Dogbone space

File:Bing's Dogbone.tiff In geometric topology, the dogbone space, constructed by R. H. Bing (1957), 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 R. H. Bing's paper and a dog bone. (Bing 1959) 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 ${\mathbb{ℝ}}^3$.

References

• 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 
• Bing, R. H. (1957), "A decomposition of E³ into points and tame arcs such that the decomposition space is topologically different from E³", Annals of Mathematics, Second Series 65 (3): 484–500, doi:10.2307/1970058, ISSN 0003-486X 
• Bing, R. H. (1959), "The cartesian product of a certain nonmanifold and a line is E⁴", Annals of Mathematics, Second Series 70 (3): 399–412, doi:10.2307/1970322, ISSN 0003-486X, http://projecteuclid.org/euclid.bams/1183522317