Cyclic cover

In algebraic topology and algebraic geometry, a cyclic cover or cyclic covering is a covering space for which the set of covering transformations forms a cyclic group.[1][2] As with cyclic groups, there may be both finite and infinite cyclic covers.[3]

Cyclic covers have proven useful in the descriptions of knot topology[1][3] and the algebraic geometry of Calabi–Yau manifolds.[2]

In classical algebraic geometry, cyclic covers are a tool used to create new objects from existing ones through, for example, a field extension by a root element.[4] The powers of the root element form a cyclic group and provide the basis for a cyclic cover. A line bundle over a complex projective variety with torsion index may induce a cyclic Galois covering with cyclic group of order .

References

  1. 1 2 Seifert and Threlfall, A Textbook of Topology. Academic Press. 1980. p. 292. ISBN 9780080874050. Retrieved 25 August 2017.
  2. 1 2 Rohde, Jan Christian (2009). Cyclic coverings, Calabi-Yau manifolds and complex multiplication ([Online-Ausg.]. ed.). Berlin: Springer. pp. 59–62. ISBN 978-3-642-00639-5.
  3. 1 2 Milnor, John. "Infinite cyclic coverings" (PDF). Conference on the Topology of Manifolds. Vol. 13. 1968. Retrieved 25 August 2017.
  4. ↑ Ambro, Florin (2013). "Cyclic covers and toroidal embeddings". arXiv:1310.3951 [math.AG].

Further reading

  • Fedorchuk, Maksym (2011-05-13). "Cyclic covering morphisms on M0,n". arXiv:1105.0655 [math.AG].
  • Singh, Anurag K. (2002-08-28). "Cyclic covers of rings with rational singularities". arXiv:math/0208226.
  • "what is the cyclic cover trick?". MathOverflow. 19 June 2013. Retrieved 2017-08-26.


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