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

Harvey, Shelly L¹

¹ Department of Mathematics, Rice University, PO Box 1892, MS 136, Houston TX 77005-1892, USA

Geometry & topology, Tome 12 (2008) no. 1, pp. 387-430.

Résumé

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 $\rho$–invariant, we obtain new real-valued homology cobordism invariants ${\rho }_n$ for closed $\left(4k-1\right)$–dimensional manifolds. For $3$–dimensional manifolds, we show that $\left\{{\rho }_n|n\in \mathbb{N}\right\}$ is a linearly independent set and for each $n\ge 0$, the image of ${\rho }_n$ 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 ${\mathcal{ℱ}}_{\left(n\right)}^m$ of the $m$–component (string) link concordance group, called the $\left(n\right)$–solvable filtration. They also define a grope filtration ${\mathcal{G}}_n^m$. We show that ${\rho }_n$ vanishes for $\left(n+1\right)$–solvable links. Using this, and the nontriviality of ${\rho }_n$, we show that for each $m\ge 2$, the successive quotients of the $\left(n\right)$–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 $\left(n\right)$–solvable filtration are known to be infinite. However, for knots, it is unknown if these quotients have infinite rank when $n\ge 3$.

Détail

DOI : 10.2140/gt.2008.12.387

Keywords: link concordance, derived series, homology cobordism

Bibliographie

[1] S E Cappell, J L Shaneson, Link cobordism, Comment. Math. Helv. 55 (1980) 20

[2] J C Cha, K H Ko, Signature invariants of covering links, Trans. Amer. Math. Soc. 358 (2006) 3399

[3] J C Cha, C Livingston, Knot signature functions are independent, Proc. Amer. Math. Soc. 132 (2004) 2809

[4] 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.

[5] S Chang, S Weinberger, On invariants of Hirzebruch and Cheeger–Gromov, Geom. Topol. 7 (2003) 311

[6] T D Cochran, Derivatives of links: Milnor's concordance invariants and Massey's products, Mem. Amer. Math. Soc. 84 (1990)

[7] T D Cochran, Noncommutative knot theory, Algebr. Geom. Topol. 4 (2004) 347

[8] 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

[9] T D Cochran, S Harvey, Homology and derived series of groups, Geom. Topol. 9 (2005) 2159

[10] T D Cochran, K E Orr, Not all links are concordant to boundary links, Ann. of Math. $(2)$ 138 (1993) 519

[11] T D Cochran, K E Orr, P Teichner, Knot concordance, Whitney towers and $L^2$-signatures, Ann. of Math. $(2)$ 157 (2003) 433

[12] T D Cochran, K E Orr, P Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004) 105

[13] T D Cochran, P Teichner, Knot concordance and von Neumann $\rho$-invariants, Duke Math. J. 137 (2007) 337

[14] P M Cohn, Free rings and their relations, London Mathematical Society Monographs 19, Academic Press [Harcourt Brace Jovanovich Publishers] (1985)

[15] M H Freedman, F Quinn, Topology of 4-manifolds, Princeton Mathematical Series 39, Princeton University Press (1990)

[16] M H Freedman, P Teichner, $4$-manifold topology. I. Subexponential groups, Invent. Math. 122 (1995) 509

[17] S Friedl, Link concordance, boundary link concordance and eta-invariants, Math. Proc. Cambridge Philos. Soc. 138 (2005) 437

[18] K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1

[19] S L Harvey, Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm, Topology 44 (2005) 895

[20] J C Hausmann, On the homotopy of nonnilpotent spaces, Math. Z. 178 (1981) 115

[21] D M Kan, W P Thurston, Every connected space has the homology of a $K(\pi ,1)$, Topology 15 (1976) 253

[22] K H Ko, Seifert matrices and boundary link cobordisms, Trans. Amer. Math. Soc. 299 (1987) 657

[23] J Y Le Dimet, Cobordisme d'enlacements de disques, Mém. Soc. Math. France (N.S.) (1988)

[24] J Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969) 229

[25] J Levine, Concordance of Boundary Links, J. Knot Theory Ramifications 16 (2007) 1111

[26] W Lück, T Schick, Various $L^2$-signatures and a topological $L^2$-signature theorem, from: "High-dimensional manifold topology", World Sci. Publ., River Edge, NJ (2003) 362

[27] W Mio, On boundary-link cobordism, Math. Proc. Cambridge Philos. Soc. 101 (1987) 259

[28] H Murakami, Y Nakanishi, On a certain move generating link-homology, Math. Ann. 284 (1989) 75

[29] D S Passman, The algebraic structure of group rings, Robert E. Krieger Publishing Co. (1985)

[30] D Sheiham, Invariants of boundary link cobordism, Mem. Amer. Math. Soc. 165 (2003)

[31] J Stallings, Homology and central series of groups, J. Algebra 2 (1965) 170

[32] B Stenström, Rings of quotients, Springer (1975)

[33] R Strebel, Homological methods applied to the derived series of groups, Comment. Math. Helv. 49 (1974) 302