Finite type invariants in low degrees and the Johnson filtration

Abstract

We study the behaviour of 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 since $\lambda$ 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 they never vanish on any level of this filtration. In contrast we prove 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 $\mathcal{M}_{g,1}(5)$. Finally we determine a surgery formula for Othsuki's second invariant $\lambda_2$.