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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Integral homology $ 3$-spheres and the Johnson filtration


Author: Wolfgang Pitsch
Journal: Trans. Amer. Math. Soc. 360 (2008), 2825-2847
MSC (2000): Primary 57M99; Secondary 20F38, 20F12
Published electronically: January 4, 2008
MathSciNet review: 2379777
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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.


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Additional Information

Wolfgang Pitsch
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
Email: pitsch@mat.uab.es

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04208-6
PII: S 0002-9947(08)04208-6
Received by editor(s): November 7, 2005
Published electronically: January 4, 2008
Additional Notes: The author was supported by MEC grant MTM2004-06686 and by the program Ramón y Cajal, MEC, Spain
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.