Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.38735

The Quillen and sharbly Hopf algebra structures on Steinberg homology coincide

Urshita Pal·Sam Payne
View original source ↗Read source document ↗Export metadata ↓

Abstract

Two recent constructions independently give bigraded commutative Hopf algebra structures on the homology of $\mathrm{GL}(\mathbb{Z})$ 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 $BK(\mathbb{Z})$. We prove that the two structures coincide. Both are induced by join and parabolic restriction operations on Steinberg modules.

Explore related subjects

math.ATmath.KT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Urshita Pal, Sam Payne. 2026-09-30. The Quillen and sharbly Hopf algebra structures on Steinberg homology coincide. https://arxiv.org/abs/2609.38735

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On rectangular Toda brackets and Oda's extension problems

This paper tackles \textit{N. Oda}'s extension problems for the homotopy groups $π_{39}(S^{6})$, $π_{40}(S^{7})$, and $π_{41}(S^{8})$ localized at 2, the issues having eluded resolution for more than four decades. We introduce a tool for the theory of determinations of unstable homotopy groups, namely, the rectangular Toda bracket, by which we are able to solve the extension problems with respect to these three homotopy groups.

math.AT↗
arXiv2026

On Fico's Lemmata and the Homotopy Type of Certain Gyrations

We undertake to determine the homotopy type of gyrations of sphere products and of connected sums, thereby generalising results known in earlier literature as ''Fico's Lemmata'' which underpin gyrations in their original formulation from geometric topology. We provide applications arising from recasting these results into the modern homotopy theoretic setting.

math.AT↗
arXiv2026

Homological critical points for hypergraphs and simplicial complexes

In this paper, we study the homological critical points for persistence hypergraphs and persistence simplicial complexes. With the help of the relative homology groups, we define the homological critical points for persistence morphisms between persistence hypergraphs and persistence simplicial maps between persistence simplicial complexes. We prove some commutative diagrams of subset relations for the homological critical points of persistence hypergraphs as well as persistence morphisms between them. We also prove some commutative diagrams of subset relations for the homological critical points of persistence simplicial complexes as well as persistence simplicial maps between them. As examples, we use the parametric configuration spaces to construct the parametric independence complexes as the space of parametric packings and construct the parametric dominating hypergraphs as the space of parametric coverings.

math.AT↗