Dogbone space

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.

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

• 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{{citation}}: CS1 maint: work parameter with ISBN (link)