#### Volume 22, issue 3 (2022)

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Smooth constructions of homotopy-coherent actions

### Yong-Geun Oh and Hiro Lee Tanaka

Algebraic & Geometric Topology 22 (2022) 1177–1216
##### Abstract

We prove that, for nice classes of infinite-dimensional smooth groups $G$, natural constructions 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.

##### Keywords
smooth approximation, Lie groups, group actions
##### Mathematical Subject Classification
Primary: 58B05, 58D05