Equivariant intersection forms, knots in 𝑆⁴, and rotations in 2-spheres

Plotnick, Steven P.

doi:10.1090/S0002-9947-1986-0846597-6

Equivariant intersection forms, knots in $S^ 4$, and rotations in $2$-spheres
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Trans. Amer. Math. Soc. 296 (1986), 543-575 Request permission

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

Homology of spaces with operators

65

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, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. 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

Homotopy of spaces of automorphisms of some three-dimensional manifolds

20

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

On the

type of a complex

36

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

manifolds

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

twisting, spinning, rolling, surgery, and branching

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