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Transactions of the American Mathematical Society

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Co-Hopficity of Seifert-bundle groups

Authors: F. González-Acuña, R. Litherland and W. Whitten
Journal: Trans. Amer. Math. Soc. 341 (1994), 143-155
MSC: Primary 57M05; Secondary 20C99, 55R05, 55R10, 57N10
MathSciNet review: 1123454
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Abstract: A group $ G$ is cohopfian, if every monomorphism $ G \to G$ is an automorphism. In this paper, we answer the cohopficity question for the fundamental groups of compact Seifert fiber spaces (or Seifert bundles, in the current vernacular). If $ M$ is a closed Seifert bundle, then the following are equivalent: (a) $ {\pi _1}M$ is cohopfian; (b) $ M$ does not cover itself nontrivially; (c) $ M$ admits a geometric structure modeled on $ {S^3}$ or on $ {\tilde{\text{SL}_2\mathbf{R}}}$. If $ M$ is a compact Seifert bundle with nonempty boundary, then $ {\pi _1}M$ is not cohopfian.

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