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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
DOI: https://doi.org/10.1090/S0002-9947-1994-1118824-6
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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Keywords: Orbifold, fundamental region
Article copyright: © Copyright 1994 American Mathematical Society