mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Bing–Borsuk conjecture states that every

n

-dimensional

homogeneous Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...

absolute neighborhood retract In topology, a retraction is a continuous mapping from a topological space into a Subspace topology, subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformation ret ...

space is a

topological manifold In topology, a topological manifold is a topological space that locally resembles real ''n''- dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds ...

. The conjecture has been proved for dimensions 1 and 2, and it is known that the 3-dimensional version of the conjecture implies the

Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured b ...

Definitions

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

is ''homogeneous'' if, for any two points

m_1, m_2 \in M

, there is a

homeomorphism 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 ...

M

which takes

m_1

m_2

. A

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

M

is an

(ANR) if, for every closed embedding

f: M \rightarrow N

(where

N

is a metric space), there exists an

open neighbourhood In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a po ...

U

of the image

f(M)

which retracts to

f(M)

. There is an alternate statement of the Bing–Borsuk conjecture: suppose

M

is embedded in

\mathbb^

for some

m \geq 3

and this embedding can be extended to an embedding of

M \times (-\varepsilon, \varepsilon)

. If

M

has a mapping cylinder neighbourhood

N=C_\varphi

of some map

\varphi: \partial N \rightarrow M

with mapping cylinder projection

\pi: N \rightarrow M

, then

\pi

is an

approximate fibration In algebraic topology, a branch of mathematics, an approximate fibration is a sort of fibration such that the homotopy lifting property In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also ...

History

The conjecture was first made in a paper by R. H. Bing and

Karol Borsuk Karol Borsuk (8 May 1905 – 24 January 1982) was a Polish mathematician. His main area of interest was topology. He made significant contributions to Shape theory (mathematics), shape theory, a term which he coined. He also obtained important r ...

in 1965, who proved it for

n=1

and 2. Włodzimierz Jakobsche showed in 1978 that, if the Bing–Borsuk conjecture is true in dimension 3, then the Poincaré conjecture must also be true. The Busemann conjecture states that every Busemann

G

-space is a topological manifold. It is a special case of the Bing–Borsuk conjecture. The Busemann conjecture is known to be true for dimensions 1 to 4.

References

{{DEFAULTSORT:Bing-Borsuk conjecture Topology Conjectures Unsolved problems in mathematics Manifolds