CHAPTER 1 Dyer-Lashof operations and the identity functor Johnson’s computation [Joh95] of the derivatives of the identity ∂i(Id) admit a description in terms of the poset of partitions of the set i [AK98], [AM99]. Ching observed that this description yields an equivalence ∂i(Id) B(1, Comm, 1)i∨ where B(1, Comm, 1)i is the ith space of the operadic bar construction on the commutative operad in spectra [Chi05]. Using this description, Ching gives ∂∗(Id) the structure of an operad, and therefore the layers ∂∗(Id) Σ∞X∧∗ form a left module over ∂∗(Id). More generally, the work of Arone and Ching [AC] gives ∂∗(F ) the structure of a bimodule over ∂∗(Id) for any reduced finitary homotopy functor F : Top Top . In Section 1.1 we review Ching’s topological model for the operadic bar con- struction. In Section 1.2 we recall Ching’s cooperad structure on this bar construc- tion. Dualizing this gives Ching’s operadic structure on ∂∗(Id), as explained in Section 1.3. We then use this model to define Dyer-Lashof-like operations ¯j on H∗(D∗(F )(X)), and relate these to the actual Dyer-Lashof operations which appear in the Arone-Mahowald computation of H∗(D∗(Sn)). We review some well-known aspects of the homology of extended powers in Section 1.4. In Section 1.5 we con- struct the operations ¯j, and explain how the Arone-Mahowald computation of the stable homology of the layers of the Goodwillie tower of the identity evaluated on spheres is given by these operations. 1.1. The operadic bar construction Our symmetric sequences shall be regarded as functors Fin Top where Fin is the category of finite sets and bijections. Let Φ be a reduced operad in Top∗. Ching gives a topological model for the realization of the operadic bar construction B(Φ) := B(1, Φ, 1). For a finite set A, a point in B(Φ)(A) is given by a tuple (T, α, (xv), l) consisting of: (1) A rooted tree T with one root and |A| leaves. Each v in V (T ), the set of internal vertices, has a set I(v) of incoming edges, with |I(v)| 2, and a single outgoing edge. A vertex v is the source s(e) of its outgoing edge e, and is the target t(e ) of each of its incoming edges e . 1
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