Improved homological stability for certain general linear groups

Abstract

We prove that the general linear groups of the integers, Gaussian integers, and Eisenstein integers satisfy homological stability of slope 1 when using double-struckZfalse[1/2false]$\mathbb {Z}[{\nicefrac{1}{2}}]$‐coefficients and of slope 1false2/1false3${\nicefrac{2}{3}}$ when using Z$\mathbb {Z}$‐coefficients.

Actions of Automorphism Groups of Free Groups on Spaces of Jacobi Diagrams. Ii

“…21 by Theorem 8.6, we obtainH 1 (IO n , A ′′ 2 (3)) ∼ = V 1,31 ⊕ V 0,21 . □ By Theorems 10.2, 10.3 and 10.4, the surjectivity of the map between the second homology of IA n and IO n follows from an argument similar to Lemma 10.1.Proposition 10.5.…”

“…Before stating our result, we will recall Borel's stability and vanishing theorem. Theorem 9.3 (Borel [4,5], Li-Sun [22] and Kupers-Miller-Patzt [21]). We have…”

The stable cohomology of self-equivalences of connected sums of products of spheres

“…Combining Propositions 5.3 and 5.6, we obtain, in the stable range an isomorphism of graded -modules which is compatible with the stabilization maps and the algebra/module structures. It follows from work of Borel [Bor74] (see also [KMP22, Theorem A] 3 ) that the canonical map is an isomorphism for ; in particular, we can replace with in the formula above.…”

Some rational homology computations for diffeomorphisms of odd‐dimensional manifolds

“…A similar bound can also deduced from our argument. The referee pointed us to the papers [48] and [49]. Using the results of these instead of the more classical [77] and [6], one can improve the bound in B.…”