# Is there a “higher Segal conjecture”?

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.

## 1 Answer

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.)

• 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! – Tim Campion Apr 15 '19 at 23:19