Geometry & Topology

Homology cobordism invariants and the Cochran–Orr–Teichner filtration of the link concordance group

Shelly L Harvey

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Abstract

For any group G, we define a new characteristic series related to the derived series, that we call the torsion-free derived series of G. Using this series and the Cheeger–Gromov ρ–invariant, we obtain new real-valued homology cobordism invariants ρn for closed (4k1)–dimensional manifolds. For 3–dimensional manifolds, we show that {ρn|n} is a linearly independent set and for each n0, the image of ρn is an infinitely generated and dense subset of .

In their seminal work on knot concordance, T Cochran, K Orr and P Teichner define a filtration (n)m of the m–component (string) link concordance group, called the (n)–solvable filtration. They also define a grope filtration Gnm. We show that ρn vanishes for (n+1)–solvable links. Using this, and the nontriviality of ρn, we show that for each m2, the successive quotients of the (n)–solvable filtration of the link concordance group contain an infinitely generated subgroup. We also establish a similar result for the grope filtration. We remark that for knots (m=1), the successive quotients of the (n)–solvable filtration are known to be infinite. However, for knots, it is unknown if these quotients have infinite rank when n3.

Article information

Source
Geom. Topol., Volume 12, Number 1 (2008), 387-430.

Dates
Received: 9 April 2007
Revised: 2 September 2007
Accepted: 15 November 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800023

Digital Object Identifier
doi:10.2140/gt.2008.12.387

Mathematical Reviews number (MathSciNet)
MR2390349

Zentralblatt MATH identifier
1157.57006

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 20F14: Derived series, central series, and generalizations

Keywords
link concordance derived series homology cobordism

Citation

Harvey, Shelly L. Homology cobordism invariants and the Cochran–Orr–Teichner filtration of the link concordance group. Geom. Topol. 12 (2008), no. 1, 387--430. doi:10.2140/gt.2008.12.387. https://projecteuclid.org/euclid.gt/1513800023


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References

  • S E Cappell, J L Shaneson, Link cobordism, Comment. Math. Helv. 55 (1980) 20–49
  • J C Cha, K H Ko, Signature invariants of covering links, Trans. Amer. Math. Soc. 358 (2006) 3399–3412
  • J C Cha, C Livingston, Knot signature functions are independent, Proc. Amer. Math. Soc. 132 (2004) 2809–2816
  • J Cha, C Livingston, D Ruberman, Algebraic and Heegaard Floer invariants of knots with slice Bing doubles, to appear in Math. Proc. Camb. Phil. Soc.
  • S Chang, S Weinberger, On invariants of Hirzebruch and Cheeger-Gromov, Geom. Topol. 7 (2003) 311–319
  • T D Cochran, Derivatives of links: Milnor's concordance invariants and Massey's products, Mem. Amer. Math. Soc. 84 (1990) x+73
  • T D Cochran, Noncommutative knot theory, Algebr. Geom. Topol. 4 (2004) 347–398
  • T D Cochran, R E Gompf, Applications of Donaldson's theorems to classical knot concordance, homology $3$-spheres and property $P$, Topology 27 (1988) 495–512
  • T D Cochran, S Harvey, Homology and derived series of groups, Geom. Topol. 9 (2005) 2159–2191
  • T D Cochran, K E Orr, Not all links are concordant to boundary links, Ann. of Math. $(2)$ 138 (1993) 519–554
  • T D Cochran, K E Orr, P Teichner, Knot concordance, Whitney towers and $L\sp 2$-signatures, Ann. of Math. $(2)$ 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
  • T D Cochran, P Teichner, Knot concordance and von Neumann $\rho$-invariants, Duke Math. J. 137 (2007) 337–379
  • P M Cohn, Free rings and their relations, second edition, London Mathematical Society Monographs 19, Academic Press [Harcourt Brace Jovanovich Publishers], London (1985)
  • M H Freedman, F Quinn, Topology of 4-manifolds, Princeton Mathematical Series 39, Princeton University Press, Princeton, NJ (1990)
  • M H Freedman, P Teichner, $4$-manifold topology. I. Subexponential groups, Invent. Math. 122 (1995) 509–529
  • S Friedl, Link concordance, boundary link concordance and eta-invariants, Math. Proc. Cambridge Philos. Soc. 138 (2005) 437–460
  • K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1–83
  • S L Harvey, Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm, Topology 44 (2005) 895–945
  • J-C Hausmann, On the homotopy of nonnilpotent spaces, Math. Z. 178 (1981) 115–123
  • D M Kan, W P Thurston, Every connected space has the homology of a $K(\pi ,1)$, Topology 15 (1976) 253–258
  • K H Ko, Seifert matrices and boundary link cobordisms, Trans. Amer. Math. Soc. 299 (1987) 657–681
  • J-Y Le Dimet, Cobordisme d'enlacements de disques, Mém. Soc. Math. France (N.S.) (1988) ii+92
  • J Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969) 229–244
  • J Levine, Concordance of Boundary Links, J. Knot Theory Ramifications 16 (2007) 1111–1119
  • W Lück, T Schick, Various $L\sp 2$-signatures and a topological $L\sp 2$-signature theorem, from: “High-dimensional manifold topology”, World Sci. Publ., River Edge, NJ (2003) 362–399
  • W Mio, On boundary-link cobordism, Math. Proc. Cambridge Philos. Soc. 101 (1987) 259–266
  • H Murakami, Y Nakanishi, On a certain move generating link-homology, Math. Ann. 284 (1989) 75–89
  • D S Passman, The algebraic structure of group rings, Robert E. Krieger Publishing Co., Melbourne, FL (1985) Reprint of the 1977 original
  • D Sheiham, Invariants of boundary link cobordism, Mem. Amer. Math. Soc. 165 (2003) x+110
  • J Stallings, Homology and central series of groups, J. Algebra 2 (1965) 170–181
  • B Stenstr öm, Rings of quotients, Springer, New York (1975) Die Grundlehren der Mathematischen Wissenschaften, Band 217, An introduction to methods of ring theory
  • R Strebel, Homological methods applied to the derived series of groups, Comment. Math. Helv. 49 (1974) 302–332