Let $G$ be a finite group. Let me call a connected finite CW-complex $X$ an imitation $BG$ relative to a prime $p$ if
- $\pi_1(X) \cong G$
- $\pi_k(X)$ is p-adically trivial for $k > 1$
Do there exist nontrivial imitation BG's?
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$\begingroup$ @A.S. mathoverflow.net/questions/79741/… $\endgroup$– user1092847Oct 21, 2022 at 5:42
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$\begingroup$ @user1092847 Sorry, I thought of finitely-generated groups like Higman's group, not of finite groups. $\endgroup$– user164898Oct 21, 2022 at 5:52
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$\begingroup$ What happens for the trivial group? Are there any finite simply connected CW-complexes with $pi_k$ p-adically trivial? Or can that not happen? $\endgroup$– Chris Schommer-PriesOct 23, 2022 at 19:04
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$\begingroup$ @ChrisSchommer-Pries I guess by taking covers you can probably find one with G trivial if there exists one for any G. But I don't know how to find such spaces $\endgroup$– davikOct 25, 2022 at 1:06
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