Equivariant intersection forms, knots in $S^ 4$, and rotations in $2$-spheres
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- by Steven P. Plotnick
- Trans. Amer. Math. Soc. 296 (1986), 543-575
- DOI: https://doi.org/10.1090/S0002-9947-1986-0846597-6
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Abstract:
We consider the problem of distinguishing the homotopy types of certain pairs of nonsimply-connected four-manifolds, which have identical three-skeleta and intersection pairings, by the equivariant isometry classes of the intersection pairings on their universal covers. As applications of our calculations, we: (i) construct distinct homology four-spheres with the same three-skeleta, (ii) generalize a theorem of Gordon to show that any nontrivial fibered knot in ${S^4}$ with odd order monodromy is not determined by its complement, and (iii) give a more constructive proof of a theorem of Hendriks concerning rotations in two-spheres embedded in threemanifolds.References
- 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
- 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 S. Eilenberg and S. Mac Lane, Homology of spaces with operators. II, Trans. Amer. Math. Soc. 65 (1949), 49-99.
- D. B. A. Epstein, The degree of a map, Proc. London Math. Soc. (3) 16 (1966), 369–383. MR 192475, DOI 10.1112/plms/s3-16.1.369
- Ronald Fintushel, Locally smooth circle actions on homotopy $4$-spheres, Duke Math. J. 43 (1976), no. 1, 63–70. MR 394716
- John L. Friedman and Donald M. Witt, Homotopy is not isotopy for homeomorphisms of $3$-manifolds, Topology 25 (1986), no. 1, 35–44. MR 836722, DOI 10.1016/0040-9383(86)90003-0
- Herman Gluck, The embedding of two-spheres in the four-sphere, Trans. Amer. Math. Soc. 104 (1962), 308–333. MR 146807, DOI 10.1090/S0002-9947-1962-0146807-0
- C. McA. Gordon, Knots in the $4$-sphere, Comment. Math. Helv. 51 (1976), no. 4, 585–596. MR 440561, DOI 10.1007/BF02568175
- John Hempel, $3$-Manifolds, Annals of Mathematics Studies, No. 86, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. MR 0415619
- Harrie Hendriks, Applications de la théorie d’obstruction en dimension $3$, Bull. Soc. Math. France Mém. 53 (1977), 81–196 (French). MR 474305, DOI 10.24033/msmf.237
- P. J. Hilton, On the homotopy groups of the union of spheres, J. London Math. Soc. 30 (1955), 154–172. MR 68218, DOI 10.1112/jlms/s1-30.2.154 N. V. Ivanov, Homotopy of spaces of automorphisms of some three-dimensional manifolds, Soviet Math. Dokl. 20 (1979).
- Mitsuyoshi Kato, A concordance classification of $\textrm {PL}$ homeomorphisms of $S^{p}\times S^{q}$, Topology 8 (1969), 371–383. MR 256401, DOI 10.1016/0040-9383(69)90023-8
- 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
- François Laudenbach, Topologie de la dimension trois: homotopie et isotopie, Astérisque, No. 12, Société Mathématique de France, Paris, 1974 (French). With an English summary and table of contents. MR 0356056
- Ronnie Lee, Semicharacteristic classes, Topology 12 (1973), 183–199. MR 362367, DOI 10.1016/0040-9383(73)90006-2 S. Mac Lane and J. H. C. Whitehead, On the $3$-type of a complex, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 41-48.
- Darryl McCullough, Connected sums of aspherical manifolds, Indiana Univ. Math. J. 30 (1981), no. 1, 17–28. MR 600029, DOI 10.1512/iumj.1981.30.30002
- William H. Meeks III and Shing Tung Yau, Topology of three-dimensional manifolds and the embedding problems in minimal surface theory, Ann. of Math. (2) 112 (1980), no. 3, 441–484. MR 595203, DOI 10.2307/1971088
- John Milnor, Groups which act on $S^n$ without fixed points, Amer. J. Math. 79 (1957), 623–630. MR 90056, DOI 10.2307/2372566 —, On simply connected $4$-manifolds, Symposium Internacional Topologia Algebraica, Mexico, 1958, pp. 122-128.
- José M. Montesinos, On twins in the four-sphere. I, Quart. J. Math. Oxford Ser. (2) 34 (1983), no. 134, 171–199. MR 698205, DOI 10.1093/qmath/34.2.171
- Paul Olum, Mappings of manifolds and the notion of degree, Ann. of Math. (2) 58 (1953), 458–480. MR 58212, DOI 10.2307/1969748
- Paul Olum, Homotopy type and singular homotopy type, Ann. of Math. (2) 60 (1954), 317–325. MR 63668, DOI 10.2307/1969635
- Peter Orlik, Seifert manifolds, Lecture Notes in Mathematics, Vol. 291, Springer-Verlag, Berlin-New York, 1972. MR 0426001, DOI 10.1007/BFb0060329
- Peter Orlik and Frank Raymond, Actions of $\textrm {SO}(2)$ on 3-manifolds, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 297–318. MR 0263112
- Peter Sie Pao, Nonlinear circle actions on the $4$-sphere and twisting spun knots, Topology 17 (1978), no. 3, 291–296. MR 508892, DOI 10.1016/0040-9383(78)90033-2
- Steven Plotnick, Circle actions and fundamental groups for homology $4$-spheres, Trans. Amer. Math. Soc. 273 (1982), no. 1, 393–404. MR 664051, DOI 10.1090/S0002-9947-1982-0664051-8
- Steven Plotnick, Homotopy equivalences and free modules, Topology 21 (1982), no. 1, 91–99. MR 630883, DOI 10.1016/0040-9383(82)90044-1
- Steven P. Plotnick, Finite group actions and nonseparating $2$-spheres, Proc. Amer. Math. Soc. 90 (1984), no. 3, 430–432. MR 728363, DOI 10.1090/S0002-9939-1984-0728363-9 —, Fibered knots in ${S^4}$-twisting, spinning, rolling, surgery, and branching, Four-Manifold Theory, AMS Summer Conference, Univ. of New Hampsphire, 1982: Contemporary Math., vol. 5, Amer. Math. Soc., Providence, R.I., 1984, pp. 437-459.
- Steven P. Plotnick and Alexander I. Suciu, Fibered knots and spherical space forms, J. London Math. Soc. (2) 35 (1987), no. 3, 514–526. MR 889373, DOI 10.1112/jlms/s2-35.3.514
- J. H. Rubinstein, On $3$-manifolds that have finite fundamental group and contain Klein bottles, Trans. Amer. Math. Soc. 251 (1979), 129–137. MR 531972, DOI 10.1090/S0002-9947-1979-0531972-6
- Joseph A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York-London-Sydney, 1967. MR 0217740
- E. C. Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471–495. MR 195085, DOI 10.1090/S0002-9947-1965-0195085-8
Bibliographic Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 296 (1986), 543-575
- MSC: Primary 57Q45; Secondary 57M10, 57M99, 57R50
- DOI: https://doi.org/10.1090/S0002-9947-1986-0846597-6
- MathSciNet review: 846597