Configuration space integrals and formal smooth structures

TL;DR

This paper introduces a new version of configuration space integrals based on formal smooth structures, leading to new insights into the topology of 4-manifolds, the structure of homeomorphism groups, and obstructions to smooth structures.

Contribution

It defines a formal smooth structure-based integral, providing new bounds on topological groups and introducing a generalized Miller-Morita-Mumford class as an obstruction.

Findings

Topological groups $ extrm{Top}(4)$ and $ extrm{Homeo}(S^4)$ are not rationally equivalent to finite CW complexes.

A new obstruction class $ heta$ prevents certain bundles from having formal smooth structures.

The space of smooth structures on a 4-manifold has nontrivial rational homotopy in dimension 2.

Abstract

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.