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A nilpotent Whitehead theorem for \(\mathsf{TQ} \)-homology of structured ring spectra. (English) Zbl 1440.55006
Let \(\mathcal{O}\) be an operad. Topological Quillen homology, i.e., TQ-homology, is a fundamental homology theory for \(\mathcal{O}\)-algebras. The classical Whitehead theorem says that a continuous map between two simply-connected spaces is a weak homotopy equivalence if it is a homology equivalence. This result was generalized to nilpotent spaces by E. Dror in [Lect. Notes Math. 249, 13–22 (1971; Zbl 0243.55018)]. Motivated by Dror’s result, the authors prove a version of the TQ-Whitehead theorem for nilpotent \(\mathcal{O}\)-algebras. An earlier result for connected \(\mathcal{O}\)-algebras was proved by J. Harper and K. Hess [Geom. Topol. 17, No. 3, 1325–1416 (2013; Zbl 1270.18025)]. The authors also prove retract theorems for the TQ-completion and homotopy completion of nilpotent structured ring spectra.

55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
55U35 Abstract and axiomatic homotopy theory in algebraic topology
55P48 Loop space machines and operads in algebraic topology
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