$. Is there a similar description of $DSigma^infty_+ K(G,n)$ when $n geq 2$ when $G$ is finite (and abelian)?



      Notes:



      • I'd be happy to understand the case of cyclic groups $G = C_p$.


      • $K(G,n)$ can be modeled by an abelian topological group, but I'm not sure it falls under the umbrella of other known generalizations of the Segal conjecture, although when $G = mathbb Z$ and $n=2$ there is a known decomposition (see Ravenel). For $G = mathbb Z^n$ and $n=2$ there is also this.



      • Let me recall that the Segal conjecture (proved by Carlsson) says that when $G$ is finite, the Spanier-Whitehead dual $DSigma^infty_+ BG$ is a certain completion of $vee_(H) subseteq G Sigma^infty_+ BW_G(H)$ where $(H) subseteq G$ ranges over conjugacy classes of subgroups and $W_G(H) = N_G(H) / H$ is the Weyl group of $H$ in $G$. In particular, when $G = C_p$ it says that



        $$DSigma^infty_+ BC_p = mathbb S vee(Sigma^infty_+ BC_p )^wedge_p$$



        where $mathbb S$ is the sphere spectrum (corresponding to the subgroup $C_p subseteq C_p$; the other term corresponds to the trivial subgroup $0 subseteq C_p$) and $(-)^wedge_p$ is $p$-completion.



      • Lin showed that $D H G = 0$ when $G$ is a finite abelian group, where $H$ indicates taking Eilenberg-MacLane spectra. Since $HG = varinjlim_n Sigma^infty-n K(G,n)$, we have $0 = DHG = varprojlim_n Sigma^n DSigma^infty K(G,n)$, and from the Milnor exact sequence we conclude that $varprojlim_n pi_ast-n DSigma^infty K(G,n) = varprojlim^1_n pi_ast-n D Sigma^infty K(G,n) = 0$. But I'm not sure how much information that is, really.


      • If we work in the $K(h)$-local or the $T(h)$-local category then by ambidexterity we have $F(Sigma^infty_+ K(G,n), Lmathbb S) = L Sigma^infty_+ K(G,n)$ where $L$ is the relevant localization. But it seems that the relevant limit does not commute with localization here.










      share|cite|improve this question











      $endgroup$




      The Segal conjecture describes the Spanier-Whitehead dual $D Sigma^infty_+ BG$ for certain $G$. Is there a similar description of $DSigma^infty_+ K(G,n)$ when $n geq 2$ when $G$ is finite (and abelian)?



      Notes:



      • I'd be happy to understand the case of cyclic groups $G = C_p$.


      • $K(G,n)$ can be modeled by an abelian topological group, but I'm not sure it falls under the umbrella of other known generalizations of the Segal conjecture, although when $G = mathbb Z$ and $n=2$ there is a known decomposition (see Ravenel). For $G = mathbb Z^n$ and $n=2$ there is also this.



      • Let me recall that the Segal conjecture (proved by Carlsson) says that when $G$ is finite, the Spanier-Whitehead dual $DSigma^infty_+ BG$ is a certain completion of $vee_(H) subseteq G Sigma^infty_+ BW_G(H)$ where $(H) subseteq G$ ranges over conjugacy classes of subgroups and $W_G(H) = N_G(H) / H$ is the Weyl group of $H$ in $G$. In particular, when $G = C_p$ it says that



        $$DSigma^infty_+ BC_p = mathbb S vee(Sigma^infty_+ BC_p )^wedge_p$$



        where $mathbb S$ is the sphere spectrum (corresponding to the subgroup $C_p subseteq C_p$; the other term corresponds to the trivial subgroup $0 subseteq C_p$) and $(-)^wedge_p$ is $p$-completion.



      • Lin showed that $D H G = 0$ when $G$ is a finite abelian group, where $H$ indicates taking Eilenberg-MacLane spectra. Since $HG = varinjlim_n Sigma^infty-n K(G,n)$, we have $0 = DHG = varprojlim_n Sigma^n DSigma^infty K(G,n)$, and from the Milnor exact sequence we conclude that $varprojlim_n pi_ast-n DSigma^infty K(G,n) = varprojlim^1_n pi_ast-n D Sigma^infty K(G,n) = 0$. But I'm not sure how much information that is, really.


      • If we work in the $K(h)$-local or the $T(h)$-local category then by ambidexterity we have $F(Sigma^infty_+ K(G,n), Lmathbb S) = L Sigma^infty_+ K(G,n)$ where $L$ is the relevant localization. But it seems that the relevant limit does not commute with localization here.







      at.algebraic-topology homotopy-theory






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      edited 1 hour ago







      Tim Campion

















      asked 1 hour ago









      Tim CampionTim Campion

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          1 Answer
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          $begingroup$

          In the 1980's, Chun Nip Lee showed that the Spanier Whitehead dual of (the suspension spectrum of) $K(mathbb Z/p, n)$ is contractible for $n >1$. (The key case is $n=2$. The idea" view $K(A,n+1)$ as the bar construction on $K(A,n)$ and



          (No time right now to write more ... but maybe this is enough.)






          share|cite|improve this answer









          $endgroup$








          • 1




            $begingroup$
            Ah, perfect, thanks so much! Here's a link. I was starting to wonder if this might be true... It's oddly difficult to search for basic data about Eilenberg-MacLane spaces, since they're so fundamental and typically used to study other things!
            $endgroup$
            – Tim Campion
            55 mins ago












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          active

          oldest

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          active

          oldest

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          4












          $begingroup$

          In the 1980's, Chun Nip Lee showed that the Spanier Whitehead dual of (the suspension spectrum of) $K(mathbb Z/p, n)$ is contractible for $n >1$. (The key case is $n=2$. The idea" view $K(A,n+1)$ as the bar construction on $K(A,n)$ and



          (No time right now to write more ... but maybe this is enough.)






          share|cite|improve this answer









          $endgroup$








          • 1




            $begingroup$
            Ah, perfect, thanks so much! Here's a link. I was starting to wonder if this might be true... It's oddly difficult to search for basic data about Eilenberg-MacLane spaces, since they're so fundamental and typically used to study other things!
            $endgroup$
            – Tim Campion
            55 mins ago
















          4












          $begingroup$

          In the 1980's, Chun Nip Lee showed that the Spanier Whitehead dual of (the suspension spectrum of) $K(mathbb Z/p, n)$ is contractible for $n >1$. (The key case is $n=2$. The idea" view $K(A,n+1)$ as the bar construction on $K(A,n)$ and



          (No time right now to write more ... but maybe this is enough.)






          share|cite|improve this answer









          $endgroup$








          • 1




            $begingroup$
            Ah, perfect, thanks so much! Here's a link. I was starting to wonder if this might be true... It's oddly difficult to search for basic data about Eilenberg-MacLane spaces, since they're so fundamental and typically used to study other things!
            $endgroup$
            – Tim Campion
            55 mins ago














          4












          4








          4





          $begingroup$

          In the 1980's, Chun Nip Lee showed that the Spanier Whitehead dual of (the suspension spectrum of) $K(mathbb Z/p, n)$ is contractible for $n >1$. (The key case is $n=2$. The idea" view $K(A,n+1)$ as the bar construction on $K(A,n)$ and



          (No time right now to write more ... but maybe this is enough.)






          share|cite|improve this answer









          $endgroup$



          In the 1980's, Chun Nip Lee showed that the Spanier Whitehead dual of (the suspension spectrum of) $K(mathbb Z/p, n)$ is contractible for $n >1$. (The key case is $n=2$. The idea" view $K(A,n+1)$ as the bar construction on $K(A,n)$ and



          (No time right now to write more ... but maybe this is enough.)







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 1 hour ago









          Nicholas KuhnNicholas Kuhn

          3,7101221




          3,7101221







          • 1




            $begingroup$
            Ah, perfect, thanks so much! Here's a link. I was starting to wonder if this might be true... It's oddly difficult to search for basic data about Eilenberg-MacLane spaces, since they're so fundamental and typically used to study other things!
            $endgroup$
            – Tim Campion
            55 mins ago













          • 1




            $begingroup$
            Ah, perfect, thanks so much! Here's a link. I was starting to wonder if this might be true... It's oddly difficult to search for basic data about Eilenberg-MacLane spaces, since they're so fundamental and typically used to study other things!
            $endgroup$
            – Tim Campion
            55 mins ago








          1




          1




          $begingroup$
          Ah, perfect, thanks so much! Here's a link. I was starting to wonder if this might be true... It's oddly difficult to search for basic data about Eilenberg-MacLane spaces, since they're so fundamental and typically used to study other things!
          $endgroup$
          – Tim Campion
          55 mins ago





          $begingroup$
          Ah, perfect, thanks so much! Here's a link. I was starting to wonder if this might be true... It's oddly difficult to search for basic data about Eilenberg-MacLane spaces, since they're so fundamental and typically used to study other things!
          $endgroup$
          – Tim Campion
          55 mins ago


















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