In
geometric topology
In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topo ...
, the dogbone space, constructed by , is a
quotient space of three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
such that all
inverse image
In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y.
More generally, evaluating f at each ...
s of points are points or
tame arcs, yet it is not
homeomorphic
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to
. 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 of the dogbone space with
is
homeomorphic
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to
.
Although the dogbone space is not a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
, it is a
generalized homological manifold and a
homotopy manifold.
See also
*
List of topologies
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, ...
*
Whitehead manifold, a contractible 3-manifold not homeomorphic to
.
References
*
*
*{{Citation , last1=Bing , first1=R. H. , authorlink = R. H. Bing , title=The cartesian product of a certain nonmanifold and a line is E
4 , jstor=1970322 , mr=0107228 , year=1959 , journal=
Annals of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
History
The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...
, series=Second Series , issn=0003-486X , volume=70 , issue=3 , pages=399–412 , doi=10.2307/1970322, url=http://projecteuclid.org/euclid.bams/1183522317
Geometric topology
Topological spaces