Abstract: I will describe work in progress with A. Kupers in which we construct a rational pullback decomposition of (a variant of) the space of homeomorphisms of a high-dimensional simply connected spin manifold. As one of the applications, we strengthen a result of Weiss by showing that the stabilisation map Top(d) -> Top of the homeomorphism group of Euclidean d-space has rationally a section as long as d>5.