Integral homology $3$-spheres and the Johnson filtration

Abstract

The mapping class group of an oriented surface Σg, 1 of genus g with one boundary component has a natural decreasing filtration Mg,1 ⊃ Mg,i(⊃) 3 Me , i(⊃) D M g,1 (3) ⊃..., where M g,1 (k) is the kernel of the action of M g,1 on the k th nilpotent quotient of π 1 (Σ g,1 ). Using a tree Lie algebra approximating the graded Lie algebra ⊕ k M g,1 (k)/M g,1 (k + 1) we prove that any integral homology sphere of dimension 3 has for some g a Heegaard decomposition of the form M = Hg? lgΦ -H g , where Φ ∈ Μ g,1 (3) and lg is such that H g ∏ lg -H g = S 3 . This proves a conjecture due to S. Morita and shows that the core of the Casson invariant is indeed the Casson invariant.