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Bing–Borsuk conjecture

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In mathematics, the Bing–Borsuk conjecture states that every -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

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A topological space is homogeneous if, for any two points , there is a homeomorphism of which takes to .

A metric space is an absolute neighborhood retract (ANR) if, for every closed embedding (where is a metric space), there exists an open neighbourhood of the image which retracts to .[1]

There is an alternate statement of the Bing–Borsuk conjecture: suppose is embedded in for some and this embedding can be extended to an embedding of . If has a mapping cylinder neighbourhood of some map with mapping cylinder projection , then is an approximate fibration.[2]

History

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The conjecture was first made in a paper by R. H. Bing and Karol Borsuk in 1965, who proved it for 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 -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

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  1. ^ M., Halverson, Denise; Dušan, Repovš (23 December 2008). "The Bing–Borsuk and the Busemann conjectures". Mathematical Communications. 13 (2). arXiv:0811.0886. ISSN 1331-0623.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  2. ^ Daverman, R. J.; Husch, L. S. (1984). "Decompositions and approximate fibrations". The Michigan Mathematical Journal. 31 (2): 197–214. doi:10.1307/mmj/1029003024. ISSN 0026-2285.
  3. ^ Bing, R. H.; Armentrout, Steve (1998). The Collected Papers of R. H. Bing. American Mathematical Soc. p. 167. ISBN 9780821810477.
  4. ^ Jakobsche, W. "The Bing–Borsuk conjecture is stronger than the Poincaré conjecture". Fundamenta Mathematicae. 106 (2). ISSN 0016-2736.