Fibrancy of Symplectic Homology in Cotangent Bundles

TL;DR

This paper extends previous work to establish a Serre type spectral sequence converging to symplectic homology of cotangent bundle domains, linking local fiber-wise symplectic homology to the topology of the base manifold.

Contribution

It introduces a fiber-wise symplectic homology concept and constructs a spectral sequence relating symplectic homology to the base manifold's homology.

Findings

Spectral sequence converges to symplectic homology SH_*(M).

Page two of the spectral sequence is isomorphic to the homology of N with local coefficients.

Defines a graded local coefficient system on N based on fiber-wise symplectic homology.

Abstract

We describe how the result in [1] extends to prove the existence of a Serre type spectral sequence converging to the symplectic homology SH_*(M) of an exact Sub-Liouville domain M in a cotangent bundle T*N. We will define a notion of a fiber-wise symplectic homology SH_*(M,q) for each point q in N, which will define a graded local coefficient system on N. The spectral sequence will then have page two isomorphic to the homology of N with coefficients in this graded local system.