The Quillen and sharbly Hopf algebra structures on Steinberg homology coincide
Urshita Pal, Sam Payne
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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