Pontryagin-Weiss classes and a rational decomposition of spaces of homeomorphisms

We construct a rational homotopy pullback decomposition for variants of the classifying space of the group of homeomorphisms for a large class of manifolds. This has various applications, including a rational section of the stabilisation map $\mathrm{T}\mathrm{o}\mathrm{p}(d)\to \mathrm{T}\mathrm{o}\mathrm{p}$ of the space of homeomorphisms of ${\mathbf{R}}^d$ for $d\ge 6$, or a new method to construct topological bundles and detect characteristic classes thereof. Some steps in the proofs may be of independent interest, such as the construction of a nullhomotopy of the twice-iterated stabilisation map for the space of orientation-preserving derived automorphisms of the rational $E_d$-operad, results on recovering boundaries of manifolds from the interior in the context of embedding calculus, or a treatment of tensor products of truncated $\mathrm{\infty }$-operads.