High-dimensional knots with 𝜋₁≅𝑍 are determined by their complements in one more dimension than Farber’s range

Richter, William

doi:10.1090/S0002-9939-1994-1195730-8

High-dimensional knots with $\pi _ 1\cong \textbf {Z}$ are determined by their complements in one more dimension than Farber’s range
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Proc. Amer. Math. Soc. 120 (1994), 285-294 Request permission

Abstract:

The surgery theory of Browder, Lashof, and Shaneson reduces the study of high-dimensional smooth knots ${\Sigma ^n} \to {S^{n + 2}}$ with ${\pi _1} \cong \mathbb {Z}$ to homotopy theory. We apply Williams’s Poincaré embedding theorem to a highly connected Seifert surface. Then such knots are determined by their complements if the $\mathbb {Z}$-cover of the complement is $[(n + 2)/3]$-connected; we improve Farber’s work by one dimension.

References

Simple knots and the third equivariant Hopf invariant

J. M. Boardman and B. Steer, On Hopf invariants, Comment. Math. Helv. 42 (1967), 180–221. MR 221503, DOI 10.1007/BF02564417
William Browder, Diffeomorphisms of $1$-connected manifolds, Trans. Amer. Math. Soc. 128 (1967), 155–163. MR 212816, DOI 10.1090/S0002-9947-1967-0212816-0

Embedding

connected manifolds

72

William Browder, Embedding smooth manifolds, Proc. Internat. Congr. Math. (Moscow, 1966) Izdat. “Mir”, Moscow, 1968, pp. 712–719. MR 0238335
William Browder, Poincaré spaces, their normal fibrations and surgery, Invent. Math. 17 (1972), 191–202. MR 326743, DOI 10.1007/BF01425447
Sylvain E. Cappell and Julius L. Shaneson, There exist inequivalent knots with the same complement, Ann. of Math. (2) 103 (1976), no. 2, 349–353. MR 413117, DOI 10.2307/1970942
M. Š. Farber, An algebraic classification of some even-dimensional spherical knots. I, II, Trans. Amer. Math. Soc. 281 (1984), no. 2, 507–527, 529–570. MR 722762, DOI 10.1090/S0002-9947-1984-99927-2
M. Š. Farber, Isotopy types of knots of codimension two, Trans. Amer. Math. Soc. 261 (1980), no. 1, 185–209. MR 576871, DOI 10.1090/S0002-9947-1980-0576871-7
C. McA. Gordon, Knots in the $4$-sphere, Comment. Math. Helv. 51 (1976), no. 4, 585–596. MR 440561, DOI 10.1007/BF02568175
Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR 148075, DOI 10.1090/S0273-0979-2015-01504-1
M. Kervaire and C. Weber, A survey of multidimensional knots, Knot theory (Proc. Sem., Plans-sur-Bex, 1977) Lecture Notes in Math., vol. 685, Springer, Berlin, 1978, pp. 61–134. MR 521731
C. Kearton, An algebraic classification of some even-dimensional knots, Topology 15 (1976), no. 4, 363–373. MR 442948, DOI 10.1016/0040-9383(76)90030-6
C. Kearton, Blanchfield duality and simple knots, Trans. Amer. Math. Soc. 202 (1975), 141–160. MR 358796, DOI 10.1090/S0002-9947-1975-0358796-3
Sadayoshi Kojima, Classification of simple knots by Levine pairings, Comment. Math. Helv. 54 (1979), no. 3, 356–367. MR 543336, DOI 10.1007/BF02566280
Richard K. Lashof and Julius L. Shaneson, Classification of knots in codimension two, Bull. Amer. Math. Soc. 75 (1969), 171–175. MR 242175, DOI 10.1090/S0002-9904-1969-12197-X
J. Levine, An algebraic classification of some knots of codimension two, Comment. Math. Helv. 45 (1970), 185–198. MR 266226, DOI 10.1007/BF02567325
Jerome Levine, Knot modules. I, Trans. Amer. Math. Soc. 229 (1977), 1–50. MR 461518, DOI 10.1090/S0002-9947-1977-0461518-0
J. Levine, Unknotting spheres in codimension two, Topology 4 (1965), 9–16. MR 179803, DOI 10.1016/0040-9383(65)90045-5
Frank Quinn, Surgery on Poincaré and normal spaces, Bull. Amer. Math. Soc. 78 (1972), 262–267. MR 296955, DOI 10.1090/S0002-9904-1972-12950-1
Andrew Ranicki, Exact sequences in the algebraic theory of surgery, Mathematical Notes, vol. 26, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981. MR 620795
William Richter, A homotopy-theoretic proof of Williams’s metastable Poincaré embedding theorem, Duke Math. J. 88 (1997), no. 3, 435–447. MR 1455528, DOI 10.1215/S0012-7094-97-08818-9
Alexander I. Suciu, Inequivalent frame-spun knots with the same complement, Comment. Math. Helv. 67 (1992), no. 1, 47–63. MR 1144613, DOI 10.1007/BF02566488
C. T. C. Wall, Poincaré complexes. I, Ann. of Math. (2) 86 (1967), 213–245. MR 217791, DOI 10.2307/1970688
C. T. C. Wall, Surgery on compact manifolds, London Mathematical Society Monographs, No. 1, Academic Press, London-New York, 1970. MR 0431216
C. T. C. Wall, Classification problems in differential topology. IV. Thickenings, Topology 5 (1966), 73–94. MR 192509, DOI 10.1016/0040-9383(66)90005-X
George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508, DOI 10.1007/978-1-4612-6318-0
Bruce Williams, Applications of unstable normal invariants. I, Compositio Math. 38 (1979), no. 1, 55–66. MR 523263
Bruce Williams, Hopf invariants, localization and embeddings of Poincaré complexes, Pacific J. Math. 84 (1979), no. 1, 217–224. MR 559639, DOI 10.2140/pjm.1979.84.217