2021/05/29 by Quentin Faes, Faes, Quentin · 1 citation
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models
paper · pdf · doi:10.48550/arxiv.2105.14253
We prove that all homology 3-spheres are J4-equivalent, i.e. that any homology 3-sphere can be obtained from one another by twisting one of its Heegaard splittings by an element of the mapping class group acting trivially on the fourth nilpotent quotient of the fundamental group of the gluing surface. We do so by exhibiting an element of J4, the fourth term of the Johnson filtration of the mapping class group, on which (the core of) the Casson invariant takes the value 1. In particular, this provides an explicit example of an element of J4 that is not a commutator of length 2 in the Torelli group.