Adelic Asphericity and One-Relator Pro-p Groups
Abstract
We give a direct Fox-calculus proof of the known rational asphericity theorem for every finite presentation complex with one nontrivial relator, including proper powers. The proof uses completed Fox derivatives and the absence of zero divisors in the completed algebra of a prounipotent group. Exact scalar extension to the finite adeles then yields adelic asphericity and a product decomposition of the higher homotopy groups of integral Bousfield--Kan completion. We also prove rational and pro-p analogues of Mannan's formula for finite two-complexes, with their natural topologies and fundamental group actions. The rational cohomological Whitehead criterion is proved directly. For a discrete one-relator group with relator not a proper power, we show that the tower L2νn(G) is Mittag--Leffler. Consequently limonenL2νn(G)=0, and π2(Z∞ BG) is naturally the inverse limit of the nilpotent Baer invariants. We identify this Baer limit canonically and topologically with the kernel of the integral augmentation-completed Fox operator and with the product of its local pro-p kernels. The comparison yields a torsion-free profinite homotopy module, vanishing inverse limits of the finite-stage transgression kernels, and explicit lifting obstructions. These results provide a homotopy-theoretic framework for the question of Serre in Gildenhuys's corrected formulation.
Related
BibTeX
@misc{mikhovich2026adelic,
title = {{Adelic Asphericity and One-Relator Pro-\(p\) Groups}},
author = {Andrey Mikhovich},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26288},
eprint = {2610.26288},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26288},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Adelic Asphericity and One-Relator Pro-p Groups
AU - Andrey Mikhovich
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26288
UR - https://arxiv.org/abs/2610.26288
AB - We give a direct Fox-calculus proof of the known rational asphericity theorem for every finite presentation complex with one nontrivial relator, including proper powers. The proof uses completed Fox derivatives and the absence of zero divisors in the completed algebra of a prounipotent group. Exact scalar extension to the finite adeles then yields adelic asphericity and a product decomposition of the higher homotopy groups of integral Bousfield--Kan completion. We also prove rational and pro-p analogues of Mannan's formula for finite two-complexes, with their natural topologies and fundamental group actions. The rational cohomological Whitehead criterion is proved directly. For a discrete one-relator group with relator not a proper power, we show that the tower L_2ν_n(G) is Mittag--Leffler. Consequently limone_nL_2ν_n(G)=0, and π_2(Z_∞ BG) is naturally the inverse limit of the nilpotent Baer invariants. We identify this Baer limit canonically and topologically with the kernel of the integral augmentation-completed Fox operator and with the product of its local pro-p kernels. The comparison yields a torsion-free profinite homotopy module, vanishing inverse limits of the finite-stage transgression kernels, and explicit lifting obstructions. These results provide a homotopy-theoretic framework for the question of Serre in Gildenhuys's corrected formulation.
ER -