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The Bousfield-Kuhn functor and topological André-Quillen cohomology. (English) Zbl 1444.55003
This paper in chromatic homotopy theory works at a prime $$p$$ and height $$h\geq 1$$, using $$E$$, height $$h$$ Morava $$E$$-theory, and $$K$$, even-periodic height $$h$$ Morava $$K$$-theory.
Recall that the Bousfield-Kuhn functor $$\Phi_T$$ from pointed spaces to spectra is defined using $$T$$, a $$v_h$$-telescope on a finite type $$h$$ complex. It has the fundamental property that, on the infinite loop space associated to a spectrum $$Y$$, $$\Phi_T (\Omega^\infty Y) \simeq Y_T$$; moreover, the homotopy groups $$\pi_* (\Phi_T X)$$ calculate the completed unstable $$v_h$$-periodic homotopy of the pointed space $$X$$.
The authors work with the $$K$$-localization $$\Phi X:= (\Phi_T X)_K$$ and, for $$R$$ a $$K$$-local commutative $$S$$-algebra, use the Basterra-McCarthy description [M. Basterra and R. McCarthy, Topology Appl. 121, No. 3, 551–566 (2002; Zbl 1004.55003)] of topological André-Quillen homology to give a comparison map $c_R : \mathrm{TAQ}^R(R^{X_+}) \rightarrow R^{\Phi(X)}$ and the induced dual $$c^R : (R \wedge \Phi (X))_K \rightarrow \mathrm{TAQ}_R (R^{X_+})$$. Taking $$R=S_K$$, the $$K$$-local sphere spectrum, gives $c^{S_K} : \Phi (X) \rightarrow \mathrm{TAQ}_{S_K} (S^{X_+}_K),$ which relates unstable $$v_h$$-periodic homotopy to André-Quillen cohomology.
The class of finite pointed spaces for which $$c^{S_K}$$ is an equivalence is thus of significant interest. The authors prove the fundamental result that, for $$q$$ a positive odd integer, the sphere $$S^q$$ lies in this class. This has been developed (see [M. Behrens and C. Rezk, “Spectral algebra models of unstable $$v_n$$-periodic homotopy theory, Bousfield Classes and Ohkawa’s Theorem”, Proc. Math. Stat. 309, 275–323 (2020; doi:10.1007/978-981-15-1588-0_10)]) and placed in a wider context by other authors.
The proof presented here uses the comparison of $$K$$-local Weiss towers for functors from $$\mathbb{C}$$-vector spaces to spectra, for the functors $$V \mapsto \Phi (\Sigma S^V)$$ and $$V \mapsto \mathrm{TAQ}_{S_K} (S_K^{(\Sigma S^V)_+})$$. The Weiss tower of the former is related to the Goodwillie tower of the identity and of the latter to the dual of the Kuhn filtration of $$\mathrm{TAQ}$$. The comparison relies on an in-depth analysis of the $$E$$-cohomology of $$QX$$.
The authors then consider the finite resolution $\Phi (S^q)\rightarrow (L(0)_q)_K\rightarrow (L(1)_q)_K\rightarrow\dots\rightarrow (L(h)_q)_K$ obtained from the Goodwillie tower of the identity for $$S^q$$, where $$L(k)_q$$ is the Steinberg summand of the Thom spectrum of $$q$$ copies of the reduced regular representation of $$(\mathbb{Z}/p)^k$$.
They show that the $$E$$-homology of this resolution is isomorphic to the dual of the Koszul resolution of the degree $$q$$ Dyer-Lashof algebra for Morava $$E$$-theory and give a modular interpretation of this Koszul complex, based on [C. Rezk, Algebr. Geom. Topol. 12, No. 3, 1373–1403 (2012; Zbl 1254.14030)].

MSC:
 55N22 Bordism and cobordism theories and formal group laws in algebraic topology 55Q51 $$v_n$$-periodicity 55P60 Localization and completion in homotopy theory
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