UC Irvine

Symplectic manifolds have many connections to complex manifolds but lack many of thenice properties that come with a complex structure, especially the fact that symplectic manifolds have no local properties that can differentiate them from other symplectic manifolds. One of the most common global invariants of manifolds is the de Rham cohomology, but this can only detect the topological structure of the manifold and discards the additional symplectic structure. This work expands upon the definition of a cohomology theory specific to symplectic manifolds, specifically based on a viewpoint which comes from the concept of a mapping cone from homological algebra. We demonstrate how common operations on differential forms can be extended to these cone cohomologies with particular focus on the blowup along a symplectic submanifold. Additionally, we show how this approach can be extended to other types of manifolds using nilmanifolds as an example.