Given a finite field $\mathbb{F}$ with $|\mathbb{F}|=q=p^m\geq4$ where $p=\text{char}(\mathbb{F})$, I'm wondering is there a characterization of the kernel of the map $f:H_3(\text{GL}_3(\mathbb{F}))\to H_3(\text{GL}_4(\mathbb{F}))$?
Is it an isomorphism?(Here $H_n(G)$ means the $n$-th integral homology of the group $G$)
Here is some material I have known. Sprehn&Wahl says $f$ must be a surjection. In the paper [1], it is shown that $f$ would be an isomorphism if $\mathbb{F}$ is an infinite field. But in the talk [2], it is asserted that the techniques in [1] doesn't apply for finite fields. And with the result of Galatius-Kupers-Randal-Williams, it is deduced that $f$ induces an isomorphism on $p$-primary part. I'm wondering what happens with the $\text{mod } l$ homology where $l\neq p$.
The reason I raise this question is that in the Chapter.VI in Weibel's K-book, remark 5.12.1, he asserts the map $\varphi$ could extend to a map $H_3(\text{GL}(\mathbb{F}))\to B(F)$. There one works on a general field $F$ with $|F|>3$. But all I can get now is $ H_3(\text{GL}(\mathbb{F}))=H_3(\text{GL}_4(\mathbb{F})) $ and if we want to extend $\varphi$ to $H_3(\text{GL}(\mathbb{F}))$ we have to work on $\ker(f)$.
reference:
[1]Yu. P. Nesterenko and A. A. Suslin, Homology of the general linear group over a
local ring, and Milnor’s K-theory, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1,
121–146. MR 992981
[2]Alexander Kupers, NRW TALK: $E_\infty$-CELLS AND THE HOMOLOGY OF GENERAL
LINEAR GROUPS OVER FINITE FIELDS
Best Answer
The original paper by Suslin is available here and with some effort you should be able to read it.
If $|\mathbb{F}| = p^r$ then the homology groups $H_*(GL_n(\mathbb{F});\mathbb{F}_\ell)$ with $\mathbb{F}_\ell$-coefficients for $\ell \neq p$ were computed completely by Quillen, in Theorem 3 of On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field. You'll see that the stabilisation map $$H_*(GL_{n-1}(\mathbb{F});\mathbb{F}_\ell) \to H_*(GL_n(\mathbb{F});\mathbb{F}_\ell)$$ is always injective and surjective in a range $* < 2n-1$. In particular, the map $$H_3(GL_3(\mathbb{F});\mathbb{F}_\ell) \to H_3(GL_4(\mathbb{F});\mathbb{F}_\ell)$$ is an isomorphism. Combined with our improvement or Sprehn-Wahl's this gives what you want as long as $p^r \geq 4$.