This question is about the computation of $H_2(Sp(2g, \mathbb{Z}_2), \mathbb{Z})$$H_2(Sp(2g, \mathbb{Z}/2), \mathbb{Z})$, where $Sp(2g, \mathbb{Z}_2)$$Sp(2g, \mathbb{Z}/2)$ is the group of symplectic $2g \times 2g$ matrices over $\mathbb{Z}_2$$\mathbb{Z}/2$. With respect to this computation I have seen quotations to the following two papers:
- Stein "Surjective stability in dimension $0$ for $K_2$ and related functors" (https://www.dropbox.com/s/jcy7sep2hpxy7gt/Stein.pdf?dl=0)
- Steinberg "Générateurs, relations et revêtements de groupes algébriques" (https://www.dropbox.com/s/jidchwwjssnmmlm/Steinberg1.pdf?dl=0)
According to the quotations of Stein's and Steinberg's papers the result is that $H_2(Sp(2g, \mathbb{Z}_2), \mathbb{Z}) = 0$$H_2(Sp(2g, \mathbb{Z}/2), \mathbb{Z}) = 0$ for $g \geq 3$.
My two concerns are:
The papers by Stein and Steinberg do not directly give the computation of $H_2(Sp(2g, \mathbb{Z}_2), \mathbb{Z})$$H_2(Sp(2g, \mathbb{Z}/2), \mathbb{Z})$. So my first question is if it is really possible to extract this result from these papers?
There is a discrepancy in the case $g=3$ between the quotations of Stein's and Steinberg's papers - which say that $H_2(Sp(6, \mathbb{Z}_2), \mathbb{Z}) =0$$H_2(Sp(6, \mathbb{Z}/2), \mathbb{Z}) =0$ - and the computation of this group done with GAP (the computational system for discrete algebra). GAP gives $H_2(Sp(6, \mathbb{Z}_2), \mathbb{Z}) = \mathbb{Z}_2$$H_2(Sp(6, \mathbb{Z}/2), \mathbb{Z}) = \mathbb{Z}/2$. Checking on GAP, the process that the computer goes through to get the computation looks correct. Is there an error in Stein's and/or Steinberg's paper or is it a misinterpretation of the case $g=3$ in the quotation?
