Degree one maps between hyperbolic small 3-manifolds

Boileau, Michel; Wang, Shicheng

doi:10.1090/S0002-9939-07-09265-9

Degree one maps between hyperbolic small 3-manifolds
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by Michel Boileau and Shicheng Wang

Proc. Amer. Math. Soc. 135 (2007), 4023-4029

DOI: https://doi.org/10.1090/S0002-9939-07-09265-9

Published electronically: September 5, 2007

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Abstract:

We construct the first examples of degree one maps between non-homeomorphic closed hyperbolic small 3-manifolds.

References

Michel Boileau and Joan Porti, Geometrization of 3-orbifolds of cyclic type, Astérisque 272 (2001), 208 (English, with English and French summaries). Appendix A by Michael Heusener and Porti. MR 1844891
Michel Boileau and Shicheng Wang, Non-zero degree maps and surface bundles over $S^1$, J. Differential Geom. 43 (1996), no. 4, 789–806. MR 1412685
Michel Boileau and Shicheng Wang, Degree one maps between small 3-manifolds and Heegaard genus, Algebr. Geom. Topol. 5 (2005), 1433–1450. MR 2171816, DOI 10.2140/agt.2005.5.1433
Michel Boileau and Bruno Zimmermann, Symmetries of nonelliptic Montesinos links, Math. Ann. 277 (1987), no. 3, 563–584. MR 891592, DOI 10.1007/BF01458332
Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR 808776
Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987), no. 2, 237–300. MR 881270, DOI 10.2307/1971311
Nathan M. Dunfield, A table of boundary slopes of Montesinos knots, Topology 40 (2001), no. 2, 309–315. MR 1808223, DOI 10.1016/S0040-9383(99)00064-6
Nathan M. Dunfield, Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds, Invent. Math. 136 (1999), no. 3, 623–657. MR 1695208, DOI 10.1007/s002220050321
David Gabai, Detecting fibred links in $S^3$, Comment. Math. Helv. 61 (1986), no. 4, 519–555. MR 870705, DOI 10.1007/BF02621931
C. McA. Gordon and R. A. Litherland, Incompressible surfaces in branched coverings, The Smith conjecture (New York, 1979) Pure Appl. Math., vol. 112, Academic Press, Orlando, FL, 1984, pp. 139–152. MR 758466, DOI 10.1016/S0079-8169(08)61639-6
A. E. Hatcher, On the boundary curves of incompressible surfaces, Pacific J. Math. 99 (1982), no. 2, 373–377. MR 658066, DOI 10.2140/pjm.1982.99.373
A. Hatcher and W. Thurston, Incompressible surfaces in $2$-bridge knot complements, Invent. Math. 79 (1985), no. 2, 225–246. MR 778125, DOI 10.1007/BF01388971
Claude Hayat-Legrand, Shicheng Wang, and Heiner Zieschang, Minimal Seifert manifolds, Math. Ann. 308 (1997), no. 4, 673–700. MR 1464916, DOI 10.1007/s002080050096
Akio Kawauchi, An imitation theory of manifolds, Osaka J. Math. 26 (1989), no. 3, 447–464. MR 1021425
William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450, DOI 10.1090/cbms/043
John Milnor, Collected papers. Vol. 1, Publish or Perish, Inc., Houston, TX, 1994. Geometry. MR 1277810
José M. Montesinos, Surgery on links and double branched covers of $S^{3}$, Knots, groups, and $3$-manifolds (Papers dedicated to the memory of R. H. Fox), Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N.J., 1975, pp. 227–259. MR 0380802
Ulrich Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984), no. 1, 209–230. MR 732067, DOI 10.2140/pjm.1984.111.209
A. Hatcher and U. Oertel, Boundary slopes for Montesinos knots, Topology 28 (1989), no. 4, 453–480. MR 1030987, DOI 10.1016/0040-9383(89)90005-0
T. Ohtsuki, R. Riley and M Sakuma, Epimorphisms between 2-bridge link groups, Manuscript, June 2006.
Alan W. Reid and Shicheng Wang, Non-Haken $3$-manifolds are not large with respect to mappings of non-zero degree, Comm. Anal. Geom. 7 (1999), no. 1, 105–132. MR 1674109, DOI 10.4310/CAG.1999.v7.n1.a4
Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
Daniel Ruberman, Seifert surfaces of knots in $S^4$, Pacific J. Math. 145 (1990), no. 1, 97–116. MR 1066400, DOI 10.2140/pjm.1990.145.97
W. Thurston, Topology and Geometry of 3-manifolds, Princeton Lecture Notes, 1978.
Shicheng Wang, Non-zero degree maps between 3-manifolds, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 457–468. MR 1957056