2011/11/26 by Pierre Derbez, Derbez, Pierre, Shicheng Wang +1
Mathematics · #51H20 #57M50 #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.DG #math.GT #msc:51H20 #msc:57M50
paper · pdf · doi:10.48550/arxiv.1111.6153
34 pages
arxiv created 2011/11/26 · openalex publication_date 2011/11/26 · arxiv updated 2011/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ρ\coπ1N→ G, where N is a closed oriented three-manifold and G is either \rm Isoe\t\rm SL2(\R), the isometry group of the Seifert geometry, or \rm Iso+\Hi3, 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+\Hi3 or \rm Isoe\t\rm SL2(\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+\Hi3)≠ \0\. On the other hand, among such class, there are some N with positive simplicial volume but \rm vol(N,\rm Iso+\Hi3)=\0\, yielding a negative answer to question (2) for hyperbolic volume.