Cyclic cover explained

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 and the algebraic geometry of Calabi–Yau manifolds.

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

r may induce a cyclic Galois covering with cyclic group of order r .

Further reading

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

Notes and References

1. Book: Seifert and Threlfall, A Textbook of Topology. 1980. Academic Press. 9780080874050. 292. registration. cyclic covering.. 25 August 2017. en.
2. Book: Rohde. Jan Christian. Cyclic coverings, Calabi-Yau manifolds and complex multiplication. 2009. Springer. Berlin. 978-3-642-00639-5. 59–62. [Online-Ausg.]..
3. Web site: Milnor. John. Infinite cyclic coverings. Conference on the Topology of Manifolds. Vol. 13. 1968.. 25 August 2017.
4. 1310.3951. Ambro. Florin. Cyclic covers and toroidal embeddings. math.AG. 2013.