Whitney towers and gropes in 4–manifolds

Schneiderman, Rob

doi:10.1090/S0002-9947-05-03768-2

Whitney towers and gropes in 4–manifolds
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Trans. Amer. Math. Soc. 358 (2006), 4251-4278 Request permission

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

Dror Bar-Natan, Vassiliev homotopy string link invariants, J. Knot Theory Ramifications 4 (1995), no. 1, 13–32. MR 1321289, DOI 10.1142/S021821659500003X
Tim D. Cochran, $k$-cobordism for links in $S^3$, Trans. Amer. Math. Soc. 327 (1991), no. 2, 641–654. MR 1055569, DOI 10.1090/S0002-9947-1991-1055569-2
Tim D. Cochran, Geometric invariants of link cobordism, Comment. Math. Helv. 60 (1985), no. 2, 291–311. MR 800009, DOI 10.1007/BF02567416
Tim D. Cochran, Derivatives of links: Milnor’s concordance invariants and Massey’s products, Mem. Amer. Math. Soc. 84 (1990), no. 427, x+73. MR 1042041, DOI 10.1090/memo/0427
Tim D. Cochran, Kent E. Orr, and Peter Teichner, Knot concordance, Whitney towers and $L^2$-signatures, Ann. of Math. (2) 157 (2003), no. 2, 433–519. MR 1973052, DOI 10.4007/annals.2003.157.433
J. Conant, R. Schneiderman and P. Teichner. Jacobi identities in low-dimensional topology, Preprint (2004).
James Conant and Peter Teichner, Grope cobordism of classical knots, Topology 43 (2004), no. 1, 119–156. MR 2030589, DOI 10.1016/S0040-9383(03)00031-4
James Conant and Peter Teichner, Grope cobordism and Feynman diagrams, Math. Ann. 328 (2004), no. 1-2, 135–171. MR 2030373, DOI 10.1007/s00208-003-0477-y
Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geometry 17 (1982), no. 3, 357–453. MR 679066
Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR 1201584
Michael H. Freedman and Peter Teichner, $4$-manifold topology. II. Dwyer’s filtration and surgery kernels, Invent. Math. 122 (1995), no. 3, 531–557. MR 1359603, DOI 10.1007/BF01231455
Kiyoshi Igusa and Kent E. Orr, Links, pictures and the homology of nilpotent groups, Topology 40 (2001), no. 6, 1125–1166. MR 1867241, DOI 10.1016/S0040-9383(00)00002-1
Vyacheslav S. Krushkal, Exponential separation in 4-manifolds, Geom. Topol. 4 (2000), 397–405. MR 1796497, DOI 10.2140/gt.2000.4.397
Vyacheslav S. Krushkal and Frank Quinn, Subexponential groups in 4-manifold topology, Geom. Topol. 4 (2000), 407–430. MR 1796498, DOI 10.2140/gt.2000.4.407
Vyacheslav S. Krushkal and Peter Teichner, Alexander duality, gropes and link homotopy, Geom. Topol. 1 (1997), 51–69. MR 1475554, DOI 10.2140/gt.1997.1.51
W. Magnus, A. Karass and D. Solitar, Combinatoral group theory, Dover Publications Inc. (1976).
Kent E. Orr, Homotopy invariants of links, Invent. Math. 95 (1989), no. 2, 379–394. MR 974908, DOI 10.1007/BF01393902
Frank Quinn, The embedding theorem for towers, Four-manifold theory (Durham, N.H., 1982) Contemp. Math., vol. 35, Amer. Math. Soc., Providence, RI, 1984, pp. 461–471. MR 780593, DOI 10.1090/conm/035/780593
R. Schneiderman Simple Whitney towers, half-gropes and the Arf invariant of a knot, Preprint (2002) http://front.math.ucdavis.edu/math.GT/0310304.
Rob Schneiderman and Peter Teichner, Higher order intersection numbers of 2-spheres in 4-manifolds, Algebr. Geom. Topol. 1 (2001), 1–29. MR 1790501, DOI 10.2140/agt.2001.1.1
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).
R. Schneiderman and P. Teichner. Pulling apart 2–spheres in 4–manifolds, in preparation.
R. Schneiderman and P. Teichner. Grope concordance of classical links, in preparation.
P. Teichner, What is ... a grope?, To appear in the Notices of the A.M.S.
Hassler Whitney, The self-intersections of a smooth $n$-manifold in $2n$-space, Ann. of Math. (2) 45 (1944), 220–246. MR 10274, DOI 10.2307/1969265
Masayuki Yamasaki, Whitney’s trick for three $2$-dimensional homology classes of $4$-manifolds, Proc. Amer. Math. Soc. 75 (1979), no. 2, 365–371. MR 532167, DOI 10.1090/S0002-9939-1979-0532167-8