Erratum to `Isotopies of homeomorphisms of Riemann surfaces’

@article{Birman2017ErratumT,
  title={Erratum to `Isotopies of homeomorphisms of Riemann surfaces’},
  author={Joan S. Birman and Hugh M. Hilden},
  journal={Annals of Mathematics},
  year={2017},
  volume={185},
  pages={345-345}
}
There is an error in the statement and proof of Lemma 5.1 of [1]. The Lemma in question is true in some cases and false in others. The error does not affect the main body of [1], that is Theorems 1,2,3 and 4, but it does imply that Theorem 5, the proof of which uses Lemma 5.1, is true precisely when Lemma 5.1 is true and must be modified when it is false. Theorem 5 of [1] is well-known to be true in the case of hyperelliptic covering spaces of the punctured sphere. That application has been… 

Liftable homeomorphisms of rank two finite abelian branched covers

We investigate branched regular finite abelian A -covers of the 2-sphere, where every homeomorphism of the base (preserving the branch locus) lifts to a homeomorphism of the covering surface. In this

Configuration space, moduli space and 3-fold covering space

A function from configuration space to moduli space of surface may induce a homomorphism between their fundamental groups which are braid groups and mapping class groups of surface, respectively.

The Liftable Mapping Class Group

Broadly, this thesis lies at the interface of mapping class groups and covering spaces. The foundations of this area were laid down in the early 1970s by Birman and Hilden. Building on these

Configuration spaces, moduli spaces and 3-fold covering spaces

We have, in this paper, constructed a new non-geometric embedding of some braid group into the mapping class group of a surface which is induced by the 3-fold branched covering over a disk with some

MAPPING CLASS GROUPS AND ORDERED GROUPS

My research is in the area of geometric group theory and low dimensional topology. In particular, I am interested in the mapping class group of both infinite-type and finite-type surfaces, and

Lifting Homeomorphisms and Cyclic Branched Covers of Spheres

We characterize the cyclic branched covers of the 2-sphere where every homeomorphism of the sphere lifts to a homeomorphism of the covering surface. This answers a question that appeared in an early

The liftable mapping class group of balanced superelliptic covers

The hyperelliptic mapping class group has been studied in various contexts within topology and algebraic geometry. What makes this study tractable is that there is a surjective map from the

An infinite family of braid group representations

The $d$-fold ($d \geq 3$) branched coverings on a disk give an infinite family of nongeometric embeddings of braid groups into mapping class groups. We, in this paper, give new explicit expressions

LIFTING HOMEOMORPHISMS AND FINITE ABELIAN BRANCHED COVERS OF THE 2-SPHERE

  • Haimiao Chen
  • Mathematics
    Bulletin of the Australian Mathematical Society
  • 2022
<jats:p>We completely determine finite abelian regular branched covers of the 2-sphere <jats:inline-formula> <jats:alternatives> <jats:inline-graphic

References

SHOWING 1-3 OF 3 REFERENCES

Isotopies of homeomorphisms of Riemann surfaces and a theorem about Artin's braid group

Let X, X be orientable surfaces. Let (p, X, X) be a regular covering space, possibly branched, with finitely many branch points and a finite group of covering transformations. We require also that

Symmetry, isotopy, and irregular covers

We say that a covering space of surfaces $$S\rightarrow X$$S→X has the Birman–Hilden property if the subgroup of the mapping class group of $$X$$X consisting of mapping classes that have

Lifting Homeomorphisms and Cyclic Branched Covers of Spheres

We characterize the cyclic branched covers of the 2-sphere where every homeomorphism of the sphere lifts to a homeomorphism of the covering surface. This answers a question that appeared in an early