Algebraic & Geometric Topology

Invariants and structures of the homology cobordism group of homology cylinders

Minkyoung Song

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The homology cobordism group of homology cylinders is a generalization of the mapping class group and the string link concordance group. We study this group and its filtrations by subgroups by developing new homomorphisms. First, we define extended Milnor invariants by combining the ideas of Milnor’s link invariants and Johnson homomorphisms. They give rise to a descending filtration of the homology cobordism group of homology cylinders. We show that each successive quotient of the filtration is free abelian of finite rank. Second, we define Hirzebruch-type intersection form defect invariants obtained from iterated p–covers for homology cylinders. Using them, we show that the abelianization of the intersection of our filtration is of infinite rank. Also we investigate further structures in the homology cobordism group of homology cylinders which previously known invariants do not detect.

Article information

Algebr. Geom. Topol., Volume 16, Number 2 (2016), 899-943.

Received: 30 December 2014
Revised: 20 April 2015
Accepted: 6 May 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds 57N10: Topology of general 3-manifolds [See also 57Mxx]

homology cylinder homology cobordism Milnor invariant Hirzebruch-type invariant


Song, Minkyoung. Invariants and structures of the homology cobordism group of homology cylinders. Algebr. Geom. Topol. 16 (2016), no. 2, 899--943. doi:10.2140/agt.2016.16.899.

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  • J,C Cha, Injectivity theorems and algebraic closures of groups with coefficients, Proc. Lond. Math. Soc. 96 (2008) 227–250
  • J,C Cha, Structure of the string link concordance group and Hirzebruch-type invariants, Indiana Univ. Math. J. 58 (2009) 891–927
  • J,C Cha, Link concordance, homology cobordism, and Hirzebruch-type defects from iterated $p$–covers, J. Eur. Math. Soc. 12 (2010) 555–610
  • J,C Cha, Amenable $L\sp 2$–theoretic methods and knot concordance, Int. Math. Res. Not. 2014 (2014) 4768–4803
  • J,C Cha, Symmetric Whitney tower cobordism for bordered $3$–manifolds and links, Trans. Amer. Math. Soc. 366 (2014) 3241–3273
  • J,C Cha, S Friedl, T Kim, The cobordism group of homology cylinders, Compos. Math. 147 (2011) 914–942
  • T,D Cochran, S Harvey, P,D Horn, Higher-order signature cocycles for subgroups of mapping class groups and homology cylinders, Int. Math. Res. Not. 2012 (2012) 3311–3373
  • T,D Cochran, K,E Orr, Not all links are concordant to boundary links, Ann. of Math. 138 (1993) 519–554
  • T,D Cochran, K,E Orr, P Teichner, Knot concordance, Whitney towers and $L\sp 2$–signatures, Ann. of Math. 157 (2003) 433–519
  • T,D Cochran, K,E Orr, P Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004) 105–123
  • W,G Dwyer, Homology, Massey products and maps between groups, J. Pure Appl. Algebra 6 (1975) 177–190
  • M,H Freedman, P Teichner, $4$–manifold topology II: Dwyer's filtration and surgery kernels, Invent. Math. 122 (1995) 531–557
  • S Garoufalidis, J Levine, Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism, preprint (1999)
  • H Goda, T Sakasai, Homology cylinders and sutured manifolds for homologically fibered knots, Tokyo J. Math. 36 (2013) 85–111
  • M Goussarov, Finite type invariants and $n$–equivalence of $3$–manifolds, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 517–522
  • K,W Gruenberg, Residual properties of infinite soluble groups, Proc. London Math. Soc. 7 (1957) 29–62
  • N Habegger, X-S Lin, The classification of links up to link-homotopy, J. Amer. Math. Soc. 3 (1990) 389–419
  • N Habegger, X-S Lin, On link concordance and Milnor's $\bar {\mu}$ invariants, Bull. London Math. Soc. 30 (1998) 419–428
  • K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1–83
  • D Johnson, An abelian quotient of the mapping class group ${\cal I}\sb{g}$, Math. Ann. 249 (1980) 225–242
  • T Kitayama, Homology cylinders of higher-order, Algebr. Geom. Topol. 12 (2012) 1585–1605
  • J,P Levine, Link concordance and algebraic closure, II, Invent. Math. 96 (1989) 571–592
  • J,P Levine, Algebraic closure of groups, from: “Combinatorial group theory”, (B Fine, A Gaglione, F,C,Y Tang, editors), Contemp. Math. 109, Amer. Math. Soc. (1990) 99–105
  • J Levine, Homology cylinders: an enlargement of the mapping class group, Algebr. Geom. Topol. 1 (2001) 243–270
  • W Magnus, A Karrass, D Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, 2nd edition, Dover Publications, New York (1976)
  • S Morita, Symplectic automorphism groups of nilpotent quotients of fundamental groups of surfaces, from: “Groups of diffeomorphisms”, (R Penner, D Kotschick, T Tsuboi, N Kawazumi, T Kitano, Y Mitsumatsu, editors), Adv. Stud. Pure Math. 52, Math. Soc. Japan, Tokyo (2008) 443–468
  • K,E Orr, Homotopy invariants of links, Invent. Math. 95 (1989) 379–394
  • T Sakasai, The Magnus representation and higher-order Alexander invariants for homology cobordisms of surfaces, Algebr. Geom. Topol. 8 (2008) 803–848
  • T Sakasai, A survey of Magnus representations for mapping class groups and homology cobordisms of surfaces, from: “Handbook of Teichmüller theory, Vol. III”, (A Papadopoulos, editor), IRMA Lect. Math. Theor. Phys. 17, Eur. Math. Soc., Zürich (2012) 531–594
  • N Smythe, Boundary links, from: “Topology Seminar Wisconsin, 1965”, (R,H Bing, R,J Bean, editors), Ann. Math. Stud. 60, Princeton Univ. Press (1966) 69–72
  • J Stallings, Homology and central series of groups, J. Algebra 2 (1965) 170–181