Link homotopy with one codimension two component
HTML articles powered by AMS MathViewer
- by Paul A. Kirk
- Trans. Amer. Math. Soc. 319 (1990), 663-688
- DOI: https://doi.org/10.1090/S0002-9947-1990-0970268-X
- PDF | Request permission
Abstract:
Link maps with one codimension two component are studied and an invariant of link maps modulo link homotopy is constructed using ideas from knot theory and immersion theory. This invariant is used to give examples of nontrivial link homotopy classes and to show that there are infinitely many distinct link homotopy classes in many dimensions. A link map with the codimension two component embedded is shown to be nullhomotopic. These ideas are applied to the special case of $2$-spheres in ${S^4}$ to give simple examples of the failure of the Whitney trick in dimension $4$.References
- Roger Fenn and Dale Rolfsen, Spheres may link homotopically in $4$-space, J. London Math. Soc. (2) 34 (1986), no. 1, 177–184. MR 859159, DOI 10.1112/jlms/s2-34.1.177
- Nathan Habegger, On linking coefficients, Proc. Amer. Math. Soc. 96 (1986), no. 2, 353–359. MR 818471, DOI 10.1090/S0002-9939-1986-0818471-8 G. T. Jin, Invariants of two component links, Thesis, Brandeis University, 1988.
- André Haefliger, Plongements différentiables de variétés dans variétés, Comment. Math. Helv. 36 (1961), 47–82 (French). MR 145538, DOI 10.1007/BF02566892
- Allen Hatcher and Frank Quinn, Bordism invariants of intersections of submanifolds, Trans. Amer. Math. Soc. 200 (1974), 327–344. MR 353322, DOI 10.1090/S0002-9947-1974-0353322-6
- Morris W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR 119214, DOI 10.1090/S0002-9947-1959-0119214-4
- Philip S. Hirschhorn, On the “stable” homotopy type of knot complements, Illinois J. Math. 23 (1979), no. 1, 128–134. MR 516575 J. Hughes, Thesis, University of California, Berkeley, 1983.
- Paul A. Kirk, Link maps in the four sphere, Differential topology (Siegen, 1987) Lecture Notes in Math., vol. 1350, Springer, Berlin, 1988, pp. 31–43. MR 979332, DOI 10.1007/BFb0081467 —, Embedded links with one codimension two component are nullhomotopic in the metastable range, J. London Math. Soc. (to appear).
- Ulrich Koschorke, Higher order homotopy invariants for higher-dimensional link maps, Algebraic topology, Göttingen 1984, Lecture Notes in Math., vol. 1172, Springer, Berlin, 1985, pp. 116–129. MR 825777, DOI 10.1007/BFb0074427
- Ulrich Koschorke, Link maps and the geometry of their invariants, Manuscripta Math. 61 (1988), no. 4, 383–415. MR 952086, DOI 10.1007/BF01258596
- Ulrich Koschorke, Multiple point invariants of link maps, Differential topology (Siegen, 1987) Lecture Notes in Math., vol. 1350, Springer, Berlin, 1988, pp. 44–86. MR 979333, DOI 10.1007/BFb0081468 —, On link maps and their homotopy classification, preprint, 1988.
- Ulrich Koschorke and Dale Rolfsen, Higher-dimensional link operations and stable homotopy, Pacific J. Math. 139 (1989), no. 1, 87–106. MR 1010788
- Ulrich Koschorke and Brian Sanderson, Geometric interpretations of the generalized Hopf invariant, Math. Scand. 41 (1977), no. 2, 199–217. MR 474289, DOI 10.7146/math.scand.a-11714
- Jerome Levine, Knot modules. I, Trans. Amer. Math. Soc. 229 (1977), 1–50. MR 461518, DOI 10.1090/S0002-9947-1977-0461518-0
- W. S. Massey and D. Rolfsen, Homotopy classification of higher-dimensional links, Indiana Univ. Math. J. 34 (1985), no. 2, 375–391. MR 783921, DOI 10.1512/iumj.1985.34.34022
- John Milnor, Link groups, Ann. of Math. (2) 59 (1954), 177–195. MR 71020, DOI 10.2307/1969685
- John Milnor, A procedure for killing homotopy groups of differentiable manifolds. , Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, pp. 39–55. MR 0130696
- G. P. Scott, Homotopy links, Abh. Math. Sem. Univ. Hamburg 32 (1968), 186–190. MR 236912, DOI 10.1007/BF02993127
- 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
Bibliographic Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 319 (1990), 663-688
- MSC: Primary 57Q45
- DOI: https://doi.org/10.1090/S0002-9947-1990-0970268-X
- MathSciNet review: 970268