Formality is preserved under domination

If a closed orientable manifold (resp. rational Poincaré 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 $A_{\mathrm{\infty }}$-algebras $A\to B$ admitting a homotopy $A$-bimodule retract, formality of $B$ implies that of $A$.