The Birman–Hilden property of covering spaces of nonorientable surfaces

F.  Atalan; E.  Medetogullari

doi:10.37863/umzh.v72i3.6044

F. Atalan Dep. Math., Atilim Univ., Ankara, Turke
E. Medetogullari Dep. Math., TED Univ., Ankara, Turkey

DOI: https://doi.org/10.37863/umzh.v72i3.6044

Abstract

UDC 517.5

Let $p: \widetilde{N} \rightarrow N$ be a finite covering space of nonorientable surfaces, where $\chi(\widetilde{N}) < 0$. We search whether or not $p$ has the Birman–Hilden property.

References

Aramayona, Javier; Leininger, Christopher J.; Souto, Juan. Injections of mapping class groups. Geom. Topol. 13 (2009), no. 5, 2523--2541. doi: 10.2140/gt.2009.13.2523

Birman, Joan S.; Hilden, Hugh M. Lifting and projecting homeomorphisms. Arch. Math. (Basel) 23 (1972), 428--434. doi: 10.1007/BF01304911

Birman, Joan S.; Hilden, Hugh M. On isotopies of homeomorphisms of Riemann surfaces. Ann. of Math. (2) 97 (1973), 424--439. doi: 10.2307/1970830

Birman, Joan S.; Wajnryb, Bronislaw. $3$-fold branched coverings and the mapping class group of a surface. Geometry and topology (College Park, Md., 1983/84), 24--46, Lecture Notes in Math., 1167, Springer, Berlin, 1985. doi: 10.1007/BFb0075214

Fuller, Terry. On fiber-preserving isotopies of surface homeomorphisms. Proc. Amer. Math. Soc. 129 (2001), no. 4, 1247--1254. doi: 10.1090/S0002-9939-00-05642-2

Maclachlan, C.; Harvey, W. J. On mapping-class groups and Teichmüller spaces. Proc. London Math. Soc. (3) 30 (1975), no. part, part 4, 496--512. doi: 10.1112/plms/s3-30.4.496

D. Margalit, R. R. Winarski,The Birman – Hilden Theory, (2017).https://arxiv.org/abs/1703.03448 .

Winarski, Rebecca R. Symmetry, isotopy, and irregular covers. Geom. Dedicata 177 (2015), 213--227. doi: 10.1007/s10711-014-9986-y

Wu, Ying Qing. Canonical reducing curves of surface homeomorphism. Acta Math. Sinica (N.S.) 3 (1987), no. 4, 305--313. doi: /10.1007/BF02559911 DOI: https://doi.org/10.1007/BF02559911