Chern Simons Theory and the volume of 3-manifolds

We give some applications of the Chern Simons gauge theory to the study of the set $\mathrm{v}\mathrm{o}\mathrm{l}(N,G)$ of volumes of all representations $\rho\co\pi_1N\to G$,where$N$isaclosedorientedthree-manifoldand$G$iseither${\rm Iso}e\t{\rm SL_2(\R)}$,theisometrygroupoftheSeifertgeometry,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 $\mathrm{v}\mathrm{o}\mathrm{l}(N,G)$? or weakly how to find non-zero elements in $\mathrm{v}\mathrm{o}\mathrm{l}(\text{t}N,G)$ for some finite cover $\text{t}N$ 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 $\mathrm{v}\mathrm{o}\mathrm{l}(N,G)$ when $N$ supports the Seifert geometry, and we find some non-zero values in $\mathrm{v}\mathrm{o}\mathrm{l}(N,G)$ for certain 3-manifolds with non-trivial geometric decomposition for either $G={\mathrm{I}\mathrm{s}\mathrm{o}}_{+}{\text{Hi}}^3$ 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 $\text{t}N$ with $\mathrm{v}\mathrm{o}\mathrm{l}(\text{t}N,{\mathrm{I}\mathrm{s}\mathrm{o}}_{+}{\text{Hi}}^3)\mathrm{\ne }\{0\}$. On the other hand, among such class, there are some $N$ with positive simplicial volume but $\mathrm{v}\mathrm{o}\mathrm{l}(N,{\mathrm{I}\mathrm{s}\mathrm{o}}_{+}{\text{Hi}}^3)=\{0\}$, yielding a negative answer to question (2) for hyperbolic volume.