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The Segal conjecture describes the Spanier-Whitehead dual $D \Sigma^\infty_+ BG$ for certain $G$. Is there a similar description of $D\Sigma^\infty_+ K(G,n)$ 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 $D\Sigma^\infty_+ BG$ 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

    $$D\Sigma^\infty_+ BC_p = \mathbb S \vee(\Sigma^\infty_+ BC_p )^{\wedge}_p$$

    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 $0 = DHG = \varprojlim_n \Sigma^n D\Sigma^\infty K(G,n)$, and from the Milnor exact sequence we conclude that $\varprojlim_n \pi_{\ast-n} D\Sigma^\infty K(G,n) = \varprojlim^1_n \pi_{\ast-n} D \Sigma^\infty K(G,n) = 0$. 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 $F(\Sigma^\infty_+ K(G,n), L\mathbb S) = L \Sigma^\infty_+ K(G,n)$ where $L$ is the relevant localization. But it seems that the relevant limit does not commute with localization here.

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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 $K(A,n)$.)

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

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  • 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 Apr 15 at 23:19

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