论文标题
在$ \ mathrm {sl} _n（\ mathbb {z}} $
On the codimension-two cohomology of $\mathrm{SL}_n(\mathbb{Z})$
论文作者
论文摘要
Borel-serre证明了$ \ Mathrm {sl} _n（\ Mathbb {Z}）$是一个虚拟二重性$ n \选择2 $的虚拟二重性组，而Steinberg Module $ \ Mathrm {st} _n（\ Mathbb {Q}）$是其固定模块。该模块是与$ \ mathrm {sl} _n（\ Mathbb {q}）$相关的山雀构建的顶级同源组。我们确定了Steinberg模块的“关系之间的关系”。也就是说，我们构建了$ \ mathrm {sl} _n（\ Mathbb {z}）$ - 模块$ \ Mathrm {st} _n（\ Mathbb {q}）$的$ \ mathrm {sl} _n（\ mathbb {z}）$两个长度的明确分辨率。我们使用此部分决议来显示Codimension -2合理共同体学组$ H^{{n \ select 2} -2}（\ Mathrm {Slrm {Slrm {sl} _n（\ MathBb {Z}; \ Mathbb {Q}）$ \ Mathrm {s sl sl} 3 $。这解决了一个猜想的教堂 - 弗拉布 - 警察的案例。我们还为$ \ mathrm {sl} _n（\ Mathbb {z}）$的某些一致性子组的Codimension-1共同体产生了下限。
Borel-Serre proved that $\mathrm{SL}_n(\mathbb{Z})$ is a virtual duality group of dimension $n \choose 2$ and the Steinberg module $\mathrm{St}_n(\mathbb{Q})$ is its dualizing module. This module is the top-dimensional homology group of the Tits building associated to $\mathrm{SL}_n(\mathbb{Q})$. We determine the "relations among the relations" of this Steinberg module. That is, we construct an explicit partial resolution of length two of the $\mathrm{SL}_n(\mathbb{Z})$-module $\mathrm{St}_n(\mathbb{Q})$. We use this partial resolution to show the codimension-2 rational cohomology group $H^{{n \choose 2} -2}(\mathrm{SL}_n(\mathbb{Z});\mathbb{Q})$ of $\mathrm{SL}_n(\mathbb{Z})$ vanishes for $n \geq 3$. This resolves a case of a conjecture of Church-Farb-Putman. We also produce lower bounds for the codimension-1 cohomology of certain congruence subgroups of $\mathrm{SL}_n(\mathbb{Z})$.
