The method of "configuration space integrals" gives invariants of certain smooth fiber bundles. It was shown by Watanabe (2018) that these invariants can detect exotic fiber bundles. I will discuss the relationship between this and the philosophy "a diffeomorphism lifts to maps between (compactified) configuration spaces, but a homeomorphism does not".
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/TopologySeminar