# On the Symplectic Structures in Frame Bundles and the Finite Dimension of Basic Symplectic Cohomologies

## TL;DR

This paper explores symplectic structures on frame bundles, classifies invariant 2-forms, and addresses finiteness properties of basic symplectic cohomologies, providing new constructions and a valid proof.

## Contribution

It introduces a classification of invariant 2-forms on the symplectic group and establishes a valid proof for finiteness theorems in basic symplectic cohomologies.

## Key findings

- Constructed invariant 2-forms on the symplectic group.
- Defined symplectic forms on quotient spaces by maximal tori.
- Provided a valid proof for finiteness of basic symplectic cohomologies.

## Abstract

We present a construction (and classification) of certain invariant 2-forms on the real symplectic group. They are used to define a symplectic form on the quotient by a maximal torus and to "lift" a symplectic structure from a symplectic manifold to the bundle of frames. This is a by-product of a failed attempt to prove certain finiteness theorems for basic symplectic cohomologies. In the last part of the paper we include a valid proof.