1.3. OPERAD STRUCTURE ON ∂∗(Id) 3 1.2. The cooperadic structure on B(Φ) Ching showed that B(Φ) is a cooperad in Top . The cooperadic structure map ◦a : B(Φ)(A ∪a B) B(Φ)(A) B(Φ)(B) sends a point (T, α, (xv), l) to the point ⎪(T , α , (xv), l ) (T , α , (xv), l ) if (T, α) is obtained by grafting (T , α ) onto the leaf labelled by a of (T , α ), ∗, otherwise, where: The labellings (x v ) and (x v ) arise from the bijection V (T ) V (T ) V (T ). We have l (e) = h(T ), s(e) = a, l(e), otherwise, where h(T ) is the height of T , when viewed as a subtree of the metric tree (T, l). The metric l is given by l (e) = l(e)/h(T ). 1.3. Operad structure on ∂∗(Id) Let Comm be the commutative operad in Top , with Comm(i) = S0. Ching observes that the partition poset description of ∂i(Id) gives rise to an equiv- alence ∂i(Id) Σ∞B(Comm)(i)∨ of Σi-spectra. As B(Comm) is a cooperad in Top , Ching deduces that ∂∗(Id) is an operad in spectra. Arone and Ching also show that, for any reduced finitary homotopy functor F : Top Top , the derivatives ∂∗(F ) are a bimodule over ∂∗(Id) [AC]. In the case where F = Id, this bimodule structures is the one given by letting ∂∗(Id) act on itself on both sides. Our Dyer-Lashof-like operations will rely on a good understanding of ∂2(Id). There is only one rooted tree with 2 leaves •❄❄ ❄❄❄ ☎☎☎☎ and the space of weightings on this tree is homeomorphic to an interval I. Taking the identifications into account, we deduce that B(Comm)(2) is homeomorphic to I/∂I = S1, and thus ∂2(Id) = S−1 (with trivial Σ2-action).
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