Homology invariants of cyclic coverings with application to links

Shinohara, Y.; Sumners, D. W.

doi:10.1090/S0002-9947-1972-0284999-X

Homology invariants of cyclic coverings with application to links
HTML articles powered by AMS MathViewer

by Y. Shinohara and D. W. Sumners PDF

Trans. Amer. Math. Soc. 163 (1972), 101-121 Request permission

Abstract:

The main purpose of this paper is to study the homology of cyclic covering spaces of a codimension two link. The integral (rational) homology groups of an infinite cyclic cover of a finite complex can be considered as finitely generated modules over the integral (rational) group ring of the integers. We first describe the properties of the invariants of these modules for certain finite complexes related to the complementary space of links. We apply this result to the homology invariants of the infinite cyclic cover of a higher dimensional link. Further, we show that the homology invariants of the infinite cyclic cover detect geometric splittability of a link. Finally, we study the homology of finite unbranched and branched cyclic covers of a link.

References

Zur Isotopies zweidimensionaler Flächen im

4

J. J. Andrews and M. L. Curtis, Knotted 2-spheres in the 4-sphere, Ann. of Math. (2) 70 (1959), 565–571. MR 107239, DOI 10.2307/1970330
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
Richard C. Blanchfield, Intersection theory of manifolds with operators with applications to knot theory, Ann. of Math. (2) 65 (1957), 340–356. MR 85512, DOI 10.2307/1969966
R. H. Crowell, $H_{2}$ of subgroups of knots groups, Illinois J. Math. 14 (1970), 665–673. MR 266191
D. S. Cochran and R. H. Crowell, $H_{2}(G^{\prime } )$ for tamely embedded graphs, Quart. J. Math. Oxford Ser. (2) 21 (1970), 25–27. MR 258013, DOI 10.1093/qmath/21.1.25
Richard H. Crowell and Ralph H. Fox, Introduction to knot theory, Ginn and Company, Boston, Mass., 1963. Based upon lectures given at Haverford College under the Philips Lecture Program. MR 0146828
D. B. A. Epstein, Linking spheres, Proc. Cambridge Philos. Soc. 56 (1960), 215–219. MR 117740, DOI 10.1017/s0305004100034496
Shin’ichi Kinoshita, Alexander polynomials as isotopy invariants. I, Osaka Math. J. 10 (1958), 263–271. MR 102819

Elementary ideals of linear graphs in a $3$-sphere

J. Levine, A characterization of knot polynomials, Topology 4 (1965), 135–141. MR 180964, DOI 10.1016/0040-9383(65)90061-3
J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244. MR 246314, DOI 10.1007/BF02564525
J. Levine, Polynomial invariants of knots of codimension two, Ann. of Math. (2) 84 (1966), 537–554. MR 200922, DOI 10.2307/1970459
J. Levine, Unknotting spheres in codimension two, Topology 4 (1965), 9–16. MR 179803, DOI 10.1016/0040-9383(65)90045-5
John W. Milnor, Infinite cyclic coverings, Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967) Prindle, Weber & Schmidt, Boston, Mass., 1968, pp. 115–133. MR 0242163
Kunio Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387–422. MR 171275, DOI 10.1090/S0002-9947-1965-0171275-5
L. P. Neuwirth, Knot groups, Annals of Mathematics Studies, No. 56, Princeton University Press, Princeton, N.J., 1965. MR 0176462
C. D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots, Ann. of Math. (2) 66 (1957), 1–26. MR 90053, DOI 10.2307/1970113
D. W. Sumners, On an unlinking theorem, Proc. Cambridge Philos. Soc. 71 (1972), 1–4. MR 290352, DOI 10.1017/s0305004100050180
D. W. Sumners, $H_{2}$ of the commutator subgroup of a knot group, Proc. Amer. Math. Soc. 28 (1971), 319–320. MR 275416, DOI 10.1090/S0002-9939-1971-0275416-9

Zur Isotopie zweidimensionaler Flächen im

6

Hans J. Zassenhaus, The theory of groups, Chelsea Publishing Co., New York, 1958. 2nd ed. MR 0091275
E. C. Zeeman, Linking spheres, Abh. Math. Sem. Univ. Hamburg 24 (1960), 149–153. MR 117739, DOI 10.1007/BF02942027