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The Taylor (Goodwillie) tower of the identity functor on based spaces has as its $j$-th layer the infinite loop space-valued functor $$ X\mapsto \Omega^\infty (W_j \wedge_{h\Sigma_j} X^{[j]}) $$ in which $W_j$ is a spectrum with (naive) $\Sigma_j$-action, $X^{[j]}$ is the $j$-fold smash power and ${}_{h\Sigma}$ denotes homotopy orbits. The underlying unequivariant homotopy type of the coefficient spectrum $W_j$ is that of a wedge of $(j-1)!$ spheres of dimension $S^{1-j}$. The restriction of the action to the subgroup $\Sigma_{j-1} \subset \Sigma_j$ is known to be given by the permutation action.

I have been told by several folks in the past that the $\Sigma_j$-action actually extends to an action by $\Sigma_{j+1}$. The way this is somehow done, if I recall, is by considering a certain space of metric trees with $j+1$ leaves.

Does anyone know of such a construction? Does anyone know a reference?

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    $\begingroup$ This seems like it might be contained in M. Ching's thesis. There the description of the operadic structure on the Taylor tower of the identity is described by means of (duals to) spaces of metric trees. At least, if I recall correctly (I'm out of the office, so can't check my copy). $\endgroup$ – Juan Villeta-Garcia May 10 '16 at 14:10
  • $\begingroup$ @JuanVilleta-Garcia I checked the published version of Ching's thesis. I could not find it there. $\endgroup$ – John Klein May 10 '16 at 14:37
  • $\begingroup$ Ooops, sorry about the false start. $\endgroup$ – Juan Villeta-Garcia May 10 '16 at 15:14
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    $\begingroup$ Ching does use the connection with metric trees in his thesis, but not for the purpose of describing the extra action. By the way, here is a natural followup question: do the derivatives of the identity have a structure of a cyclic operad in spectra? If yes, it would combine both the extra symmetries and Ching's operad structure. And a further followup question: if such a cyclic operad structure exists, then what is the calculus meaning of it? $\endgroup$ – Gregory Arone May 10 '16 at 15:20
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This is done by identifying the partition complex of the set with j elements with the space of fully grown trees with j+1 leaves. Reference: "Partition complexes, duality and integral tree representations" by Alan Robinson, Algebr. Geom. Topol. 4. See specifically proposition 2.7 and corollary 2.8. http://msp.org/agt/2004/4-2/agt-v4-n2-p12-s.pdf

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  • $\begingroup$ Thanks Greg! By the way, I think you were the one who first told me about the extension of the action. $\endgroup$ – John Klein May 10 '16 at 15:29
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Jean-Louis Loday told me about the extended action by $\Sigma_{j+1}$ in the fall of 1992, after an Oberwolfach talk I gave about the rank filtration of algebraic $K$-theory, where the $\Sigma_j$-representations given by the integral homology of the Goodwillie derivative spectra $W_j$ played a role. I had shown that these representations were freely generated by $j$-fold Lie brackets, and Loday knew the connection to spaces of trees. Maybe you were there, too?

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  • $\begingroup$ It's quite possible. But...maybe not: I arrived in Bielefeld at the end of '92 from Siegen and I was not interacting with the K-theory gang until late '92. $\endgroup$ – John Klein May 10 '16 at 18:07

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