Publication: Constructions of two-fold branched covering spaces
Loading...
Official URL
Full text at PDC
Publication Date
1986-12
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Pacific Journal of Mathematics
Citations
Exportar
Abstract
By equivariantly pasting together exteriors of links in S3 that are invariant under several different involutions of S3, we construct closed orientable 3-manifolds that are two-fold branched covering spaces of S3 in distinct ways, that is, with different branch sets. Sufficient conditions are given to guarantee when the constructed manifold M admits an induced involution, h, and when M∕h≅S3. Using the theory of characteristic submanifolds for Haken manifolds with incompressible boundary components, we also prove that doubles, D(K,ρ), of prime knots that are not strongly invertible are characterized by their two-fold branched covering spaces, when ρ≠0. If, however, K is strongly invertible, then the manifold branch covers distinct knots. Finally, the authors characterize the type of a prime knot by the double covers of the doubled knots, D(K;ρ,η) and D(K∗;ρ,η), of K and its mirror image K∗ when ρ and η are fixed, with ρ≠0 and η ∈{−2,2}.
Description
UCM subjects
Unesco subjects
Keywords
Citation
J. Birman, F. Gonzalez-Acuna and J. M. Montesinos, Heegaard splittings of prime 3-manifolds are not unique, Michigan Math. J., 23 (1976), 97-103.
Steven A. Bleiler, Knots prime on many strings, Trans. Amer. Math. Soc, (to appear).
M. Boileau, F. Gonzalez-Acuha and J. M. Montesinos, Surgery on double knots and symmetries, (to appear).
R. H. Bing and J. M. Martin, Cubes with knotted holes, Trans. Amer. Math. Soc, 155 (1971), 217-231.
G. Burde and K. Murasugi, Links and Seifert fiber spaces, Duke Math. J., 37 (1970), 89-93.
S. Furusawa and M. Sakuma, Dehn surgery on symmetric knots, Math. Sem. Notes, 11 (1983), 179-198.
F. Gonzalez Acuña, Ph.D. Thesis, Princeton University, 1969.
C. McA. Gordon and Wolfgang Heil, Cyclic normal subgroups of fundamental groups of 3-manifolds,Topology,14 (1975), 305-309.
C. McA. Gordon and R. A. Litherland, Incompressible surfaces in branched coverings, Proceedings of the Symposium on the Smith Conjecture, Columbia University, (1978), 139-152.
R. Hartley, Knots and involutions, Math. Zeit., 171 (1980),175-185.
C. D. Hodgson, Involutions and isotopies of lens spaces, MS Thesis, University of Melbourne,1981.
W. Jaco, Lectures on Three-Manifold Topology, Regional Conference Series43, Amer. Math. Soc,Providence, R. L,1980.
W. Jaco and P. Shalen, Seifert Fibered Spaces in 3-Manifolds, Memoirs of the Amer. Math. Soc,Vol. 21,No. 220,Amer. Math. Soc,Providence, R. I.,1979.
P. K. Kim, Involutions on Klein spaces M(p, q), Trans. Amer. Math. Soc,268 (1981), 377-409.
Kim and J. L. Tollefson, PL involutions of fibered 3-manifolds, Trans. Amer. Math. Soc,232 (1977),221-237.
Kim and J. L. Tollefson, Splitting the PL involutions of non-prime 3-manifolds, Michigan Math. J., 27 (1980),259-274.
K. W. Kwun and J. L. Tollefson, Extending PL involutions of a compact manifold, Amer. J. Math., 99 (1977),995-1001.
Jose M. Montesinos, Variedades de Seifert que son recubridores ciclicos ramificados de dos hojas, Bol. Soc. Math. Mexicana (2), 18 (1973), 1-32.
Jose M. Montesinos, Sugery on links and double branched covers, Knots, Groups, and 3-Manifolds, Ann. of Math. Studies, 84, 227-259, Princeton University Press, Princeton, N.J.,1975.
Jose M. Montesinos, Sobre la representacion de variedades tridimensionales, unpublished preprint (1975).
W. H. Meeks and P. Scott, Finite group actions on 3-manifolds, preprint.
Jose M. Montesinos and W. Whitten, Constructions of two-fold branchedcovering spaces, Abstracts Amer. Math. Soc, 4 (Mar., 1983), Abstract 802-57-11, p. 178.
J. H. Rubinstein, Representations of some 3-manifolds as 2-fold cyclicbranched covers ofS3, Notices Amer. Math. Soc,25 (1978), Abstract 78T-G7, A-18.
M. Sakuma, Surface bundles over S1 which are 2-fold branched cyclic coverings ofS3, Math. Sem.notes, 9 (1981), 159-180.
H. Schubert,Knotenund Vollringe, Acta Math., 90 (1953), 131-286.
M. Takahashi, Two knots with the same two fold branched covering space, Yokohama Math. J., 25 (1977), 91-99.
M. Takahashi, On homology spheres obtained by surgery on the figure-eight knot, Proc Sympos. Res. Inst. Math. Sci. Univ. Kyoto, 309 (1977) (in Japanese).
W. Thurston, The Geometry and Topology of 3-Manifolds, Lecture notes, Princeton University, 1978-1979.
Tollefson, involutions on S X S2 and other 3-manifolds, Trans. Amer. Math. Soc, 183 (1973), 139-152.
Tollefson, Involutions of Seifert fiber spaces, Pacific J. Math., 74 (1978), 519-529.
Tollefson, Involutions of sufficiently large 3-manifolds, Topoloy, 20 (1981), 323-352.
F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten I, II, Invent. Math., 3 (1967), 308-333; and Invent. Math., 4 (1967), 87-117.
F. Waldhausen, Uber Involutionen der 3-Sphre, Topology,8 (1969), 81-91.
W. Whitten, Algebraic and geometric characterizationsof knots, Invent. Math., 26 (1974), 259-270.
W. Whitten, Inverting doubleknots, Pacific J. Math,97 (1981), 209-216.
H. Zieschang, On extensions of fundamental groups of surfaces and related groups, Bull. Amer. Math. Soc,77 (1971), 1116-1119.
H. Zieschang, Addendum to: On extensions of fundamental groups of surfaces and related groups, Bull. Amer. Math. Soc,80 (1974), 366-367.