Symmetric invariants and cohomology of groups

Modular coinvariants and the modphomology ofQSk

“…From relation (ii) in Theorem 2.2, we see that the above monomials can be written in the form (up to a sign) In this section, we use results of the previous sections to give a complete set of relations for {H * QS k } k≥0 as a Hopf ring. Let [1] ∈ H * QS 0 be the image of non-base point generator of H 0 S 0 under the map induced by the canonical inclusion S 0 ֒→ QS 0 and let σ ∈ H * QS 1 be the image of the generator of H 1 S 1 under the homomorphism induced by the inclusion S 1 ֒→ QS 1 . Note that the element σ is usually known as the homology suspension element because σ • x is the homology suspension of x.…”

“…Thus, the elements E (ǫ,s) = (−1) s β ǫ Q s [1] and σ generate {H * QS k } k≥0 as a Hopf ring. The problem is to find a complete set of relations.…”

“…Let [1] ∈ H 0 QS 0 denote the image of the non-base point generator of H 0 S 0 under the homomorphism induced by the obvious inclusion S 0 ֒→ QS 0 and let σ denote the image of the generator of H 1 S 1 under the homomorphism induced by the inclusion S 1 ֒→ QS 1 . It follows from work of Araki-Kudo [2], Dyer-Lashof [9] and May [7] that the mod p homology of {QS k } k≥0 is generated as a Hopf ring by the elements Q i [1], i ≥ 0, σ (for p = 2) and by Q i [1], i ≥ 0, βQ i [1], i ≥ 1, σ (for p odd), where Q i denote the Dyer-Lashof operations [9].…”

“…It follows from work of Araki-Kudo [2], Dyer-Lashof [9] and May [7] that the mod p homology of {QS k } k≥0 is generated as a Hopf ring by the elements Q i [1], i ≥ 0, σ (for p = 2) and by Q i [1], i ≥ 0, βQ i [1], i ≥ 1, σ (for p odd), where Q i denote the Dyer-Lashof operations [9]. This actually corresponds to the fact that the Quillen's approximation map of the symmetric group by elementary abelian subgroups is a monomorphism [33].…”

“…The relation (4.4) occurs when a homology class is detected by both subgroups. 1 According to Madsen and Milgram [26,p. 49], Dyer-Lashof were the first to prove this result in an unpublished version of [9].…”