PreprintCC BY 4.0arXiv
The Quillen and sharbly Hopf algebra structures on Steinberg homology coincide
Abstract
Two recent constructions independently give bigraded commutative Hopf algebra structures on the homology of GL(ℤ) with rational Steinberg coefficients. One, by Ash--Miller--Patzt, uses the sharbly resolution. The other, by Brown--Chan--Galatius--Payne, uses the Quillen spectral sequence and rank-filtered maps on BK(ℤ). We prove that the two structures coincide. Both are induced by join and parabolic restriction operations on Steinberg modules.
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BibTeX
@misc{pal2026quillen,
title = {{The Quillen and sharbly Hopf algebra structures on Steinberg homology coincide}},
author = {Urshita Pal and Sam Payne},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38735},
eprint = {2609.38735},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38735},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Quillen and sharbly Hopf algebra structures on Steinberg homology coincide
AU - Urshita Pal
AU - Sam Payne
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.38735
UR - https://arxiv.org/abs/2609.38735
AB - Two recent constructions independently give bigraded commutative Hopf algebra structures on the homology of GL(ℤ) with rational Steinberg coefficients. One, by Ash--Miller--Patzt, uses the sharbly resolution. The other, by Brown--Chan--Galatius--Payne, uses the Quillen spectral sequence and rank-filtered maps on BK(ℤ). We prove that the two structures coincide. Both are induced by join and parabolic restriction operations on Steinberg modules.
ER -