Subalgebras of group cohomology defined by infinite loop spaces

We study natural subalgebras Ch E ( BG ; R ) of group cohomology H * ( BG ; R ) defined in terms of the infinite loop spaces in spectra E and give representation theoretic descriptions of those based on QS 0 and the Johnson–Wilson theories E ( n ) . We describe the subalgebras arising from the Brown–Peterson spectra BP and as a result give a simple reproof of Yagitaʹs theorem that the image of BP * ( BG ) in H * ( BG ; F p ) is F-isomorphic to the whole cohomology ring; the same result is shown to hold with BP replaced by any complex oriented theory E with a non-trivial map of ring spectra E → H F p . We also extend our constructions to define subalgebras of H * ( X ; R ) for any space X; when X is a finite CW complex we show that the subalgebras Ch E ( n ) ( X ; R ) give a natural unstable chromatic filtration of H * ( X ; R ) .