Smooth constructions of homotopy-coherent actions
arXiv:2003.06033 · doi:10.2140/agt.2022.22.1177
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 [OT19], 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.
arXiv comment23 pages. Comments welcome! Portions of this work previously appeared in arXiv:1911.00349v2; that previous work has been split into multiple papers (including this one) to better explicate the ingredients
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