  If a closed orientable manifold (resp. rational Poincar\'e duality space) $X$
receives a map $Y \to X$ from a formal manifold (resp. space) $Y$ that hits a
fundamental class, then $X$ is formal. The main technical ingredient in the
proof states that given a map of $A_\infty$-algebras $A\to B$ admitting a
homotopy $A$-bimodule retract, formality of $B$ implies that of $A$.
