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Birman-Hilden theory concerns results that relate the mapping class groups of surfaces to those of their branched covers, suitably restricted.
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]
Beware that the statements of Lem. 5.1 and Thm. 5 of Birman & Hilden 1973 are lacking a crucial assumption, discovered 43 years later:
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]
Joan Birman, Hugh Hilden: Erratum to “Isotopies of homeomorphisms of Riemann surfaces”, Ann. of Math. 185 1 (2017) 345-345 [doi:10.4007/annals.2017.185.1.9, pdf]
Historical recollection:
Early further discussion:
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]
The case of 3-manifolds instead of surfaces:
Last revised on March 1, 2026 at 17:37:18. See the history of this page for a list of all contributions to it.