In mathematics, the Bing–Borsuk conjecture states that every formula_1-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 formula_2, there is a homeomorphism of formula_3 which takes formula_4 to formula_5.
A metric space formula_3 is an absolute neighborhood retract (ANR) if, for every closed embedding formula_7 (where formula_8 is a metric space), there exists an open neighbourhood formula_9 of the image formula_10 which retracts to formula_10.
There is an alternate statement of the Bing–Borsuk conjecture: suppose formula_3 is embedded in formula_13 for some formula_14 and this embedding can be extended to an embedding of formula_15. If formula_3 has a mapping cylinder neighbourhood formula_17 of some map formula_18 with mapping cylinder projection formula_19, then formula_20 is an approximate fibration.
History.
The conjecture was first made in a paper by R. H. Bing and Karol Borsuk in 1965, who proved it for formula_21 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 formula_22-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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