Fundamental classes of 3-manifold group representations in SL(4,R)

TL;DR

This paper computes the fundamental classes of certain 3-manifold group representations in SL(4,R), explores their implications for character varieties, and reveals the existence of knots with many components having zero Chern-Simons invariant.

Contribution

It introduces methods to compute fundamental classes for SL(4,R) representations factoring through SL(2,C) and analyzes the structure of SL(n,C)-character varieties.

Findings

Computed fundamental classes in the extended Bloch group for specific representations.

Identified multiple connected components in SL(4,R)-character varieties.

Found knots with arbitrarily many components of vanishing Chern-Simons invariant.

Abstract

We compute the fundamental class (in the extended Bloch group) for representations of fundamental groups of 3-manifolds to SL(4,R) that factor over SL(2,C), in particular for those factoring over the isomorphism PSL(2,C) = S0(3,1). We also discuss consequences for the number of connected components of SL(4,R)-character varieties, and we show that there are knots with arbitrarily many components of vanishing Chern-Simons invariant in their SL(n,C)-character varieties