  The cohomology $H^*(\Gamma, E) $ of a torsion-free arithmetic subgroup
$\Gamma$ of the special linear $\mathbb{Q}$-group $\mathsf{G} = SL_n$ may be
interpreted in terms of the automorphic spectrum of $\Gamma$. Within this
framework, there is a decomposition of the cohomology into the cuspidal
cohomology and the Eisenstein cohomology. The latter space is decomposed
according to the classes $\{\mathsf{P}\}$ of associate proper parabolic
$\mathbb{Q}$-subgroups of $\mathsf{G}$. Each summand
$H^*_{\mathrm{\{P\}}}(\Gamma, E)$ is built up by Eisenstein series (or residues
of such) attached to cuspidal automorphic forms on the Levi components of
elements in $\{\mathsf{P}\}$. The cohomology $H^*(\Gamma, E) $ vanishes above
the degree given by the cohomological dimension $\mathrm{cd}(\Gamma) =
\frac{n(n-1)}{2}$. We are concerned with the internal structure of the
cohomology in this top degree. On the one hand, we explicitly describe the
associate classes $\{\mathsf{P}\}$ for which the corresponding summand
$H^{\mathrm{cd}(\Gamma)}_{\mathrm{\{\mathsf{P}\}}}(\Gamma, E)$ vanishes. On the
other hand, in the remaining cases of associate classes we construct various
families of non-vanishing Eisenstein cohomology classes which span
$H^{\mathrm{cd}(\Gamma)}_{\mathrm{\{\mathsf{Q}\}}}(\Gamma, \mathbb{C})$.
Finally, in the case of a principal congruence subgroup $\Gamma(q)$, $q =
p^{\nu} > 5$, $p\geq 3$ a prime, we give lower bounds for the size of these
spaces if not even a precise formula for its dimension for certain associate
classes $\{\mathsf{Q}\}$.
