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](https://arxiv.org/abs/1812.08742v2) 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]](https://people.math.harvard.edu/%7Ekupers/notes/nrwtalk.pdf), it is asserted that the techniques in [1] doesn't apply for finite fields. And with the result of [Galatius-Kupers-Randal-Williams](https://arxiv.org/abs/1810.11931v1), 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](https://sites.math.rutgers.edu/%7Eweibel/Kbook/Kbook.VI.pdf), 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](https://people.math.harvard.edu/%7Ekupers/notes/nrwtalk.pdf)*



The third homology stability of general linear groups over finite fields
The original paper by Suslin is available [here](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tm&paperid=1914&option_lang=eng) 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](https://www.jstor.org/stable/1970825?seq=1#metadata_info_tab_contents). 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 $p^r \geq 4$.


