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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
References
Publication
Received: 15 April 2020
Revised: 16 April 2021
Accepted: 1 June 2021
Published: 25 August 2022
Authors
Yong-Geun Oh
Center for Geometry and Physics
Institute for Basic Science
Pohang
South Korea
Department of Mathematics
Pohang University of Science and Technology
Pohang
South Korea
Hiro Lee Tanaka
Department of Mathematics
Texas State University
San Marcos, TX
United States
http://www.hiroleetanaka.com