Integral Homology 3-spheres and the Johnson Filtration
نویسنده
چکیده
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,1(1) ⊃ Mg,1(2) ⊃ Mg,1(3) ⊃ · · · , where Mg,1(k) is the kernel of the action of Mg,1 on the kth nilpotent quotient of π1(Σg,1). Using a tree Lie algebra approximating the graded Lie algebra ⊕ k Mg,1(k)/Mg,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 ∐ ιgφ −Hg, where φ ∈ Mg,1(3) and ιg is such that Hg ∐ ιg −Hg = S3. This proves a conjecture due to S. Morita and shows that the “core” of the Casson invariant is indeed the Casson invariant.
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