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Homology cylinders and the acyclic closure of a free group. (English) Zbl 1149.57001

Summary: We give a Dehn-Nielsen type theorem for the homology cobordism group of homology cylinders by considering its action on the acyclic closure, which was defined by J. P. Levine [in Invent. Math. 96, No. 3, 571–592 (1989; Zbl 0692.57010) and in Combinatorial group theory, Proc. AMS Spec. Sess., College Park, MD, USA 1988, Contemp. Math. 109, 99–105 (1990; Zbl 0755.20007)], of a free group. Then we construct an additive invariant of those homology cylinders which act on the acyclic closure trivially. We also describe some tools for studying the automorphism group of the acyclic closure of a free group generalizing those for the automorphism group of a free group or the homology cobordism group of homology cylinders.

MSC:

57M05 Fundamental group, presentations, free differential calculus
20E05 Free nonabelian groups
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
20F28 Automorphism groups of groups
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References:

[1] K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1982) · Zbl 0584.20036
[2] P M Cohn, Free rings and their relations, London Mathematical Society Monographs 19, Academic Press (1985) · Zbl 0659.16001
[3] S Garoufalidis, J Levine, Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism, Proc. Sympos. Pure Math. 73, Amer. Math. Soc. (2005) 173 · Zbl 1086.57013
[4] N Habegger, Milnor, Johnson, and tree level perturbative invariants, preprint
[5] K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1 · Zbl 0941.57015 · doi:10.2140/gt.2000.4.1
[6] A Heap, Bordism invariants of the mapping class group, preprint · Zbl 1156.57017 · doi:10.1016/j.top.2006.06.001
[7] J Hillman, Algebraic invariants of links, Series on Knots and Everything 32, World Scientific Publishing Co. (2002) · Zbl 1007.57001
[8] K Igusa, K E Orr, Links, pictures and the homology of nilpotent groups, Topology 40 (2001) 1125 · Zbl 1002.57012 · doi:10.1016/S0040-9383(00)00002-1
[9] N Kawazumi, Cohomological Aspects of Magnus expansions, preprint
[10] P Kirk, C Livingston, Z Wang, The Gassner representation for string links, Commun. Contemp. Math. 3 (2001) 87 · Zbl 0989.57005 · doi:10.1142/S0219199701000299
[11] J Y Le Dimet, Enlacements d’intervalles et représentation de Gassner, Comment. Math. Helv. 67 (1992) 306 · Zbl 0759.57010
[12] J P Levine, Link concordance and algebraic closure. II, Invent. Math. 96 (1989) 571 · Zbl 0692.57010 · doi:10.1007/BF01393697
[13] J P Levine, Algebraic closure of groups, Contemp. Math. 109, Amer. Math. Soc. (1990) 99 · Zbl 0755.20007
[14] J Levine, Homology cylinders: an enlargement of the mapping class group, Algebr. Geom. Topol. 1 (2001) 243 · Zbl 0978.57015 · doi:10.2140/agt.2001.1.243
[15] S Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993) 699 · Zbl 0801.57011 · doi:10.1215/S0012-7094-93-07017-2
[16] K E Orr, Homotopy invariants of links, Invent. Math. 95 (1989) 379 · Zbl 0668.57014 · doi:10.1007/BF01393902
[17] J Stallings, Homology and central series of groups, J. Algebra 2 (1965) 170 · Zbl 0135.05201 · doi:10.1016/0021-8693(65)90017-7
[18] Y Yokomizo, private communication,
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