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- W1489962015 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 $rhocopi_1Nto G$, where $N$ is a closed oriented three-manifold and $G$ is either ${rm Iso}_et{rm SL_2(R)}$, 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 ${rm Iso}_et{rm SL_2(R)}$. 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." @default.
- W1489962015 created "2016-06-24" @default.
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- W1489962015 date "2011-11-26" @default.
- W1489962015 modified "2023-09-27" @default.
- W1489962015 title "Chern Simons Theory and the volume of 3-manifolds" @default.
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