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

 

Nonfibering spherical $ 3$-orbifolds


Author: William D. Dunbar
Journal: Trans. Amer. Math. Soc. 341 (1994), 121-142
MSC: Primary 57M50; Secondary 57S25
MathSciNet review: 1118824
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Abstract: Among the finite subgroups of $ SO(4)$, members of exactly $ 21$ conjugacy classes act on $ {S^3}$ preserving no fibration of $ {S^3}$ by circles. We identify the corresponding spherical $ 3$-orbifolds, i.e., for each such $ {\mathbf{G}} < SO(4)$, we describe the embedded trivalent graph $ \{ x \in {S^3}:\exists {\mathbf{I}} \ne {\mathbf{g}} \in {\mathbf{G}}$ s.t. $ {\mathbf{g}}(x) = x\} /{\mathbf{G}}$ in the topological space $ {S^3}/{\mathbf{G}}$ (which turns out to be homeomorphic to $ {S^3}$ in all cases). Explicit fundamental domains (of Dirichlet type) are described for $ 9$ of the groups, together with the identifications to be made on the boundary. The remaining $ 12$ spherical orbifolds are obtained as mirror images or (branched) covers of these.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1118824-6
PII: S 0002-9947(1994)1118824-6
Keywords: Orbifold, fundamental region
Article copyright: © Copyright 1994 American Mathematical Society