Homology cobordism and Seifert fibered $3$-manifolds
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- by Tim D. Cochran and Daniel Tanner PDF
- Proc. Amer. Math. Soc. 142 (2014), 4015-4024 Request permission
Abstract:
It is known that every closed oriented $3$-manifold is homology cobordant to a hyperbolic $3$-manifold. By contrast we show that many homology cobordism classes contain no Seifert fibered $3$-manifold. This is accomplished by determining the isomorphism type of the rational cohomology ring of all Seifert fibered $3$-manifolds with no $2$-torsion in their first homology. Then we exhibit families of examples of $3$-manifolds (obtained by surgery on links), with fixed linking form and cohomology ring, that are not homology cobordant to any Seifert fibered space (as shown by their rational cohomology rings). These examples are shown to represent distinct homology cobordism classes using higher Massey products and Milnor’s $\overline {\mu }$-invariants for links.References
- Kerstin Aaslepp, Michael Drawe, Claude Hayat-Legrand, Christian A. Sczesny, and Heiner Zieschang, On the cohomology of Seifert and graph manifolds, Proceedings of the Pacific Institute for the Mathematical Sciences Workshop “Invariants of Three-Manifolds” (Calgary, AB, 1999), 2003, pp. 3–32. MR 1953318, DOI 10.1016/S0166-8641(02)00040-8
- Glen E. Bredon, Topology and geometry, Graduate Texts in Mathematics, vol. 139, Springer-Verlag, New York, 1997. Corrected third printing of the 1993 original. MR 1700700
- J. Bryden and F. Deloup, A linking form conjecture for 3-manifolds, Advances in topological quantum field theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 179, Kluwer Acad. Publ., Dordrecht, 2004, pp. 253–265. MR 2147422, DOI 10.1007/978-1-4020-2772-7_{9}
- J. Bryden and P. Zvengrowski, The cohomology ring of the orientable Seifert manifolds. II, Proceedings of the Pacific Institute for the Mathematical Sciences Workshop “Invariants of Three-Manifolds” (Calgary, AB, 1999), 2003, pp. 213–257. MR 1953328, DOI 10.1016/S0166-8641(02)00061-5
- Tim Cochran and Paul Melvin, The Milnor degree of a 3-manifold, J. Topol. 3 (2010), no. 2, 405–423. MR 2651365, DOI 10.1112/jtopol/jtq011
- Tim D. Cochran, Derivatives of links: Milnor’s concordance invariants and Massey’s products, Mem. Amer. Math. Soc. 84 (1990), no. 427, x+73. MR 1042041, DOI 10.1090/memo/0427
- Tim D. Cochran, Amir Gerges, and Kent Orr, Dehn surgery equivalence relations on 3-manifolds, Math. Proc. Cambridge Philos. Soc. 131 (2001), no. 1, 97–127. MR 1833077, DOI 10.1017/S0305004101005151
- Allen Hatcher. Notes on basic $3$-manifold topology. Preprint, http://www.cornell.edu/ ~ hatcher/3M/3M.pdf.
- Jonathan A. Hillman, The linking pairings of orientable Seifert manifolds, Topology Appl. 158 (2011), no. 3, 468–478. MR 2754370, DOI 10.1016/j.topol.2010.11.023
- Michael Hutchings. Cup product and intersections. 2011. Preprint available at http://www.osti.gov/eprints/topicpages/documents/record/635/1949160.html.
- Sadayoshi Kojima, Milnor’s $\bar \mu$-invariants, Massey products and Whitney’s trick in $4$ dimensions, Topology Appl. 16 (1983), no. 1, 43–60. MR 702619, DOI 10.1016/0166-8641(83)90006-8
- Charles Livingston, Homology cobordisms of $3$-manifolds, knot concordances, and prime knots, Pacific J. Math. 94 (1981), no. 1, 193–206. MR 625818
- Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory, 2nd ed., Dover Publications, Inc., Mineola, NY, 2004. Presentations of groups in terms of generators and relations. MR 2109550
- Robert Myers, Homology cobordisms, link concordances, and hyperbolic $3$-manifolds, Trans. Amer. Math. Soc. 278 (1983), no. 1, 271–288. MR 697074, DOI 10.1090/S0002-9947-1983-0697074-4
- Nikolai Saveliev, Invariants for homology $3$-spheres, Encyclopaedia of Mathematical Sciences, vol. 140, Springer-Verlag, Berlin, 2002. Low-Dimensional Topology, I. MR 1941324, DOI 10.1007/978-3-662-04705-7
- John Stallings, Homology and central series of groups, J. Algebra 2 (1965), 170–181. MR 175956, DOI 10.1016/0021-8693(65)90017-7
Additional Information
- Tim D. Cochran
- Affiliation: Department of Mathematics MS-136, P.O. Box 1892, Rice University, Houston, Texas 77251-1892
- Email: cochran@rice.edu
- Daniel Tanner
- Affiliation: Epic Systems, 1979 Milky Way, Verona, Wisconsin 53593
- Email: dtanner@epic.com
- Received by editor(s): July 30, 2012
- Received by editor(s) in revised form: December 17, 2012
- Published electronically: July 22, 2014
- Additional Notes: The first author was partially supported by the National Science Foundation DMS-1006908
The second author was partially supported by the National Science Foundation DMS-0739420 - Communicated by: Daniel Ruberman
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 4015-4024
- MSC (2010): Primary 57Mxx; Secondary 57R75
- DOI: https://doi.org/10.1090/S0002-9939-2014-12122-8
- MathSciNet review: 3251741