Let $S$ be a locally symmetric space, not necessarily compact, and $\overline{S}$ be its Borel-Serre compactification. Let $\partial S$ be the boundary of $S$. Let $\widetilde{\mathbb{C}}$ be the coefficient system of locally constant sheaf of one-dimensional vector space $\mathbb{C}$ attached to $S(\mathbb{C})$. One then has a long-exact sequence of cohomology groups $\ldots \to H^\bullet_c(S(\mathbb{C}), \widetilde{\mathbb{C}}) \to H^\bullet(\overline{S},(\mathbb{C}), \widetilde{\mathbb{C}}) \to H^\bullet(\partial \overline{S}(\mathbb{C}), \widetilde{\mathbb{C}}) \to \ldots $.
Is it, in general, true that the cohomology of the boundary in degree $1$ does not vanish, i.e., $H^1_\partial \neq 0$?
