Dogbone space

File:Bing's Dogbone.tiff In geometric topology, the dogbone space, constructed by R. H. Bing, is a quotient space of three-dimensional Euclidean space ${ℝ}^3$ such that all inverse images of points are points or tame arcs, yet it is not homeomorphic to ${ℝ}^3$. 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 ${ℝ}^1$ is homeomorphic to ${ℝ}^4$.

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 ${ℝ}^3$.