2016/10/19 by Paul Gustafson, Gustafson, Paul
Mathematics · #18D10 #20F36 #57M99 #Advanced Algebra and Geometry #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #math.GT #math.QA #msc:18D10 #msc:20F36 #msc:57M99
paper · pdf · doi:10.48550/arxiv.1610.06069
Added more detail to the proof of the main theorem and to various definitions. Fixed definition of composition map. Clarified unstrictified colored graph construction
openalex publication_date 2016/10/19 · arxiv created 2017/11/12 · arxiv updated 2017/11/15 · openalex created_date 2023/03/19 · openalex updated_date 2026/07/28
We show that any twisted Dijkgraaf-Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image. This generalizes work of Etingof, Rowell and Witherspoon showing that the braid group images are finite. In particular, our result answers their question regarding finiteness of images of arbitrary mapping class group representations in the affirmative. Our approach is to translate the problem into manipulation of colored graphs embedded in the given surface. To do this translation, we use the fact that any twisted Dijkgraaf-Witten representation associated to a finite group G and 3-cocycle ω is isomorphic to a Turaev-Viro-Barrett-Westbury (TVBW) representation associated to the spherical fusion category VecGω of twisted G-graded vector spaces. As shown by Kirillov, the representation space for this TVBW representation is canonically isomorphic to a vector space spanned by VecGω-colored graphs embedded in the surface. By analyzing the action of the Birman generators on a finite spanning set of colored graphs, we find that the mapping class group acts by permutations on a slightly larger finite spanning set. This implies that the representation has finite image.