Smooth constructions of homotopy-coherent actions

Abstract

We prove that, for nice classes of infinite-dimensional smooth groups G, natural con-structions in smooth topology and symplectic topology yield homotopically coherent group actions of G. This yields a bridge between infinite-dimensional smooth groups and homotopy theory.The result relies on two computations: one showing that the diffeological homotopy groups of the Milnor classifying space BG are naturally equivalent to the (continuous) homotopy groups, and a second showing that a particular strict category localizes to yield the homotopy type of BG.We then prove a result in symplectic geometry: these methods are applicable to the group of Liouville automorphisms of a Liouville sector. The present work is written with an eye toward Oh and Tanaka (2019), where our constructions 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.