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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Seifert manifolds with fiber
spherical space forms


Authors: Jong Bum Lee, Kyung Bai Lee and Frank Raymond
Journal: Trans. Amer. Math. Soc. 348 (1996), 3763-3798
MSC (1991): Primary 57M50; Secondary 55R60, 57M05, 57M60
MathSciNet review: 1348866
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Abstract: We study the Seifert fiber spaces modeled on the product space $S^3 \times \mathbb {R}^2$. Such spaces are ``fiber bundles'' with singularities. The regular fibers are spherical space-forms of $S^3$, while singular fibers are finite quotients of regular fibers. For each of possible space-form groups $\Delta $ of $S^3$, we obtain a criterion for a group extension $\varPi $ of $\Delta $ to act on $S^3 \times \mathbb {R}^2$ as weakly $S^3$-equivariant maps, which gives rise to a Seifert fiber space modeled on $S^3 \times \mathbb {R}^2$ with weakly $S^3$-equivariant maps $ \mathrm {TOP}_{S^3}(S^3 \times \mathbb {R}^2)$ as the universal group. In the course of proving our main results, we also obtain an explicit formula for $H^2(Q; \mathbb {Z})$ for a cocompact crystallographic or Fuchsian group $Q$. Most of our methods for $S^3$ apply to compact Lie groups with discrete center, and we state some of our results in this general context.


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Additional Information

Jong Bum Lee
Affiliation: Department of Mathematics, Sogang University, Seoul 121–742, Korea
Email: jlee@ccs.sogang.ac.kr

Kyung Bai Lee
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email: kblee@.math.uoknor.edu

Frank Raymond
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: fraymond@math.lsa.umich.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01609-1
PII: S 0002-9947(96)01609-1
Keywords: Seifert fiber space, spherical space-form, Fuchsian group, crystallographic group, group extension, cohomology of groups
Received by editor(s): December 1, 1994
Received by editor(s) in revised form: September 7, 1995
Additional Notes: The first author was supported in part by the Basic Science Research Institute Program, Ministry of Education, 1994, Project No. BSRI-94-1422, and by TGRC-KOSEF, Korea.
The third author was supported in part by National Science Foundation grant DMS-9306240, U.S.A
Article copyright: © Copyright 1996 American Mathematical Society