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Journal of Topology Advance Access originally published online on September 17, 2009
Journal of Topology 2009 2(3):527-569; doi:10.1112/jtopol/jtp020
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© 2009 London Mathematical Society

Symplectic Jacobi diagrams and the Lie algebra of homology cylinders

Kazuo Habiro

RIMS, Kyoto University, Kyoto 606-8502, Japan habiro@kurims.kyoto-u.ac.jp

Gwénaël Massuyeau

IRMA, Université de Strasbourg & CNRS, 7 rue René Descartes, 67084 Strasbourg, France massuyeau@math.u-strasbg.fr


  Abstract

Let S be a compact connected oriented surface whose boundary is connected or empty. A homology cylinder over the surface S is a cobordism between S and itself, homologically equivalent to the cylinder over S. The Y-filtration on the monoid of homology cylinders over S is defined by clasper surgery. Using a functorial extension of the Le–Murakami–Ohtsuki invariant, we show that the graded Lie algebra associated to the Y-filtration is isomorphic to the Lie algebra of ‘symplectic Jacobi diagrams’. This Lie algebra consists of the primitive elements of a certain Hopf algebra whose multiplication is a diagrammatic analogue of the Moyal–Weyl product. The mapping cylinder construction embeds the Torelli group into the monoid of homology cylinders, sending the lower central series to the Y-filtration. We give a combinatorial description of the graded Lie algebra map induced by this embedding, by connecting Hain’s infinitesimal presentation of the Torelli group to the Lie algebra of symplectic Jacobi diagrams. This Lie algebra map is shown to be injective in degree 2, and the question of the injectivity in higher degrees is discussed.

Received February 15, 2008.

2000 Mathematics Subject Classification 57M27, 57R50, 20F12, 20F38, 20F40


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