Dogbone space

Short description: Quotient space in geometric topology

File:Bing's Dogbone.tiff In geometric topology, the dogbone space, constructed by R. H. Bing, is a quotient space of three-dimensional Euclidean space ${\mathbb{ℝ}}^3$ such that all inverse images of points are points or tame arcs, yet it is not homeomorphic to ${\mathbb{ℝ}}^3$. The name "dogbone space" refers to a fanciful resemblance between some of the diagrams of genus 2 surfaces in Bing's paper and a dog bone. Bing showed that the product of the dogbone space with ${\mathbb{ℝ}}^1$ is homeomorphic to ${\mathbb{ℝ}}^4$.

Although the dogbone space is not a manifold, it is a generalized homological manifold and a homotopy manifold.

See also

• List of topologies
• Whitehead manifold, a contractible 3-manifold not homeomorphic to ${\mathbb{ℝ}}^3$.

Sources

• Daverman, Robert J. (2007), "Decompositions of manifolds", Geom. Topol. Monogr. 9: 7–15, doi:10.1090/chel/362, ISBN 978-0-8218-4372-7, https://www.ams.org/bookstore-getitem/item=chel-362.h