In geometric topology

Geometric topology

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...

, the dogbone space, constructed by , is a quotient space

Quotient space

In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...

 of three-dimensional Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 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. showed that the product

Product topology

In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...

 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
  Whitehead manifold
  In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to R3. Henry Whitehead discovered this puzzling object while he was trying to prove the Poincaré conjecture....
  
  , a 3-manifold not homeomorphic to R³ whose product with R¹ is homeomorphic to R⁴.