Quasisymmetric nonparametrization and spaces associated with the Whitehead continuum

Heinonen, Juha¹ ; Wu, Jang-Mei²

¹ Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109-1043, USA
² Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61822, USA

Geometry & topology, Tome 14 (2010) no. 2, pp. 773-798.

Résumé

The decomposition space $R^3∕Wh$ associated with the Whitehead continuum $Wh$ is not a manifold, but the product $\left(R^3∕Wh\right)\times R^m$ is homeomorphic to $R^{3+m}$ for any $m\ge 1$ (known since the 1960’s). We study the quasisymmetric structure on $\left(R^3∕Wh\right)\times R^m$ and show that the space $\left(R^3∕Wh\right)\times R^m$ may be equipped with a metric resembling $R^{3+m}$ geometrically and measure theoretically—it is linearly locally contractible and Ahlfors $\left(3+m\right)$–regular—nevertheless the resulting space does not admit a quasisymmetric parametrization by $R^{3+m}$.

Détail

DOI : 10.2140/gt.2010.14.773

Keywords: Whitehead continuum, decomposition space, quasisymmetric parametrization, quasisymmetric map

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