# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Non-standard automorphisms of branched coverings of a disk and a sphere

### Tom 218 / 2012

Fundamenta Mathematicae 218 (2012), 1-11 MSC: Primary 20F36, 20F38, 57M12. DOI: 10.4064/fm218-1-1

#### Streszczenie

Let $Y$ be a closed 2-dimensional disk or a 2-sphere. We consider a simple, $d$-sheeted branched covering $\pi :X\to Y$. We fix a base point $A_0$ in $Y$ ($A_0\in \partial Y$ if $Y$ is a disk). We consider the homeomorphisms $h$ of $Y$ which fix $\partial Y$ pointwise and lift to homeomorphisms $\phi$ of $X$—the automorphisms of $\pi$. We prove that if $Y$ is a sphere then every such $\phi$ is isotopic by a fiber-preserving isotopy to an automorphism which fixes the fiber $\pi ^{-1}(A_0)$ pointwise. If $Y$ is a disk, we describe explicitly a small set of automorphisms of $\pi$ which induce all allowable permutations of $\pi ^{-1}(A_0)$. This complements our result in Fund. Math. 217 (2012), no. 2, where we found a set of generators for the group of isotopy classes of automorphisms of $\pi$ which fix the fiber $\pi ^{-1}(A_0)$ pointwise.

#### Autorzy

• Bronisław WajnrybDepartment of Mathematics
Rzeszów University of Technology
Powstańców Warszawy 12
35-959 Rzeszów, Poland
e-mail
• Agnieszka Wiśniowska-WajnrybDepartment of Mathematics
Rzeszów University of Technology
Powstańców Warszawy 12
35-959 Rzeszów, Poland
e-mail

## Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Odśwież obrazek