On tame embeddings of solenoids into 3-space

Boju Jiang¹ ; Shicheng Wang¹ ; Hao Zheng¹ ; Qing Zhou²

¹ Department of Mathematics Peking University Beijing 100871, China
² Department of Mathematics East China Normal University Shanghai 200030, China

Fundamenta Mathematicae, Tome 214 (2011) no. 1, pp. 57-75

Résumé

Solenoids are inverse limits of the circle, and the classical knot theory is the theory of tame embeddings of the circle into 3-space. We make a general study, including certain classification results, of tame embeddings of solenoids into 3-space, seen as the “inverse limits” of tame embeddings of the circle.Some applications in topology and in dynamics are discussed. In particular, there are tamely embedded solenoids $\Sigma\subset \mathbb R^3$ which are strictly achiral. Since solenoids are non-planar, this contrasts sharply with the known fact that if there is a strictly achiral embedding $Y\subset \mathbb R^3$ of a compact polyhedron $Y$, then $Y$ must be planar.

Détail

DOI : 10.4064/fm214-1-4

Keywords: solenoids inverse limits circle classical knot theory theory tame embeddings circle space make general study including certain classification results tame embeddings solenoids space seen inverse limits tame embeddings circle applications topology dynamics discussed particular there tamely embedded solenoids sigma subset mathbb which strictly achiral since solenoids non planar contrasts sharply known there strictly achiral embedding subset mathbb compact polyhedron planar