  When $\mathcal{M}$ is a smooth, oriented, compact and simply connected
manifold, Luc Menichi has shown that $HH^\ast(C^\ast(\mathcal{M};
\mathbb{F}))$, the Hochschild cohomology of the singular cochain complex of
$\mathcal{M}$ is a Batalin-Vilkovisky algebra. Using the properties of algebras
over the Barratt-Eccles operad, we show that this results holds even when the
manifold is not simply connected. Furthermore, we prove a similar result for
pseudomanifolds. Namely, we explain why $HH^\ast_\bullet(\widetilde
N^\ast_\bullet(X;\mathbb{F}))$, the Hochschild cohomology of the blown-up
intersection cochain complex of a compact, oriented pseudomanifold $X$, is
endowed with a Batalin-Vilkovisky algebra structure.
