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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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