Homology of $\GL_n$ over infinite fields outside the stability range

TL;DR

This paper investigates the homological stability range for general linear groups over infinite fields, providing conjectural estimates for kernels and cokernels and proving these for low dimensions.

Contribution

It introduces conjectural estimates for the kernels and cokernels of homology maps of GL_n over infinite fields and verifies these conjectures for n ≤ 4.

Findings

Conjectural bounds for kernels and cokernels in homology maps.

Proof of conjectures for n ≤ 4.

Insights into homology stability outside the classical range.

Abstract

For an infinite field $F$, we study the kernel of the map $H_{n}(\mathrm{GL}_{n-1}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big]) \to H_{n}(\mathrm{GL}_{n}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big])$ and the cokernel of $H_{n+1}\Big(\mathrm{GL}_{n-1}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big]\Big) \to H_{n+1}\Big(\mathrm{GL}_{n}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big]\Big)$. We give conjectural estimates of these kernels and cokernels and prove our conjectures for $n\leq 4$.