4 MARK BEHRENS 1.4. Homology of extended powers Let Y be a spectrum. We review the structure of the homology of the extended powers Y ∧2 hΣ2 . Lemma 1.3 of [May70] and Corollary I.2.3 of [BMMS86] imply the following. Lemma 1.4.1. Let {yi}i∈I be an ordered basis of H∗(Y ). Then H∗(YhΣ ∧2 2 ) = F2{ek ⊗ yi ⊗ yi}i,k ⊕ F2{e0 ⊗ yi 1 ⊗ yi 2 }i 1 i2 . Following standard conventions, for y ∈ Hd(Y ) and j ≥ d, we define the Dyer- Lashof operation Qj by Qj(y) := ej−d ⊗ y ⊗ y ∈ Hd+j(YhΣ ∧2 2 ). If a sequence (j1, . . . , jk) is allowable (i.e. js ≥ js+1 + · · · + jk + d for all s, or equivalently, (j1, . . . , jk) has excess ≥ d) then iterating the extended power construction gives elements Qj1 · · · Qjky ∈ Hd+j 1 +···+jk (Y ∧2k hΣ k 2 ). Define Rn(k) to be the F2-module Rn(k) := F2{QJ : J = (j1, . . . , jk) has excess ≥ n} where QJ := Qj1 · · · Qjk. The homology of the iterated extended power H∗(Y ∧2k hΣ k 2 ) contains a summand given by R(k) ⊗ H∗(Y ) := F2{QJyi : i ∈ I, J = (j1, . . . , jk), QJyi allowable} (so Rn(k) = R(k) ⊗ H∗(Sn)). The summand Rn(k) ⊗ H∗(Y ) is closed under the dual action of the Steenrod algebra, and the dual Steenrod action is computed by the Nishida relations Sq∗Qs r = t s − r r − 2t Qs−r+tSq∗.t For t ≥ 0, the diagonal map St → St ∧ St induces a suspension map Et : (Σ−tY )hΣ ∧2 2 → Σ−tYhΣ2∧ 2 (see, for example, [BMMS86, Sec. 3]). The following lemma follows from [BMMS86, Lem. II.5.6]. Lemma 1.4.2. The induced map on homology Et ∗ : H∗((Σ−tY )∧2 hΣ2 ) → H∗(Σ−tY ∧2 hΣ2 ) satisfies E∗Qjσ−ty t = σ−tQjy, j ≥ |y|, 0, otherwise.

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