Dogbone space

The first stage of the dogbone space construction.

In geometric topology, the dogbone space, constructed by R. H. Bing, ^[1] is a quotient space of three-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{3}}$ such that all inverse images of points are points or tame arcs, yet it is not homeomorphic to ${\displaystyle \mathbb {R} ^{3}}$. 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 ${\displaystyle \mathbb {R} ^{1}}$ is homeomorphic to ${\displaystyle \mathbb {R} ^{4}}$. ^[2]

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 ${\displaystyle \mathbb {R} ^{3}}$.

References

1. Bing, R. H. (May 1957). "A Decomposition of E 3 into Points and Tame Arcs Such That the Decomposition Space is Topologically Different from E 3" . The Annals of Mathematics. 65 (3): 484. doi:10.2307/1970058.
2. Bing, R. H. (November 1959). "The Cartesian Product of a Certain Nonmanifold and a Line is E 4" . The Annals of Mathematics. 70 (3): 399. doi:10.2307/1970322.

Sources

• Daverman, Robert J. (2007), "Decompositions of manifolds", Geom. Topol. Monogr., 9: 7–15, arXiv: 0903.3055 , doi:10.1090/chel/362, ISBN   978-0-8218-4372-7, MR   2341468