In mathematics, the Bing–Borsuk conjecture states that every

n

-dimensional

homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...

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

space is a

topological manifold In topology, a branch of mathematics, 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 ma ...

. 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 geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

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

m_1, m_2 \in M

, there is a

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...

M

which takes

m_1

m_2

. A

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...

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 ...

U

of the image

f(M)

which retracts to

f(M)

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

M

embedded Embedded or embedding (alternatively imbedded or imbedding) may refer to: Science * Embedding, in mathematics, one instance of some mathematical object contained within another instance ** Graph embedding * Embedded generation, a distributed ge ...

\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.

History

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

Karol Borsuk Karol Borsuk (May 8, 1905 – January 24, 1982) was a Polish mathematician. His main interest was topology, while he obtained significant results also in functional analysis. Borsuk introduced the theory of ''absolute retracts'' (ARs) and '' abs ...

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