The spectral Sullivan conjecture
Ishan Levy
TL;DR
The paper generalizes the Sullivan conjecture by showing that any map from an infinite loop space X to a finite dimensional, p-complete nilpotent space Y factors canonically through a union of p-adic tori, captured by a map to a disjoint union of classifying spaces of torsion-free abelian parts completed at p. The authors develop unstable and stable localization techniques, establishing the pivotal equality of connective Bousfield classes ⟨ΣS/p⟩^s = ⟨Σ^∞BC_p⟩^s and leveraging this to compare X with its L-localization, ultimately proving that mapping spaces into Y are controlled by p-adic torus data. Consequences include strong restrictions on actions of infinite loop spaces on finite dimensional spaces, reducing such actions to extensions by discrete abelian groups and p-adic tori, with integral variants under rational-vanishing hypotheses. The work also offers an alternative proof route via symmetric power filtrations and situates these results within a broader framework of unstable Bousfield theory and presentable localizations.
Abstract
We show that any map from an infinite loop space to a $p$-complete nilpotent finite dimensional space factors canonically through a union of $p$-adic tori. This is proven via bootstrapping from the case of $B\mathbb{Z}/p\mathbb{Z}$, which is the key case of the Sullivan conjecture proven by Miller. The main step in our proof is to show that the subcategory of spectra generated by the reduced suspension spectrum of $B\mathbb{Z}/p\mathbb{Z}$ under colimits and extensions agrees with that of a Moore spectrum.