# The third homology stability 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

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

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 $$p^r \geq 4$$.
• 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? Mar 1 at 3:19