Volume 14, issue 2 (2010)

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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
Abstract

The decomposition space R3Wh associated with the Whitehead continuum Wh is not a manifold, but the product (R3Wh) ×Rm is homeomorphic to R3+m for any m 1 (known since the 1960’s). We study the quasisymmetric structure on (R3Wh) ×Rm and show that the space (R3Wh) ×Rm may be equipped with a metric resembling R3+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 R3+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