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Subalgebras of group cohomology defined by infinite loop spaces
, 2001
"... We study natural subalgebras ChE(G) of group cohomology defined in terms of infinite loop spaces E and give representation theoretic descriptions of those based on QS 0 and the JohnsonWilson theories E(n). We describe the subalgebras arising from the BrownPeterson spectra BP and as a result give a ..."
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We study natural subalgebras ChE(G) of group cohomology defined in terms of infinite loop spaces E and give representation theoretic descriptions of those based on QS 0 and the JohnsonWilson theories E(n). We describe the subalgebras arising from the BrownPeterson spectra BP and as a result give a simple reproof of Yagita’s theorem that the image of BP ∗ (BG) in H ∗ (BG; Fp) is Fisomorphic to the whole cohomology ring; the same result is shown to hold with BP replaced by any complex oriented theory E with a map of ring spectra E → HFp which is nontrivial in homotopy. We also extend the constructions to define subalgebras of H ∗ (X; Fp) for any space X; when X is finite we show that the subalgebras Ch E(n)(X) give a natural unstable chromatic filtration of H ∗ (X; Fp). 1