Homotopy Colimits of Semifree DG Categories and Fukaya Categories of Cotangent Bundles of Lens Spaces
Joint work with Sangjin Lee
Preprint arXiv:2109.03411
This paper consists of two parts:
1) We describe a practical formula for homotopy colimits of semifree dg categories where dg categories are considered either up to quasi-equivalence, pretriangulated equivalence, or Morita equivalence. This has an application in symplectic geometry using the result of Ganatra-Pardon-Shende which roughly states that the wrapped Fukaya category of a Weinstein manifold W is the homotopy colimit of wrapped Fukaya categories of a (sectorial) cover of W. After a (non-characteristic) deformation, the cover can be chosen in such a way that each Weinstein sector in the cover is with an arboreal skeleton whose wrapped Fukaya category is readily described by a semifree dg category (up to pretriangulated equivalence). Hence, our homotopy colimit formula directly calculates, at least, the wrapped Fukaya categories of cotangent bundles and plumbing spaces.
2) We use our homotopy colimit formula to calculate the wrapped Fukaya categories of cotangent bundles of lens spaces. We apply various arguments to distinguish these categories and conclude that the endomorphism algebra of the cotangent fibre in the wrapped Fukaya category is a full invariant of the homotopy type of the lens spaces. By Abouzaid-Kragh, we know that cotangent bundles of homotopic but not homeomorphic lens spaces are not symplectomorphic. Hence, the quasi-equivalences we construct between wrapped Fukaya categories of such lens spaces are not coming from geometry.