Prime-localized Weinstein subdomains

Abstract

For any high-dimensional Weinstein domain and finite collection of primes, we construct a Weinstein subdomain whose wrapped Fukaya category is a localization of the original wrapped Fukaya category away from the given primes. When the original domain is a cotangent bundle, these subdomains form a decreasing lattice whose order cannot be reversed.Furthermore, we classify the possible wrapped Fukaya categories of Weinstein subdomains of a cotangent bundle of a simply connected, spin manifold, showing that they all coincide with one of these prime localizations. In the process, we describe which twisted complexes in the wrapped Fukaya category of a cotangent bundle of a sphere are isomorphic to genuine Lagrangians.

Localization and flexibilization in symplectic geometry

“…For example, our constructions are expected to give a purely symplectic construction of various localizations of the stable homotopy category, and symmetric monoidally so. We also view our results as an improvement on Murphy and Cieliebak-Eliashberg's flexibilization [2], and on work of Abouzaid-Seidel [1] and Lazarev-Sylvan [14]. Our work enables geometers to kill only certain P -torsion symplectic phenomena functorially and multiplicatively, in a process we call P -flexibilization.…”

“…Note that the construction in Theorem 1.3 differs significantly from the construction of X P of Abouzaid-Seidel [1] and related construction X P of the first two authors [14]. First, X P and X ×(T * D n ) P have different dimensions.…”

“…In our first result, we give a symplectic analog of the localization functor in classical topology. Recall that by the first two authors' previous work [14], for any collection P of primes, there exists a Weinstein sector (T * D n ) P so that T w W((T * D n ) P ) ∼ = T w W(T * D n )[ 1 P ] and (T * D n ) P is obtained by carving out certain Lagrangian disks from T * D n and hence is a subsector of T * D n . Next, we note that any Weinstein sector Z defines an endofunctor ×Z of Wein str ; this endofunctor commutes with stabilization and sends subcritical morphisms to subcritical morphisms and hence descends to Wein crit .…”

“…Remark 1.5. There are several possible models for the smooth topology of (T * D n ) P as described in the previous work [14], which all differ by subcritical or flexible Weinstein handles. In this paper, we can take a model for (T * D n ) P for which (T * D n ) P , T * D n are not diffeomorphic but are diffeomorphic up to smooth subcritical handles (but not Weinstein subcritical handles since they have different Fukaya categories).…”

“…However their construction X P depends very much on the Weinstein presentation of X and a priori is not homotopy invariant, while exhibiting little apparent functoriality. The first two authors [14] also introduced a variant construction X P by carving out certain Lagrangian disks from X located near all the co-cores of X; in particular, X P is a Weinstein subdomain of X and, by [14], the Viterbo transfer functor is a localization away from P . Since these carving disks had to be implanted near all the co-cores of X, the subdomain X P also depended very much on the Weinstein presentation of X and lacked homotopy invariance; see Example 4.2 for an illustration.…”