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Whitney towers and gropes in 4-manifolds


Author: Rob Schneiderman
Journal: Trans. Amer. Math. Soc. 358 (2006), 4251-4278
MSC (2000): Primary 57M99; Secondary 57M25
DOI: https://doi.org/10.1090/S0002-9947-05-03768-2
Published electronically: June 21, 2005
MathSciNet review: 2231378
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Abstract: Many open problems and important theorems in low-dimensional topology have been formulated as statements about certain 2-complexes called gropes. This paper describes a precise correspondence between embedded gropes in 4-manifolds and the failure of the Whitney move in terms of iterated `towers' of Whitney disks. The `flexibility' of these Whitney towers is used to demonstrate some geometric consequences for knot and link concordance connected to $n$-solvability, $k$-cobordism and grope concordance. The key observation is that the essential structure of gropes and Whitney towers can be described by embedded unitrivalent trees which can be controlled during surgeries and Whitney moves. It is shown that a Whitney move in a Whitney tower induces an IHX (Jacobi) relation on the embedded trees.


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

  • 1. D. Bar-Natan. Vassiliev homotopy string link invariants, J. Knot Theory Ramifications 4 (1995) 13-32. MR 1321289 (96b:57004)
  • 2. T. Cochran. $k$-cobordism for links in $S^3$, Trans. A.M.S. Vol. 327 No. 2 (1991) 641-654.MR 1055569 (92a:57003)
  • 3. T. Cochran. Geometric invariants of link cobordism, Comment. Math. Helv. Vol. 60 (1985) 291-311.MR 0800009 (87f:57021)
  • 4. T. Cochran. Derivatives of links, Milnor's concordance invariants and Massey products, Mem. Amer. Math. Soc. Vol. 84 No. 427 (1990). MR 1042041 (91c:57005)
  • 5. T. Cochran, K. Orr and P. Teichner. Knot concordance, Whitney towers and $L^2$-signatures, Annals of Math. Vol. 157 No.2 (2003) 433-519. MR 1973052 (2004i:57003)
  • 6. J. Conant, R. Schneiderman and P. Teichner. Jacobi identities in low-dimensional topology, Preprint (2004).
  • 7. J. Conant and P. Teichner, Grope cobordism of classical knots, Topology, Vol. 43, Issue 1 (2004); 119-156. MR 2030589 (2004k:57006)
  • 8. J. Conant and P. Teichner, Grope cobordism and Feynman diagrams, Math. Annalen Vol. 328 Nos. 1-2 (2004), 135-171. MR 2030373
  • 9. M. H. Freedman. The topology of four-dimensional manifolds, J. Diff. Geom. 17 (1982) 357-453. MR 0679066 (84b:57006)
  • 10. M. H. Freedman and F. Quinn. The topology of 4-manifolds, Princeton Math. Series 39 Princeton NJ (1990). MR 1201584 (94b:57021)
  • 11. M. H. Freedman and P. Teichner. 4-manifold topology II: Dwyer's filtration and surgery kernels, Invent.Math. 122 (1995) 531-557. MR 1359603 (96k:57016)
  • 12. K. Igusa and K. Orr. Links, pictures and the homology of nilpotent groups, Topology 40 (2001) 1125-1166. MR 1867241 (2003a:57011)
  • 13. V. Krushkal. Exponential separation in 4-manifolds, Geometry and Topology Vol. 4 (2000) 397-405. MR 1796497 (2001i:57030)
  • 14. V. Krushkal and F. Quinn. Subexponential groups in 4-manifold topology, Geometry and Topology Vol. 4 (2000) 407-430. MR 1796498 (2001i:57031)
  • 15. V. Krushkal and P. Teichner. Alexander duality, gropes and link homotopy, Geometry and Topology Vol. 1 (1997) 51-69. MR 1475554 (98i:57013)
  • 16. W. Magnus, A. Karass and D. Solitar, Combinatoral group theory, Dover Publications Inc. (1976).
  • 17. K. Orr. Homotopy invariants of links, Invent. Math. Vol. 95 (1989) 379-394. MR 0974908 (90a:57012)
  • 18. F. Quinn. Embedding towers in 4-manifolds, Four-manifold Theory AMS Contemporary Math 35 (1984) 461-472. MR 0780593 (86f:57014)
  • 19. R. Schneiderman Simple Whitney towers, half-gropes and the Arf invariant of a knot, Preprint (2002) http://front.math.ucdavis.edu/math.GT/0310304.
  • 20. R. Schneiderman and P. Teichner. Higher order intersection numbers of 2-spheres in 4-manifolds, Algebraic and Geometric Topology Vol. 1 (2001) 1-29.MR 1790501 (2002c:57036)
  • 21. R. Schneiderman and P. Teichner. Whitney towers and the Kontsevich integral, Preprint (2003) (to appear in Proceedings of the CassonFest Geometry and Topology Monograph Series).
  • 22. R. Schneiderman and P. Teichner. Pulling apart 2-spheres in 4-manifolds, in preparation.
  • 23. R. Schneiderman and P. Teichner. Grope concordance of classical links, in preparation.
  • 24. P. Teichner, What is ... a grope?, To appear in the Notices of the A.M.S.
  • 25. H. Whitney. The self intersections of a smooth $n$-manifold in $2n$-space, Annals of Math. 45 (1944) 220-246. MR 0010274 (5:273g)
  • 26. M. Yamasaki. Whitney's trick for three 2-dimensional homology classes of 4-manifolds, Proc. Amer. Math. Soc. 75 (1979) 365-371. MR 0532167 (80k:57024)

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Additional Information

Rob Schneiderman
Affiliation: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012-1185
Email: schneiderman@courant.nyu.edu

DOI: https://doi.org/10.1090/S0002-9947-05-03768-2
Keywords: Whitney tower, grope, 4--manifold, Whitney move, $n$-solvability, $k$-cobordism, grope concordance, IHX relation
Received by editor(s): March 7, 2004
Received by editor(s) in revised form: May 25, 2004
Published electronically: June 21, 2005
Additional Notes: The author is an NSF VIGRE postdoctoral fellow at the Courant Institute of Mathematical Sciences.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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