Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-29T12:42:31.667Z Has data issue: false hasContentIssue false
  • English
  • Français

The cobordism group of homology cylinders

Part of: Low-dimensional topology Topological manifolds

Published online by Cambridge University Press:  07 September 2010

Jae Choon Cha ,
Stefan Friedl  and
Taehee Kim
Jae Choon Cha
Affiliation:
Department of Mathematics and Pohang Mathematics Institute, Pohang University of Science and Technology, Pohang Gyungbuk 790–784, Republic of Korea (email: [email protected])
Stefan Friedl
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: [email protected])
Taehee Kim
Affiliation:
Department of Mathematics, Konkuk University, Seoul 143–701, Republic of Korea (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Garoufalidis and Levine introduced the homology cobordism group of homology cylinders over a surface. This group can be regarded as an enlargement of the mapping class group. Using torsion invariants, we show that the abelianization of this group is infinitely generated provided that the first Betti number of the surface is positive. In particular, this shows that the group is not perfect. This answers questions of Garoufalidis and Levine, and Goda and Sakasai. Furthermore, we show that the abelianization of the group has infinite rank for the case that the surface has more than one boundary component. These results also hold for the homology cylinder analogue of the Torelli group.

Keywords

torsion invarianthomology cylinderhomology cobordism

MSC classification

Secondary: 57M27: Invariants of knots and 3-manifolds 57N10: Topology of general $3$-manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 3 , May 2011 , pp. 914 - 942
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Birman, J. S. and Hilden, H. M., On the mapping class groups of closed surfaces as covering spaces, in Advances in the theory of Riemann surfaces, Proc. conf. (Stony Brook, NY, 1969), Annals of Mathematics Studies, vol. 66 (Princeton University Press, Princeton, NJ, 1971), 81–115.Google Scholar
[2]Benedetti, R. and Petronio, C., Reidemeister–Turaev torsion of 3-dimensional Euler structure with simple boundary tangency and pseudo-Legendrian knots, Manuscripta Math. 106 (2001), 13–61.CrossRefGoogle Scholar
[3]Cha, J. C., The structure of the rational concordance group of knots, Mem. Amer. Math. Soc. 189 (2007), x+95pp.Google Scholar
[4]Cha, J. C. and Friedl, S., Twisted torsion invariants and link concordance, arXiv:1001.0926.Google Scholar
[5]Everest, G. and Ward, T., Heights of polynomials and entropy in algebraic dynamics, Universitext (Springer, New York, 1999).CrossRefGoogle Scholar
[6]Farb, B. and Margalit, D., A primer on mapping class groups,http://www.math.utah.edu/∼margalit/primer/.Google Scholar
[7]Fintushel, R. and Stern, R., Instanton homology of Seifert fibred homology three spheres, Proc. Lond. Math. Soc. 61 (1990), 109–137.CrossRefGoogle Scholar
[8]Fox, R. H. and Milnor, J. W., Singularities of 2-spheres in 4-space and cobordism of knots, Osaka J. Math. 3 (1966), 257–267.Google Scholar
[9]Freedman, M., The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), 357–453.CrossRefGoogle Scholar
[10]Freedman, M. H. and Quinn, F., Topology of 4-manifolds, Princeton Mathematical Series, vol. 39 (Princeton University Press, Princeton, NJ, 1990).Google Scholar
[11]Friedl, S., Juhasz, A. and Rasmussen, J., The decategorification of sutured Floer homology, arXiv:0903.5287.Google Scholar
[12]Furuta, M., Homology cobordism group of homology 3-spheres, Invent. Math. 100 (1990), 339–355.CrossRefGoogle Scholar
[13]Garoufalidis, S. and Levine, J., Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism, Graphs and patterns in mathematics and theorical physics, Proc. Sympos. Pure Math. 73 (2005), 173–205.CrossRefGoogle Scholar
[14]Goda, H. and Sakasai, T., Homology cylinders in knot theory, arXiv:0807.4034.Google Scholar
[15]Goda, H. and Sakasai, T., Abelian quotients of monoids of homology cylinders, arXiv:0905.4775.Google Scholar
[16]Goussarov, M., Finite type invariants and n-equivalence of 3-manifolds, C. R. Math. Acad. Sci. Paris 329 (1999), 517–522.CrossRefGoogle Scholar
[17]Habiro, K., Claspers and finite type invariants of links, Geom. Topol. 4 (2000), 1–83.CrossRefGoogle Scholar
[18]Johnson, D., The structure of the Torelli group. I. A finite set of generators for ℐ, Ann. of Math. (2) 118 (1983), 423–442.CrossRefGoogle Scholar
[19]Johnson, D., A survey of the Torelli group, Contemp. Math. 20 (1983), 165–179.CrossRefGoogle Scholar
[20]Johnson, D., The structure of the Torelli group. III. The abelianization of ℐ, Topology 24 (1985), 127–144.CrossRefGoogle Scholar
[21]Kirk, P. and Livingston, C., Twisted Alexander invariants, Reidemeister torsion and Casson–Gordon invariants, Topology 38 (1999), 635–661.CrossRefGoogle Scholar
[22]Kirk, P., Livingston, C. and Wang, Z., The Gassner representation for string links, Commun. Contemp. Math. 3 (2001), 87–136.CrossRefGoogle Scholar
[23]Levine, J., Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244.CrossRefGoogle Scholar
[24]Levine, J., Invariants of knot cobordism, Invent. Math. 8 (1969), 98–110.CrossRefGoogle Scholar
[25]Levine, J., Homology cylinders: An enlargement of the mapping class group, Algebr. Geom. Topol. 1 (2001), 243–270.CrossRefGoogle Scholar
[26]McCool, J., Some finitely presented subgroups of the automorphism group of a free group, J. Algebra 35 (1975), 205–213.CrossRefGoogle Scholar
[27]Miller, A. and McCullough, D., The genus 2 Torelli group is not finitely generated, Topology Appl. 22 (1986), 43–49.Google Scholar
[28]Mess, G., The Torelli groups for genus 2 and 3 surfaces, Topology 31 (1992), 775–790.CrossRefGoogle Scholar
[29]Milnor, J., A duality theorem for Reidemeister torsion, Ann. of Math. (2) 76 (1962), 137–147.CrossRefGoogle Scholar
[30]Milnor, J., Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426.CrossRefGoogle Scholar
[31]Morita, S., Cohomological structure of the mapping class group and beyond, in Problems on mapping class groups and related topics, Proceedings of Symposia in Pure Mathematics, vol. 74 ed. Farb, B. (American Mathematical Society, Providence, RI, 2006).Google Scholar
[32]Morita, S., Symplectic automorphism groups of nilpotent quotients of fundamental groups of surfaces, Adv. Stud. Pure Math. 52 (2008), 443–468.CrossRefGoogle Scholar
[33]Ni, Y., Knot Floer homology detects fibred knots, Invent. Math. 170 (2007), 577–608.CrossRefGoogle Scholar
[34]Nicolaescu, L., The Reidemeister torsion of 3-manifolds, de Gruyter Studies in Mathematics, vol. 30 (Walter de Gruyter & Co., Berlin, 2003).CrossRefGoogle Scholar
[35]Powell, J., Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (1978), 347–350.CrossRefGoogle Scholar
[36]Sakasai, T., Mapping class groups, groups of homology cobordisms of surfaces and invariants of 3-manifolds, Doctoral Dissertation, The University of Tokyo (2006).Google Scholar
[37]Sakasai, T., The Magnus representation and higher-order Alexander invariants for homology cobordisms of surfaces, Algebr. Geom. Topol. 8 (2008), 803–848.CrossRefGoogle Scholar
[38]Saveliev, N., Invariants of Homology 3-spheres, Encyclopaedia of Mathematical Sciences, vol. 140 (Springer, Berlin, 2002).CrossRefGoogle Scholar
[39]Silver, D. and Williams, S., Mahler measure, links and homology growth, Topology 41 (2002), 979–991.CrossRefGoogle Scholar
[40]Silver, D. and Williams, S., Mahler measure of Alexander polynomials, J. London Math. Soc. (2) 69 (2004), 767–782.CrossRefGoogle Scholar
[41]Stallings, J., Homology and central series of groups, J. Algebra 2 (1965), 170–181.CrossRefGoogle Scholar
[42]Turaev, V., Reidemeister torsion in knot theory, Russian Math. Surveys 41 (1986), 119–182.CrossRefGoogle Scholar
[43]Turaev, V., Introduction to combinatorial torsions, Lectures in Mathematics, ETH Zürich, 2001 (Birkhäuser, Basel, 2001).CrossRefGoogle Scholar
[44]Turaev, V., Torsions of 3-manifolds, Progress in Mathematics, vol. 208 (Birkhäuser, Basel, 2002).CrossRefGoogle Scholar