Smooth approximation for classifying spaces of diffeomorphism groups

Yong-Geun Oh, Hiro Lee Tanaka

Published 2020-03-12Version 1

We prove a smooth approximation theorem for classifying spaces of certain infinite-dimensional smooth groups. More precisely, using the framework of diffeological spaces, we show that the smooth singular complex of a classifying space BG is weakly homotopy equivalent to the (continuous) singular complex of BG when G is a diffeomorphism group of a compact smooth manifold. In particular, the smooth homotopy groups of BG are naturally isomorphic to the usual (continuous) homotopy groups of BG. On top of a computation of homotopy groups, our methods yield a way to construct homotopically coherent actions of G using infinity-categorical techniques. We discuss some generalizations and consequences of this result with an eye toward [OT19], where we show that higher homotopy groups of symplectic automorphism groups map to Fukaya-categorical invariants, and where we prove a conjecture of Teleman from the 2014 ICM in the Liouville and monotone settings.