Chern Simons Theory and the volume of 3-manifolds

TL;DR

This paper explores how Chern-Simons gauge theory relates to the volumes of representations of 3-manifold fundamental groups, revealing new insights into geometric structures and their topological implications.

Contribution

It determines volume sets for certain 3-manifolds and shows these volumes encode gluing information, extending understanding beyond Gromov volume.

Findings

Identifies non-zero volume values for specific 3-manifolds.

Shows volumes satisfy Thurston's covering property in some cases.

Links positive Gromov volume to the existence of non-zero hyperbolic volume in finite covers.

Abstract

We give some applications of the Chern Simons gauge theory to the study of the set ${\rm vol}(N,G)$ of volumes of all representations $\rho\co\pi_1N\to G$, where $N$ is a closed oriented three-manifold and $G$ is either ${\rm Iso}_e\t{\rm SL_2(\R)}$, the isometry group of the Seifert geometry, or ${\rm Iso}_+{\Hi}^3$, the orientation preserving isometry group of the hyperbolic 3-space. We focus on three natural questions: (1) How to find non-zero values in ${\rm vol}(N, G)$? or weakly how to find non-zero elements in ${\rm vol}(\t N, G)$ for some finite cover $\t N$ of $N$? (2) Do these volumes satisfy the covering property in the sense of Thurston? (3) What kind of topological information is enclosed in the elements of ${\rm vol}(N, G)$? We determine ${\rm vol}(N, G)$ when $N$ supports the Seifert geometry, and we find some non-zero values in ${\rm vol}(N,G)$ for certain 3-manifolds with non-trivial geometric decomposition for either $G={\rm Iso}_+{\Hi}^3$ or ${\rm Iso}_e\t{\rm SL_2(\R)}$. Moreover we will show that unlike the Gromov simplicial volume, these non-zero elements carry the gluing information between the geometric pieces of $N$. For a large class 3-manifolds $N$, including all rational homology 3-spheres, we prove that $N$ has a positive Gromov simplicial volume iff it admits a finite covering $\t N$ with ${\rm vol}(\t N,{\rm Iso}_+{\Hi}^3)\ne \{0\}$. On the other hand, among such class, there are some $N$ with positive simplicial volume but ${\rm vol}(N,{\rm Iso}_+{\Hi}^3)=\{0\}$, yielding a negative answer to question (2) for hyperbolic volume.