A nilpotent Whitehead theorem for \(\mathsf{TQ} \)-homology of structured ring spectra (Q2299510)

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 \textit{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 \textit{J. Harper} and \textit{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.