Bing–Borsuk conjecture

In mathematics, the Bing–Borsuk conjecture states that every ${\displaystyle n}$-dimensional homogeneous absolute neighborhood retract space is a topological manifold. 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.

Definitions

A topological space is homogeneous if, for any two points ${\displaystyle m_{1},m_{2}\in M}$, there is a homeomorphism of ${\displaystyle M}$ which takes ${\displaystyle m_{1}}$ to ${\displaystyle m_{2}}$.

A metric space ${\displaystyle M}$ is an absolute neighborhood retract (ANR) if, for every closed embedding ${\displaystyle f:M\rightarrow N}$ (where ${\displaystyle N}$ is a metric space), there exists an open neighbourhood ${\displaystyle U}$ of the image ${\displaystyle f(M)}$ which retracts to ${\displaystyle f(M)}$.^[1]

There is an alternate statement of the Bing–Borsuk conjecture: suppose ${\displaystyle M}$ is embedded in ${\displaystyle \mathbb {R} ^{m+n}}$ for some ${\displaystyle m\geq 3}$ and this embedding can be extended to an embedding of ${\displaystyle M\times (-\varepsilon ,\varepsilon )}$. If ${\displaystyle M}$ has a mapping cylinder neighbourhood ${\displaystyle N=C_{\varphi }}$ of some map ${\displaystyle \varphi :\partial N\rightarrow M}$ with mapping cylinder projection ${\displaystyle \pi :N\rightarrow M}$, then ${\displaystyle \pi }$ is an approximate fibration.^[2]

History

The conjecture was first made in a paper by R. H. Bing and Karol Borsuk in 1965, who proved it for ${\displaystyle n=1}$ and 2.^[3]

Włodzimierz Jakobsche showed in 1978 that, if the Bing–Borsuk conjecture is true in dimension 3, then the Poincaré conjecture must also be true.^[4]

The Busemann conjecture states that every Busemann ${\displaystyle G}$-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