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 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, NEW TALK: $E_\infty$-CELLS AND THE HOMOLOGY OF GENERAL LINEAR GROUPS OVER FINITE FIELDS

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

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 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 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, NEW TALK: $E_\infty$-CELLS AND THE HOMOLOGY OF GENERAL LINEAR GROUPS OVER FINITE FIELDS

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 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, NEW TALK: $E_\infty$-CELLS AND THE HOMOLOGY OF GENERAL LINEAR GROUPS OVER FINITE FIELDS