The Euler class in the Simplicial de Rham Complex

TL;DR

This paper constructs a cocycle in the simplicial de Rham complex representing the Euler class and applies it to create a Lie algebra cocycle on $L\mathfrak{so}(4)$, linking topology with Lie algebra theory.

Contribution

It introduces a new cocycle in the simplicial de Rham complex for the Euler class and demonstrates its application to Lie algebra cocycles.

Findings

Constructed a cocycle representing the Euler class.

Developed a Lie algebra cocycle on $L\mathfrak{so}(4)$.

Bridged topology and Lie algebra structures.

Abstract

We exhibit a cocycle in the simplicial de Rham complex which represents the Euler class. As an application, we construct a Lie algebra cocycle on $L\mathfrak{so}(4)$.