Nonfibering spherical 3-orbifolds

Dunbar, William D.

doi:10.1090/S0002-9947-1994-1118824-6

Nonfibering spherical $3$-orbifolds
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Trans. Amer. Math. Soc. 341 (1994), 121-142 Request permission

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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