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Homology cobordism and Seifert fibered $ 3$-manifolds


Authors: Tim D. Cochran and Daniel Tanner
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
Published electronically: July 22, 2014
MathSciNet review: 3251741
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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.


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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

DOI: https://doi.org/10.1090/S0002-9939-2014-12122-8
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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.