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$\DeclareMathOperator\SL{SL}$In Soulé's paper "The cohomology of $\SL_3(\mathbb{Z})$" the cohomology ring $H^*(\SL(3,\mathbb{Z}),\mathbb{Z})_{(2)}$ is determined in Theorem 4.iv. I'm wanting to determine the ring structure of the 2-torsion (also called localization 2) $$H^*(\SL(3,\mathbb{Z}),\mathbb{Z}_2)_{(2)}.$$ It seems that the best strategy is to mimic the proof of 4.iv for the case of $\mathbb{Z}_2$ coefficients. Given the proof of 4.iv, this would seemingly require analogs of Proposition 2.iii and Proposition 4.ii in the case $\mathbb{Z}_2$ coefficients.

The cohomology ring of $D_4$ with $\mathbb{Z}_2$ coefficients is known, so we have the analog of 2.ii for $\mathbb{Z}_2$ coefficients. Furthermore, the ring structure of the symmetric group $S_4$ with $\mathbb{Z}_2$ coefficients is known, which should give us the analog of 2.iii. However, I'm not sure how to transfer things over for 4.ii. Maybe there is a more straightforward way to determine the ring structure of $H^*(\SL(3,\mathbb{Z}),\mathbb{Z}_2)_{(2)}$?

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EDIT: The below of course does not help with the ring structure and only describes the mod $2$ cohomology additively.

Since you wrote "2-torsion", I'm going to assume that $\mathbb{Z}_2$ refers to $\mathbb{Z}/2$ (rather than the $2$-adic integers, which I think is the preferred interpretation of the notation $\mathbb{Z}_2$ in those parts of algebraic topology close to algebra nowadays).

In that case, the short exact sequence $0\to \mathbb{Z} \to \mathbb{Z}\to \mathbb{Z}/2\to 0$ induces a long exact sequence on cohomology groups, exhibiting $H^*(X;\mathbb{Z}/2)$ for any $X$ as sitting in a short exact sequence

$$ 0 \to H^*(X;\mathbb{Z})/2 \to H^*(X;\mathbb{Z}/2) \to \text{2-tors in } H^{*+1}(X;\mathbb{Z})\to 0 $$

which is (noncanonically) split since all terms are $\mathbb{F}_2$-vector spaces. This allows you to describe cohomology with $\mathbb{Z}/2$-coefficients in terms of cohomology with $\mathbb{Z}$-coefficients.

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  • $\begingroup$ Thanks for the response. Yes, $\mathbb{Z}_2$ means the integers mod 2 here. Also, I should have been more clear in my original question, but I’m looking for the ring structure of $H^*(SL(3,\mathbb{Z}),\mathbb{Z}_2)_{(2)}$. $\endgroup$
    – Noah B
    Commented Jun 2, 2023 at 7:27
  • $\begingroup$ Ah, right, sorry.Yeah, the Bockstein sequence alone does not determine how elements coming from the right-hand summand multiply. $\endgroup$
    – Achim Krause
    Commented Jun 2, 2023 at 7:35
  • $\begingroup$ Quite a lot of the multiplicative structure should follow from the restriction maps to finite subgroups. But without doing the computations, it's not clear that this determines everything. $\endgroup$
    – Dave Benson
    Commented Jun 2, 2023 at 20:42
  • $\begingroup$ And yes, $\mathbb{Z}_2$ has become ambiguous. When the number theorists write this, they mean the $2$-adics, and many topoligists write it to mean the integers modulo two. Most confusing. $\endgroup$
    – Dave Benson
    Commented Jun 2, 2023 at 21:05
  • $\begingroup$ @DaveBenson That sounds interesting about restricting to finite subgroups. Could you expand a little bit so I can understand better? $\endgroup$
    – Noah B
    Commented Jun 7, 2023 at 2:15

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