Bing–Borsuk conjecture
In mathematics, the Bing–Borsuk conjecture states that every [math]\displaystyle{ n }[/math]-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 [math]\displaystyle{ m_1, m_2 \in M }[/math], there is a homeomorphism of [math]\displaystyle{ M }[/math] which takes [math]\displaystyle{ m_1 }[/math] to [math]\displaystyle{ m_2 }[/math].
A metric space [math]\displaystyle{ M }[/math] is an absolute neighborhood retract (ANR) if, for every closed embedding [math]\displaystyle{ f: M \rightarrow N }[/math] (where [math]\displaystyle{ N }[/math] is a metric space), there exists an open neighbourhood [math]\displaystyle{ U }[/math] of the image [math]\displaystyle{ f(M) }[/math] which retracts to [math]\displaystyle{ f(M) }[/math].[1]
There is an alternate statement of the Bing–Borsuk conjecture: suppose [math]\displaystyle{ M }[/math] is embedded in [math]\displaystyle{ \mathbb{R}^{m+n} }[/math] for some [math]\displaystyle{ m \geq 3 }[/math] and this embedding can be extended to an embedding of [math]\displaystyle{ M \times (-\varepsilon, \varepsilon) }[/math]. If [math]\displaystyle{ M }[/math] has a mapping cylinder neighbourhood [math]\displaystyle{ N=C_\varphi }[/math] of some map [math]\displaystyle{ \varphi: \partial N \rightarrow M }[/math] with mapping cylinder projection [math]\displaystyle{ \pi: N \rightarrow M }[/math], then [math]\displaystyle{ \pi }[/math] 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 [math]\displaystyle{ n=1 }[/math] 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 [math]\displaystyle{ G }[/math]-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
- ↑ M., Halverson, Denise; Dušan, Repovš (23 December 2008). "The Bing–Borsuk and the Busemann conjectures" (in en). Mathematical Communications 13 (2). ISSN 1331-0623. https://hrcak.srce.hr/30884.
- ↑ Daverman, R. J.; Husch, L. S. (1984). "Decompositions and approximate fibrations." (in en). The Michigan Mathematical Journal 31 (2): 197–214. doi:10.1307/mmj/1029003024. ISSN 0026-2285. https://projecteuclid.org/euclid.mmj/1029003024.
- ↑ Bing, R. H.; Armentrout, Steve (1998) (in en). The Collected Papers of R. H. Bing. American Mathematical Soc.. pp. 167. ISBN 9780821810477. https://books.google.com/books?id=NnBQ0xp_rUcC&pg=PA167.
- ↑ Jakobsche, W.. "The Bing–Borsuk conjecture is stronger than the Poincaré conjecture" (in en). Fundamenta Mathematicae 106 (2). ISSN 0016-2736. https://eudml.org/doc/211089.
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