Smooth constructions of homotopy-coherent actions

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.Comment: 23 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 ingredient

Localization and flexibilization in symplectic geometry

We introduce the critical Weinstein category -- the result of stabilizing the category of Weinstein sectors and inverting subcritical morphisms -- and for every finite collection $P$ of prime numbers, construct a $P$-flexibilization endofunctor. Our main result is that $P$-flexibilization is a localization of the critical Weinstein category, allowing us to characterize the essential image of the endofunctor by a universal property. This localization has the effect of replacing every Weinstein sector with one in which $P$ is invertible in the wrapped Fukaya category and hence we view it as a symplectic analogue of topological localization. We prove that this construction generalizes the flexibilization operation introduced by Cieliebak-Eliashberg and Murphy and is a variant of the `homologous recombination' construction of Abouzaid-Seidel. In particular, we give an h-principle-free proof that flexibilization is idempotent and independent of presentation, up to subcriticals and stabilization. Moreover, we show that $P$-flexibilization is symmetric monoidal, and hence gives rise to a new way of constructing commutative algebra objects from symplectic geometry. Our constructions work more generally for any finite collection of regular Lagrangian disks in $T^*D^n$, where the corresponding endofunctor in particular nullifies those disks as objects in the wrapped Fukaya category.Comment: Restructured introduction to improve exposition, gave greater detail on how to construct Weinstein structures on movies, and streamlined discussion of definition