Dogbone space
The first stage of the dogbone space construction.
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, MR 2341468
- 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, MR 0092961
- 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, MR 0107228
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