  Watanabe disproved the 4-dimensional Smale conjecture by constructing
topologically trivial $D^{4}$-bundles over spheres and showing that they are
smoothly nontrivial using configuration space integrals. In this paper, we
define a new version of configuration space integrals that only relies on a
formal smooth structure on the $D^{4}$-bundle (i.e., a vector bundle structure
on the vertical tangent microbundle). It coincides with Watanabe's definition
when the $D^{4}$-bundle is smooth. We obtain several applications. First, we
give a lower bound (in terms of the graph homology) on the dimension of the
rational homotopy and homology groups of $\textrm{Top}(4)$ and
$\textrm{Homeo}(S^4)$ (the homeomorphism group of $\mathbb{R}^4$ and $S^4$). In
particular, this implies that $\textrm{Top}(4)$ and $\textrm{Homeo}(S^4)$ are
not rationally equivalent to any finite-dimensional CW complexes. Second, we
discover a generalized Miller-Morita-Mumford class $\kappa_{\theta}(\pi)\in
H^{3}(B;\mathbf{Q})$, which is defined for any topological 4-manifold bundle
$X\to E\to B$. This class obstructs the existence of a formal smooth structure
on the bundle. Third, we show that for any compact, orientable, smooth
4-manifold $X$ (possibly with boundary), the inclusion map from its
diffeomorphism group to its homeomorphism group is not rationally $2$-connected
(hence not a weak homotopy equivalence). This implies that the space of smooth
structures on $X$ has a nontrivial rational homotopy group in dimension 2.
