Dogbone space

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
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[edit]

References[edit]

  • Daverman, Robert J. (2007), Decompositions of manifolds, AMS Chelsea Publishing, Providence, RI, p. 22, arXiv:0903.3055, doi:10.1090/chel/362, 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, doi:10.2307/1970058, 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, doi:10.2307/1970322, ISSN 0003-486X, JSTOR 1970322, MR 0107228