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

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