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Smooth constructions of
homotopy-coherent actions
Yong-Geun Oh and Hiro Lee Tanaka |
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Algebraic & Geometric Topology 22 (2022) 1177–1216
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Abstract |
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We prove that, for nice classes of infinite-dimensional smooth groups , natural constructions in smooth topology and symplectic topology yield homotopically coherent group actions of . 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 are naturally equivalent to the (continuous) homotopy groups, and a second showing that a particular strict category localizes to yield the homotopy type of . 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. |
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Keywords
smooth approximation, Lie groups, group actions
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Mathematical Subject Classification
Primary: 58B05, 58D05
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References |
Publication
Received: 15 April 2020
Revised: 16 April 2021
Accepted: 1 June 2021
Published: 25 August 2022
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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 |
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| http://www.hiroleetanaka.com | |
