Co-Hopficity of Seifert-bundle groups

González-Acuña, F.; Litherland, R.; Whitten, W.

doi:10.1090/S0002-9947-1994-1123454-6

Co-Hopficity of Seifert-bundle groups
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by F. González-Acuña, R. Litherland and W. Whitten PDF

Trans. Amer. Math. Soc. 341 (1994), 143-155 Request permission

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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