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The topological IHX relation, pure braids, and the Torelli group. (English) Zbl 1023.20020

Let \(T_{g,1}\) denote the Torelli group of a surface of genus \(g\) with \(1\) boundary component, and let \(P(g)\) denote the pure braid group on \(g\) strands. Recall that the lower central series of a group \(G\) is given by \(G_1=G\), and inductively by \(G_{n+1}=[G,G_n]\). The authors prove that if \(g\geq 3\), then for all \(n\geq 1\), \(P(g)_{n+1}\subset(T_{g,1})_n\). Then they explicitly give the relation obtained from the Jacobi relation via the map \(P(g)_3\to(T_{g,1})_2\).

MSC:

20F36 Braid groups; Artin groups
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
20F38 Other groups related to topology or analysis
57R50 Differential topological aspects of diffeomorphisms
20F14 Derived series, central series, and generalizations for groups
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[1] S. Garoufalidis and J. Levine, Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism , preprint 1999, · Zbl 1086.57013
[2] M. Goussarov, Finite type invariants and (n)-equivalence of (3)-manifolds , C. R. Acad. Sci. Paris Sèr. I Math. 329 (1999), 517–522. · Zbl 0938.57013 · doi:10.1016/S0764-4442(00)80053-1
[3] N. Habegger, “The Topological IHX Relation” in Knots in Hellas ’98 (Delphi, Greece), Vol. II , J. Knot Theory Ramifications 10 (2001), World Sci., Singapore, 2001, 309–329. \CMP1 822 495 · Zbl 0997.57030 · doi:10.1142/S0218216501000895
[4] N. Habegger and X.-S. Lin, On link concordance and Milnor’s (\overline\mu) invariants , Bull. London Math. Soc. 30 (1998), 419–428. · Zbl 0936.57006 · doi:10.1112/S0024609398004494
[5] N. Habegger and G. Masbaum, The Kontsevich integral and Milnor’s invariants , Topology 39 (2000), 1253–1289. · Zbl 0964.57011 · doi:10.1016/S0040-9383(99)00041-5
[6] N. Habegger and C. Sorger, An infinitesimal presentation of the Torelli group of a surface with boundary , preprint, 2000,
[7] K. Habiro, Claspers and finite type invariants of links , Geom. Topol. 4 (2000), 1–83., , · Zbl 0941.57015 · doi:10.2140/gt.2000.4.1
[8] R. Hain, Infinitesimal presentations of the Torelli groups , J. Amer. Math. Soc. 10 (1997), 597–651. JSTOR: · Zbl 0915.57001 · doi:10.1090/S0894-0347-97-00235-X
[9] A. Hatcher and W. Thurston, A presentation for the mapping class group of a closed orientable surface , Topology 19 (1980), 221–237. · Zbl 0447.57005 · doi:10.1016/0040-9383(80)90009-9
[10] D. Johnson, The structure of the Torelli group, I: A finite set of generators for (\mathcalI ), Ann. of Math. (2) 118 (1983), 423–442. JSTOR: · Zbl 0549.57006 · doi:10.2307/2006977
[11] –. –. –. –., The structure of the Torelli group, II: A characterization of the group generated by twists on bounding curves , Topology 24 (1985), 113–126. · Zbl 0571.57009 · doi:10.1016/0040-9383(85)90049-7
[12] –. –. –. –., The structure of the Torelli group, III: The abelianization of (\mathcalT ), Topology 24 (1985), 127–144. · Zbl 0571.57010 · doi:10.1016/0040-9383(85)90050-3
[13] T. T. Q. Le, “An invariant of integral homology 3-spheres which is universal for all finite type invariants” in Soliton Geometry and Topology: On the Crossroad , Amer. Math. Soc. Transl. Ser. 2 179 , Amer. Math. Soc., Providence, 1997, 75–100. · Zbl 0914.57013
[14] T. T. Q. Le, J. Murakami, and T. Ohtsuki, On a universal perturbative invariant of 3-manifolds , Topology 37 (1998), 539–574. · Zbl 0897.57017 · doi:10.1016/S0040-9383(97)00035-9
[15] J. Levine, “Pure braids, a new subgroup of the mapping class group and finite-type invariants of 3-manifolds” in Tel Aviv Topology Conference: Rothenberg Festschrift (Tel Aviv, 1998) , Contemp. Math. 231 , Amer. Math. Soc., Providence, 1999, 137–157. · Zbl 0933.57018
[16] G. Mess, The Torelli groups for genus \(2\) and \(3\) surfaces , Topology 31 (1992), 775–790. · Zbl 0772.57025 · doi:10.1016/0040-9383(92)90008-6
[17] S. Morita, Casson’s invariant for homology \(3\)-spheres and characteristic classes of surface bundles, I , Topology 28 (1989), 305–323. · Zbl 0684.57008 · doi:10.1016/0040-9383(89)90011-6
[18] –. –. –. –., On the structure of the Torelli group and the Casson invariant , Topology 30 (1991), 603–621. · Zbl 0747.57010 · doi:10.1016/0040-9383(91)90042-3
[19] –. –. –. –., “A linear representation of the mapping class group of orientable surfaces and characteristic classes of surface bundles” in Topology and Teichmüller Spaces (Katinkulta, Finland, 1995) , World Sci., River Edge, N.J., 1996, 159–186. · Zbl 0939.32011
[20] T. Oda, A lower bound for the graded modules associated with the relative weight filtration on the Teichmüller group ,
[21] T. Ohtsuki, Finite type invariants of integral homology 3-spheres , J. Knot Theory Ramifications 5 (1996), 101–115. · Zbl 0942.57009 · doi:10.1142/S0218216596000084
[22] L. Paris and D. Rolfsen, Geometric subgroups of mapping class groups , J. Reine Angew. Math. 521 (2000), 47–83. · Zbl 1007.57014 · doi:10.1515/crll.2000.030
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