Quasisymmetric nonparametrization and spaces associated with the Whitehead continuum

Heinonen, Juha; Wu, Jang-Mei

doi:10.2140/gt.2010.14.773

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Quasisymmetric nonparametrization and spaces associated with the Whitehead continuum

Juha Heinonen and Jang-Mei Wu

Geometry & Topology 14 (2010) 773–798

DOI: 10.2140/gt.2010.14.773

Abstract

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

In memory of Juha Heinonen, an extraordinary mathematician and friend, who passed away after the mathematical work in this paper was completed. – J-M W

Keywords

Whitehead continuum, decomposition space, quasisymmetric parametrization, quasisymmetric map

Mathematical Subject Classification 2000

Primary: 30C65

Secondary: 57N16, 57N45

References

Publication

Received: 5 March 2008

Revised: 28 October 2009

Accepted: 7 December 2009

Published: 19 February 2010

Proposed: Colin Rourke

Seconded: Tobias Colding, David Gabai

Authors

Juha Heinonen
Department of Mathematics University of Michigan 530 Church Street Ann Arbor, MI 48109-1043 USA

Jang-Mei Wu
Department of Mathematics University of Illinois 1409 West Green Street Urbana, IL 61822 USA

Volume 14, issue 2 (2010)