Symplectic flexibility and the Grothendieck group of the Fukaya category

Abstract

In the previous work, the author used symplectic flexibility techniques to prove an upper bound on the number of generators of the wrapped Fukaya category of a high-dimensional, simply connected Weinstein domain. In this article, we give a purely categorical proof of this result for all Weinstein domains via Thomason's theorem on split-generating subcategories and the Grothendieck group. In the process, we prove that there is a surjective map from singular cohomology to the Grothendieck group of the Fukaya category, relate the acceleration map to symplectic cohomology and the Dennis trace map and upgrade Abouzaid's splitgeneration criterion to a generation criterion for Weinstein domains. As a geometric analog of Thomason's result, we also produce the first examples of exotic presentations for cotangent bundles as Weinstein handle attachments along infinitely many non-fillable Legendrians.

Lagrangian cobordism functor in microlocal sheaf theory I

“…Remark 1.7. As there are examples whose partially wrapped Fukaya category only has higher dimensional representations [37,38], by the equivalence between Fukaya categories and sheaf categories [24] and the fact that [44,Theorem 3.21] (see Subsection 2.5)…”

“…Remark As there are examples whose partially wrapped Fukaya category only has higher dimensional representations [37, 38], by the equivalence between Fukaya categories and sheaf categories [24] and the fact that [44, Theorem 3.21] (see Subsection 2.5) $$$$\begin{equation*} \mu {{\rm Sh}}_{{\mathfrak{c}}_{X}}^{b}({\mathfrak{c}}_{X})\simeq {\mathrm{Fun}}^{\mathrm{ex}}(\mu {{\rm Sh}}_{{\mathfrak{c}}_{X}}^{c}({\mathfrak{c}}_{X})^{\mathrm{op}},\mathrm{Perf}(\mathbb{k})), \end{equation*}$$$$this corollary is expected to be stronger than the result in [8]. Note that there are also examples whose Legendrian contact homology is nontrivial but has only higher dimensional representations [58].…”