Algebraic & Geometric Topology

An expansion of the Jones representation of genus 2 and the Torelli group

Yasushi Kasahara

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We study the algebraic property of the representation of the mapping class group of a closed oriented surface of genus 2 constructed by V F R Jones [Annals of Math. 126 (1987) 335-388]. It arises from the Iwahori–Hecke algebra representations of Artin’s braid group of 6 strings, and is defined over integral Laurent polynomials [t,t1]. We substitute the parameter t with eh, and then expand the powers eh in their Taylor series. This expansion naturally induces a filtration on the Torelli group which is coarser than its lower central series. We present some results on the structure of the associated graded quotients, which include that the second Johnson homomorphism factors through the representation. As an application, we also discuss the relation with the Casson invariant of homology 3–spheres.

Article information

Algebr. Geom. Topol., Volume 1, Number 1 (2001), 39-55.

Received: 18 October 2000
Accepted: 30 November 2000
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N05: Topology of $E^2$ , 2-manifolds
Secondary: 20F38: Other groups related to topology or analysis 20C08: Hecke algebras and their representations 20F40: Associated Lie structures

Jones representation mapping class group Torelli group Johnson homomorphism


Kasahara, Yasushi. An expansion of the Jones representation of genus 2 and the Torelli group. Algebr. Geom. Topol. 1 (2001), no. 1, 39--55. doi:10.2140/agt.2001.1.39.

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