Finite type invariants in low degrees and the Johnson filtration

Wolfgang Pitsch, Ricard Riba

Published 2023-11-16Version 1

We study the behavior the Casson invariant $\lambda$, its square, and Othsuki's second invariant $\lambda_2$ as functions on the Johnson subgroup of the mapping class group. We show that $\lambda$ and $d_2 = \lambda_2 - 18 \lambda^2$ are invariants that are morphisms on respectively the second and the third level of the Johnson filtration and as a consequence never vanish on any level of this filtration. Finally we show that the invariant $\lambda_2-18\lambda^2 +3\lambda$ vanishes on the fifth level of the Johnson filtration, $\mathcal{M}_{g,1}(5)$, and as a consequence we prove that for instance the Poincar\'e homology sphere does not admit any Heegaard splitting with gluing map an element in $\mathcal{M}_{g,1}(5)$.