1. A functorial extension of the Magnus representation to the category of three-dimensional cobordisms
- Author
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Juan Serrano de Rodrigo, Gwénaël Massuyeau, Vincent Florens, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Departamento de Matemáticas, University of Zaragoza - Universidad de Zaragoza [Zaragoza], Spanish Ministry of Education. Grant number : MTM2013-45710-C2-1-P, MTM2016-76868-C2-2-P, Grupo de Geometria del Gobierno de Aragon, European Social Fund, The authors would like to thank the referee for helpful comments and suggestions. The third author is partially supported by the Spanish Ministry of Education (grants MTM2013-45710-C2-1-P and MTM2016-76868-C2-2-P), by the 'Grupo de Geometria del Gobierno de Aragon' and by the European Social Fund., Laboratoire de Mathématiques et de leurs Applications [Pau] ( LMAP ), Université de Pau et des Pays de l'Adour ( UPPA ) -Centre National de la Recherche Scientifique ( CNRS ), Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique Avancée ( IRMA ), Université de Strasbourg ( UNISTRA ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), and Universidad de Zaragoza
- Subjects
[ MATH ] Mathematics [math] ,[ MATH.MATH-GT ] Mathematics [math]/Geometric Topology [math.GT] ,Pure mathematics ,Fundamental group ,Braid group ,01 natural sciences ,Alexander polynomial ,Mathematics - Geometric Topology ,Mathematics::Category Theory ,[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] ,Mapping class group ,FOS: Mathematics ,0101 mathematics ,[MATH]Mathematics [math] ,3-manifold ,Monoidal functor ,Mathematics ,Algebra and Number Theory ,Functor ,Burau representation ,Topological quantum field theory ,010102 general mathematics ,Magnus representation ,Geometric Topology (math.GT) ,57M27, 57M10 ,16. Peace & justice ,Mathematics::Geometric Topology ,TQFT ,Cobordism ,Free abelian group ,MSC: Primary 57M27 ,Secondary 57M10 ,Group ring - Abstract
Let $R$ be an integral domain and $G$ be a subgroup of its group of units. We consider the category $\mathbf{\mathsf{Cob}}_G$ of 3-dimensional cobordisms between oriented surfaces with connected boundary, equipped with a representation of their fundamental group in $G$. Under some mild conditions on $R$, we construct a monoidal functor from $\mathbf{\mathsf{Cob}}_G$ to the category $\mathbf{\mathsf{pLagr}}_R$ consisting of "pointed Lagrangian relations" between skew-Hermitian $R$-modules. We call it the "Magnus functor" since it contains the Magnus representation of mapping class groups as a special case. Our construction is inspired from the work of Cimasoni and Turaev on the extension of the Burau representation of braid groups to the category of tangles. It can also be regarded as a $G$-equivariant version of a TQFT-like functor that has been described by Donaldson. The study and computation of the Magnus functor is carried out using classical techniques of low-dimensional topology. When $G$ is a free abelian group and $R = Z[G]$ is the group ring of $G$, we relate the Magnus functor to the "Alexander functor" (which has been introduced in a prior work using Alexander-type invariants), and we deduce a factorization formula for the latter., 36 pages; very minor changes
- Published
- 2018
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