$\DeclareMathOperator\SL{SL}$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 $H^*(\SL(3,\mathbb{Z}),\mathbb{Z}_2)_{(2)}$?

$\DeclareMathOperator\SL{SL}$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 $H^*(\SL(3,\mathbb{Z}),\mathbb{Z}_2)_{(2)}$?

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 $H^*(SL(3,\mathbb{Z}),\mathbb{Z}_2)_{(2)}$?

$\DeclareMathOperator\SL{SL}$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 $H^*(\SL(3,\mathbb{Z}),\mathbb{Z}_2)_{(2)}$?

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 $H^*(SL(3,\mathbb{Z}),\mathbb{Z}_2)_{(2)}$?

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 $H^*(SL(3,\mathbb{Z}),\mathbb{Z}_2)_{(2)}$?