Dogbone space

In geometric topology, the dogbone space, constructed by R. H. Bing (1957), is a quotient space of three-dimensional Euclidean space R³ such that all inverse images of points are points or tame arcs, yet it is not homeomorphic to R³. 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 R¹ is homeomorphic to R⁴.

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

See also

• Whitehead manifold, a 3-manifold not homeomorphic to R³ whose product with R¹ is homeomorphic to R⁴.

References

• Daverman, Robert J. (2007), Decompositions of manifolds, AMS Chelsea Publishing, Providence, RI, p. 22, ISBN 978-0-8218-4372-7, MR2341468, http://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: 484–500, ISSN 0003-486X, JSTOR 1970058, MR0092961 
• Bing, R. H. (1959), "The cartesian product of a certain nonmanifold and a line is E⁴", Annals of Mathematics. Second Series 70: 399–412, ISSN 0003-486X, JSTOR 1970322, MR0107228