Geometric interpretation of simplicial formulas for the Chern-Simons invariant

TL;DR

This paper provides a geometric interpretation of a combinatorial formula for the Chern-Simons invariant of 3-manifolds, using explicit flat connections on tetrahedra without relying on group homology or Bloch groups.

Contribution

It introduces a direct geometric approach to understanding Neumann's combinatorial formula for the Chern-Simons invariant, avoiding complex algebraic tools.

Findings

Explicit flat connections constructed for each tetrahedron.

A new geometric interpretation of the combinatorial formula.

Simplifies understanding of the Chern-Simons invariant in this context.

Abstract

We give a direct interpretation of Neumann's combinatorial formula for the Chern-Simons invariant of a 3-manifold with a representation in PSL(2,C) whose restriction to the boundary takes values in upper triangular matrices. Our construction does not involve group homology or Bloch group but is based on the construction of an explicit flat connection for each tetrahedron of a simplicial decomposition of the manifold.