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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Co-Hopficity of Seifert-bundle groups
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by F. González-Acuña, R. Litherland and W. Whitten
Trans. Amer. Math. Soc. 341 (1994), 143-155
DOI: https://doi.org/10.1090/S0002-9947-1994-1123454-6

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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Bibliographic Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 341 (1994), 143-155
  • MSC: Primary 57M05; Secondary 20C99, 55R05, 55R10, 57N10
  • DOI: https://doi.org/10.1090/S0002-9947-1994-1123454-6
  • MathSciNet review: 1123454