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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

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    $\begingroup$ Can't you get the mod-l-homology case from Quillen's compuations? $\endgroup$
    – tj_
    Feb 28 at 13:00

1 Answer 1

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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 $p^r \geq 4$.

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  • $\begingroup$ Thx a lot for your answer! I'm not very familiar with Quillen's result. So if i understand it correctly, after ruling out two cases $|F|=2,3$, we could extend Suslin's stability result that $H_n(\text{GL}_n(F))=H_n(\text{GL}(F))$ to a general field. Is that right? $\endgroup$ Mar 1 at 3:19

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