Cohomology of $GL_4 (\mathbb{Z})$ with nontrivial coefficients

Abstract

In this paper we compute the cohomology groups of GL 4 (Z) with coefficients in symmetric powers of the standard representation twisted by the determinant. This problem arises in Goncharov's approach to the study of motivic multiple zeta values of depth 4. The techniques that we use include Kostant's formula for cohomology groups of nilpotent Lie subalgebras of a reductive Lie algebra, Borel-Serre compactification, a result of Harder on Eisenstein cohomology. Finally, we need to show that the ghost class, which is present in the cohomology of the boundary of the Borel-Serre compactification, disappears in the Eisenstein cohomology of GL 4 (Z). For this we use a computationally effective version for the homological Euler characteristic of GL 4 (Z) with non-trivial coefficients. * 1 n k 1 1 . . . n km m , where k 1 + . . . + k m is called weight and m is called depth. Goncharov has described the cases of depth=2 [G2] and of depth=3 [G3]. He relates the space of motivic multiple zeta values of depth=2 and weight=n to the cohomology groups of GL 2 (Z) with coefficients in the (n − 2)-symmetric power of the standard representation V 2 , namely, toHe calls this a misterious relation between the multiple zeta values of depth=m and the "modular variety"In the paper [G3], he relates the spaces of motivic multiple zeta values of depth=3 and weight=n to the cohomology of GL 3 (Z) with coefficients in the (n−3)-symmetric power of the standard representation V 3 , namely,Goncharov has also related the case of multiple zeta values of depth=4 and weight=n to the computation of the cohomology of GL 4 (Z) with coefficients in the

Boundary and Eisenstein cohomology of $$\mathrm {SL}_3({\mathbb {Z}})$$

“…Now we will explain Eq. (19) in detail. The summation is over all possible block diagonal matrices A ∈ satisfying the following conditions:…”

“…5, we discuss this in detail. The importance of Euler charcateristic to study the space of Eisenstein cohomology has been discussed by the third author in [19]. For more details about Euler characteristic of arithmetic groups see [17,18].…”

Improved homological stability for certain general linear groups

“…For 𝑛 = 2, 3 these are folklore results (for example, [31,Theorem 42]). For 𝑛 = 4 one uses that 𝐻 * (GL 4 (ℤ); ℚ sign ) was computed by Horozov [19,Theorem 1.1]. For 𝑛 = 5, 6, 7 one uses [11,Theorem 7.3].…”

Euler characteristic and cohomology of $\mathrm{Sp}_4(\mathbb{Z})$ with nontrivial coefficients

“…Euler characteristic has been a useful tool to address the various problems in group cohomology. For example see [12]. We quickly review the basics about Euler characteristic.…”