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Transactions of the American Mathematical Society

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Higher-order Alexander invariants and filtrations of the knot concordance group

Authors: Tim D. Cochran and Taehee Kim
Journal: Trans. Amer. Math. Soc. 360 (2008), 1407-1441
MSC (2000): Primary 57M25
Published electronically: October 5, 2007
MathSciNet review: 2357701
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Abstract: We establish certain ``nontriviality'' results for several filtrations of the smooth and topological knot concordance groups. First, as regards the n-solvable filtration of the topological knot concordance group, $ \mathcal{C}$, defined by K. Orr, P. Teichner and the first author:

$\displaystyle 0\subset\cdots\subset\mathcal{F}_{(n.5)}\subset\mathcal{F}_{(n)}\... ...}_{(1.0)}\subset\mathcal{F}_{(0.5)} \subset\mathcal{F}_{(0)}\subset\mathcal{C},$

we refine the recent nontriviality results of Cochran and Teichner by including information on the Alexander modules. These results also extend those of C. Livingston and the second author. We exhibit similar structure in the closely related symmetric Grope filtration of $ \mathcal{C}$. We also show that the Grope filtration of the smooth concordance group is nontrivial using examples that cannot be distinguished by the Ozsváth-Szabó $ \tau$-invariant nor by J. Rasmussen's $ s$-invariant. Our broader contribution is to establish, in ``the relative case'', the key homological results whose analogues Cochran-Orr-Teichner established in ``the absolute case''.

We say two knots $ K_0$ and $ K_1$ are concordant modulo $ n$-solvability if $ K_0\char93 (-K_1)\in \mathcal{F}_{(n)}$. Our main result is that, for any knot $ K$ whose classical Alexander polynomial has degree greater than 2, and for any positive integer $ n$, there exist infinitely many knots $ K_i$ that are concordant to $ K$ modulo $ n$-solvability, but are all distinct modulo $ n.5$-solvability. Moreover, the $ K_i$ and $ K$ share the same classical Seifert matrix and Alexander module as well as sharing the same higher-order Alexander modules and Seifert presentations up to order $ n-1$.

References [Enhancements On Off] (What's this?)

  • [CS] S. Cappell and J. L. Shaneson, The codimension two placement problem and homology equivalent manifolds, Ann. of Math. (2) 99 (1974), 277-348. MR 0339216 (49:3978)
  • [CG] A. J. Casson and C. McA. Gordon, Cobordism of classical knots, printed notes, Orsay, 1975, published in A la recherche de la topologie perdue 62, ed. Guilllou and Marin, Progress in Mathematics, 1986, pp. 181-199.
  • [ChG] J. Cheeger and M. Gromov, Bounds on the von Neumann dimension of $ L\sp 2$-cohomology and the Gauss-Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985), no. 1, 1-34. MR 806699 (87d:58136)
  • [COT1] T. D. Cochran, K. E. Orr, and P. Teichner, Knot concordance, Whitney towers and $ L^2$-signatures, Ann. of Math. (2) 157 (2003), no. 2, 433-519. MR 1973052 (2004i:57003)
  • [COT2] T. D. Cochran, K. E. Orr, and P. Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004), no. 1, 105-123. MR 2031301 (2004k:57005)
  • [C] T. Cochran, Noncommutative Knot Theory, Algebr. Geom. Topol. 4 (2004), 347-398. MR 2077670 (2005k:57023)
  • [CT] T. D. Cochran and P. Teichner, Knot concordance and von Neumann $ \eta$-invariants, Duke Math. Journal 137 (2007), no. 2, 337-379.
  • [F] M. H. Freedman, The topology of four dimensional manifolds, J. Differential Geom. 17 (1982), no. 3, 357-453. MR 679066 (84b:57006)
  • [FQ] M. H. Freedman and F. Quinn, Topology of $ 4$-manifolds, Princeton Mathematical Series, 39, Princeton University Press, Princeton, NJ, 1990. MR 1201584 (94b:57021)
  • [Fr] S. Friedl, Eta invariants as sliceness obstructions and their relation to Casson-Gordon invariants, Algebr. Geom. Topol. 4 (2004), 893-934. MR 2100685 (2005j:57016)
  • [FT] S. Friedl and P. Teichner, New topologically slice knots, Geom. Topol. 9 (2005), 2129-2158. MR 2209368 (2007b:57007)
  • [G] P. M. Gilmer, Slice knots in $ S^3$, Quart. J. Math. Oxford Ser. (2) 34 (1983), no. 135, 305-322. MR 711523 (85d:57004)
  • [Ha] S. Harvey, Higher-order polynomial invariants of $ 3$-manifolds giving lower bounds for the Thurston norm, Topology 44 no.5 (2005) 895-945. MR 2153977 (2006g:57019)
  • [K] T. Kim, An infinite family of non-concordant knots having the same Seifert form, Comment. Math. Helv. 80 (2005), no. 1, 147-155. MR 2130571 (2006a:57007)
  • [KL] P. Kirk and C. Livingston, Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants, Topology 38 (1999), no. 3, 635-661. MR 1670420 (2000c:57010)
  • [L] J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229-244. MR 0246314 (39:7618)
  • [Le] C. F. Letsche, An obstruction to slicing knots using the eta invariant, Math. Proc. Cambridge Philos. Soc. 128 (2000), no. 2, 301-319. MR 1735303 (2001b:57017)
  • [Li] C. Livingston, Seifert forms and concordance, Geom. Topol. 6 (2002), 403-408. MR 1928840 (2003f:57019)
  • [Ma] W.S. Massey, Algebraic Topology: An Introduction, Harcourt, Brace, and World, 1967. MR 0211390 (35:2271)
  • [OS] P. Ozsváth and Z. Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003), 615-639. MR 2026543 (2004i:57036)
  • [R] M. Ramachandran, von Neumann index theorems for manifolds with boundary, J. Differential Geom. 38 (1993), no. 2, 315-349. MR 1237487 (94j:58164)
  • [Ra] J. A. Rasmussen, Khovanov homology and the slice genus, preprint, 2004, arXiv:math. GT/0402131.
  • [Ste] B. Stenström, Rings of Quotients, Springer-Verlag, 1975, New York. MR 0389953 (52:10782)
  • [Str] R. Strebel, Homological methods applied to the derived series of groups, Comment. Math. Helv. 49 (1974), 302-332. MR 0354896 (50:7373)
  • [Wa] C. T. C. Wall, Surgery on Compact Manifolds, London Math. Soc. Monographs 1, Academic Press, 1970. MR 0431216 (55:4217)

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Additional Information

Tim D. Cochran
Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005-1892

Taehee Kim
Affiliation: Department of Mathematics, Konkuk University, Seoul 143-701, Korea

Received by editor(s): January 26, 2005
Received by editor(s) in revised form: October 21, 2005
Published electronically: October 5, 2007
Additional Notes: The first author was partially supported by the National Science Foundation
Article copyright: © Copyright 2007 American Mathematical Society

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