Manifolds and cell complexes Generalized manifolds Conjecture

Bing–Borsuk conjecture

Builds on Absolute neighborhood retract Homeomorphism Lebesgue covering dimension Topological manifold

Statement

Let $n \ge 1$ and let $X$ be an absolute neighborhood retract of covering dimension $n$ . Suppose that $X$ is homogeneous: for all points $x, y \in X$ there is a homeomorphism $h \colon X \to X$ with $h(x) = y$ . Then $X$ is a topological manifold of dimension $n$ .

Remarks

Every connected topological manifold is a homogeneous absolute neighborhood retract, so the conjecture asks for a converse. The conjecture is known to hold for $n = 1$ and $n = 2$ . In dimension $3$ , it is known to imply the Poincaré conjecture. In general it is open. The finite dimension is needed: the Hilbert cube is a homogeneous compact absolute retract of infinite dimension that is not a manifold of any finite dimension.

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Tag	`VK39X07EYK`
URI	`m2://VK39X07EYK`
Kind	conjecture
Writing	W0 stub
MSC	57P05, 55M15
Status	This copy of the site has no build data. Run `m2 status VK39X07EYK`.
About	Absolute neighborhood retract, Topological manifold
Uses	Absolute neighborhood retract, Homeomorphism, Lebesgue covering dimension, Topological manifold
Same as	Wikidata Q55605472 Wikipedia Bing–Borsuk conjecture
Changed	2 November 2026

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