## Geometry & Topology

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

Shelly L Harvey

#### 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 $(4k−1)$–dimensional manifolds. For $3$–dimensional manifolds, we show that ${ρn|n∈ℕ}$ is a linearly independent set and for each $n≥0$, the image of $ρn$ is an infinitely generated and dense subset of $ℝ$.

In their seminal work on knot concordance, T Cochran, K Orr and P Teichner define a filtration $ℱ(n)m$ 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 $m≥2$, 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 $n≥3$.

#### Article information

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

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

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

#### 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

#### 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