Integral homology 3-spheres and the Johnson filtration

Pitsch, Wolfgang

doi:10.1090/S0002-9947-08-04208-6

Integral homology $3$-spheres and the Johnson filtration
HTML articles powered by AMS MathViewer

by Wolfgang Pitsch PDF

Trans. Amer. Math. Soc. 360 (2008), 2825-2847 Request permission

Abstract:

The mapping class group of an oriented surface $\Sigma _{g,1}$ of genus $g$ with one boundary component has a natural decreasing filtration $\mathcal {M}_{g,1} \supset \mathcal {M}_{g,1}(1) \supset \mathcal {M}_{g,1}(2) \supset \mathcal {M}_{g,1}(3) \supset \cdots$, where $\mathcal {M}_{g,1}(k)$ is the kernel of the action of $\mathcal {M}_{g,1}$ on the $k^{th}$ nilpotent quotient of $\pi _1(\Sigma _{g,1})$. Using a tree Lie algebra approximating the graded Lie algebra $\displaystyle \bigoplus _{k} \mathcal {M}_{g,1}(k)/\mathcal {M}_{g,1}(k+1)$ we prove that any integral homology sphere of dimension $3$ has for some $g$ a Heegaard decomposition of the form $M = \mathcal {H}_g \coprod _{\iota _g \phi } - \mathcal {H}_g$, where $\phi \in \mathcal {M}_{g,1}(3)$ and $\iota _g$ is such that $\mathcal {H}_g \coprod _{\iota _g} - \mathcal {H}_g= S^3$. This proves a conjecture due to S. Morita and shows that the “core” of the Casson invariant is indeed the Casson invariant.

References

Dror Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), no. 2, 423–472. MR 1318886, DOI 10.1016/0040-9383(95)93237-2
Robert Craggs, A new proof of the Reidemeister-Singer theorem on stable equivalence of Heegaard splittings, Proc. Amer. Math. Soc. 57 (1976), no. 1, 143–147. MR 410749, DOI 10.1090/S0002-9939-1976-0410749-9
Stavros Garoufalidis and Jerome Levine, Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism, Graphs and patterns in mathematics and theoretical physics, Proc. Sympos. Pure Math., vol. 73, Amer. Math. Soc., Providence, RI, 2005, pp. 173–203. MR 2131016, DOI 10.1090/pspum/073/2131016
H. B. Griffiths, Automorphisms of a $3$-dimensional handlebody, Abh. Math. Sem. Univ. Hamburg 26 (1963/64), 191–210. MR 159313, DOI 10.1007/BF02992786
Nathan Habegger and Gregor Masbaum, The Kontsevich integral and Milnor’s invariants, Topology 39 (2000), no. 6, 1253–1289. MR 1783857, DOI 10.1016/S0040-9383(99)00041-5
Nathan Habegger and Wolfgang Pitsch, Tree level Lie algebra structures of perturbative invariants, J. Knot Theory Ramifications 12 (2003), no. 3, 333–345. MR 1983089, DOI 10.1142/S0218216503002494
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. MR 1336822
Dennis Johnson, The structure of the Torelli group. II. A characterization of the group generated by twists on bounding curves, Topology 24 (1985), no. 2, 113–126. MR 793178, DOI 10.1016/0040-9383(85)90049-7
Jerome Levine, Addendum and correction to: “Homology cylinders: an enlargement of the mapping class group” [Algebr. Geom. Topol. 1 (2001), 243–270; MR1823501 (2002m:57020)], Algebr. Geom. Topol. 2 (2002), 1197–1204. MR 1943338, DOI 10.2140/agt.2002.2.1197
Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory, 2nd ed., Dover Publications, Inc., Mineola, NY, 2004. Presentations of groups in terms of generators and relations. MR 2109550
Shigeyuki Morita, Casson’s invariant for homology $3$-spheres and characteristic classes of surface bundles. I, Topology 28 (1989), no. 3, 305–323. MR 1014464, DOI 10.1016/0040-9383(89)90011-6
Shigeyuki Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993), no. 3, 699–726. MR 1224104, DOI 10.1215/S0012-7094-93-07017-2
J. Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen i, Acta Math 50 (1927), 189–358.
—, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen II, Acta Math 53 (1929), 1–76.
—, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen III, Acta Math 58 (1931), 87–167.
Wolfgang Pitsch, Une construction intrinsèque du cœur de l’invariant de Casson, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 6, 1741–1761 (French, with English and French summaries). MR 1871288
Wolfgang Pitsch, Extensions verselles et automorphismes des groupes nilpotents libres, J. Algebra 249 (2002), no. 2, 512–527 (French, with French summary). MR 1901170, DOI 10.1006/jabr.2001.9077
K. Reidemeister, Zur dreidimensionalen Topologie, Abh. Math. Sem. Univ. Hamburg (1933), 189–194.
James Singer, Three-dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc. 35 (1933), no. 1, 88–111. MR 1501673, DOI 10.1090/S0002-9947-1933-1501673-5
Shin’ichi Suzuki, On homeomorphisms of a 3-dimensional handlebody, Canadian J. Math. 29 (1977), no. 1, 111–124. MR 433433, DOI 10.4153/CJM-1977-011-1