×

zbMATH — the first resource for mathematics

On composants of the bucket handle. (English) Zbl 0759.54017
The class group \({\mathcal Cl}({\mathcal S}_ 2)\), where \({\mathcal S}_ 2\) denotes the 2-solenoid, is the group of base point preserving homeomorphisms of \(\mathcal{S}_ 2\) up to base point preserving isotopy. The class group \({\mathcal Cl}({\mathcal K})\), where \(\mathcal K\) denotes the bucket handle continuum, is the group of homeomorphisms of \(\mathcal K\) up to isotopy. The authors prove that \({\mathcal Cl}({\mathcal S}_ 2)\) is isomorphic to \(\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\), and that \({\mathcal Cl}({\mathcal K})\) is isomorphic to \(\mathbb{Z}\). They also show that if \(C\) and \(D\) are composants of \(\mathcal K\) not containing the end point 0, then there exists an injective continuous map \(\Phi: {\mathcal K}-\{0\}\to {\mathcal K} - \{0\}\) which maps \(C\) onto \(D\).
Reviewer: K.M.Kuperberg

MSC:
54F15 Continua and generalizations
54H20 Topological dynamics (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI EuDML