All spaces below will be assumed simply connected. A continuous map is a rational equivalence if it induces an isomorphism of the rational homology groups. Two spaces are rationally equivalent if they can be connected by a sequence of rational equivalences (not necessarily all going in the same direction). Now suppose $X$ and $Y$ are rationally equivalent smooth compact manifolds without boundary. Can one choose a sequence of rational equivalences from $X$ to $Y$ that only passes through smooth compact manifolds of dimension $\dim X=\dim Y$?
Note: if $X$ and $Y$ are CW-complexes, or more generally, compactly generated Hausdorff spaces, then $X$ is rationally equivalent to $Y$ if and only if the $\mathbb{Q}$-localizations $X_\mathbb{Q}$ and $Y_\mathbb{Q}$ are homotopy equivalent, see e.g. Felix, Halperin, Thomas, Rational homotopy theory, Proposition 9.8. If this is the case, then there is a roof $X\to X_\mathbb{Q}\gets Y$ of rational equivalences. The middle of the roof is of course an infinite complex (except when it is a point).