The first step towards symplectic homotopy theory

Abstract

We consider two categories related to symplectic manifolds: 1. Objects are symplectic manifolds and morphisms are symplectic embeddings. 2. Objects are symplectic manifolds endowed with compatible almost complex structure and morphisms are pseudoholomorphic maps. We define new homotopy theories for these categories. In particular, we give definitions of homotopy equivalent symplectic manifolds and define new cohomology theories. Theses cohomology theories are functorial, homotopy invariant and have other interesting properties. We also construct triangulated persistence category of symplectic manifolds. This allows to apply machinery developed Biran, Cornea, and Zhang and define distances between symplectic manifolds.