nLab Birman-Hilden theory

Idea

Birman-Hilden theory concerns results that relate the mapping class groups of surfaces to those of their branched covers, suitably restricted.

References

The original articles:

• Joan S. Birman, Hugh M. Hilden: On the Mapping Class Group of Closed Surfaces as Covering Spaces, in: Advances in the Theory of Riemann Surfaces, Annals of Mathematics Studies 66, Princeton University Press (1971) 81-115 [doi:10.1515/9781400822492-007]

• Joan S. Birman, Hugh M. Hilden: On isotopies of homeomorphisms of Riemann surfaces, Annals of Mathematics 97 3 (1973) 424-439 [doi:10.2307/1970830, jstor:1970830]

Historical recollection:

• Joan S. Birman: My favorite paper, Celebratio Mathematica (Jan 2018) [celebratio:470]

Early further discussion:

• C. Maclachlan, W. J. Harvey: On Mapping-Class Groups and Teichmüller Spaces, Proceedings of the London Mathematical Society s3-30 4 (1975) 496-512 [doi:10.1112/plms/s3-30.4.496, plms:s3-30/4/496/1494994]

Review:

• Dan Margalit, Rebecca R. Winarski: The Birman-Hilden theory, Celebratio Mathematics (Jan 2017) [arXiv:1703.03448, celebratio:471]

• Dan Margalit, Rebecca R. Winarski: Braid groups and mapping class groups: The Birman–Hilden theory, Bull. London Math. Soc. 53 3 (2021) 643-659 [doi:10.1112/blms.12456]

See also:

• Tyrone Ghaswala, Rebecca R. Winarski: Lifting Homeomorphisms and Cyclic Branched Covers of Spheres, Michigan Math. J. 66 4 (2017) 885-890 [arXiv:1607.06060, doi:10.1307/mmj/1508810819]

The case of 3-manifolds instead of surfaces:

• Trent Lucas: Birman-Hilden theory for 3-manifolds, Advances in Mathematics 468 (2025) 110204 [doi:10.1016/j.aim.2025.110204]