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.
A topological space is homogeneous if, for any two points
A metric space
is an absolute neighborhood retract (ANR) if, for every closed embedding
is a metric space), there exists an open neighbourhood
[1] There is an alternate statement of the Bing–Borsuk conjecture: suppose
has a mapping cylinder neighbourhood
with mapping cylinder projection
is an approximate fibration.
[2] The conjecture was first made in a paper by R. H. Bing and Karol Borsuk in 1965, who proved it for
[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.
-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.