Homology of regular coverings of spun CW pairs with applications to knot theory
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- by W. L. Motter PDF
- Proc. Amer. Math. Soc. 58 (1976), 331-338 Request permission
Abstract:
The $p$-spin of a pair of CW complexes, one a subcomplex of the other, is defined. The algebraic properties of certain tensor-product chain complexes are used to calculate the homology groups of regular coverings of such spun pairs where these groups are considered as modules over the integral group ring of the group of covering transformations. In §4, by using the free differential calculus and “geometric presentations” for fundamental groups, presentations for certain homology groups are developed. In §§5 and 6 these results are used to analyze the homology and associated invariants for coverings of complements of higher-dimensional knots and torus-like embeddings in the sphere obtained by $p$-spinning.References
- J. J. Andrews and S. J. Lomonaco, The second homotopy group of spun $2$-spheres in $4$-space, Ann. of Math. (2) 90 (1969), 199–204. MR 250308, DOI 10.2307/1970724
- J. J. Andrews and D. W. Sumners, On higher-dimensional fibered knots, Trans. Amer. Math. Soc. 153 (1971), 415–426. MR 271927, DOI 10.1090/S0002-9947-1971-0271927-5 E. Artin, Zur Isotopic zweidimensionaler Fläachen in $R_4^ \ast$, Abh. Math. Sem. Univ. Hamburg 4 (1926), 174-177.
- Sylvain Cappell, Superspinning and knot complements, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) Markham, Chicago, Ill., 1970, pp. 358–383. MR 0276972
- D. B. A. Epstein, Linking spheres, Proc. Cambridge Philos. Soc. 56 (1960), 215–219. MR 117740, DOI 10.1017/s0305004100034496 R. H. Fox, Free differential calculus. I. Derivation in the group ring, Ann. of Math. (2) 57 (1953), 547-560; II. The isomorphism problem of groups, Ann. of Math. (2) 59 (1954), 196-210; III. Subgroups, Ann. of Math. (2) 64 (1956), 407-414. MR 14, 843; 15, 931; 20 #2374.
- C. McA. Gordon, Some higher-dimensional knots with the same homotopy groups, Quart. J. Math. Oxford Ser. (2) 24 (1973), 411–422. MR 326746, DOI 10.1093/qmath/24.1.411
- Michel A. Kervaire, Les nœuds de dimensions supérieures, Bull. Soc. Math. France 93 (1965), 225–271 (French). MR 189052
- J. Levine, Polynomial invariants of knots of codimension two, Ann. of Math. (2) 84 (1966), 537–554. MR 200922, DOI 10.2307/1970459
- S. J. Lomonaco Jr., The second homotopy group of a spun knot, Topology 8 (1969), 95–98. MR 238318, DOI 10.1016/0040-9383(69)90001-9 A. T. Lundell and S. Weingram, Topology of CW complexes, Van Nostrand Reinhold, New York, 1969.
- W. S. Massey, On the normal bundle of a sphere imbedded in Euclidean space, Proc. Amer. Math. Soc. 10 (1959), 959–964. MR 109351, DOI 10.1090/S0002-9939-1959-0109351-8 W. A. McCallum, The higher homotopy groups of $k$-spun knots and links, Ph.D. Thesis, Florida State Univ., Tallahassee, Fla., 1973. W. L. Motter, Smooth embeddings of ${S^p} \times {S^q}$ in ${S^{p + q + 2}}$, Ph.D. Thesis, Florida State Univ., Tallahassee. Fla., 1973.
- Kurt Reidemeister, Complexes and homotopy chains, Bull. Amer. Math. Soc. 56 (1950), 297–307. MR 36506, DOI 10.1090/S0002-9904-1950-09404-X
- Y. Shinohara and D. W. Sumners, Homology invariants of cyclic coverings with application to links, Trans. Amer. Math. Soc. 163 (1972), 101–121. MR 284999, DOI 10.1090/S0002-9947-1972-0284999-X
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112 D. W. Sumners, The effect of spinning and twist-spinning on some knot invariants, Florida State University (preprint).
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 58 (1976), 331-338
- MSC: Primary 57C45
- DOI: https://doi.org/10.1090/S0002-9939-1976-0407852-6
- MathSciNet review: 0407852