Bryant's modified Bing–Borsuk conjecture

A locally compact space is homogeneous if its homeomorphism group acts transitively on the space, and an ANR-space is an absolute neighborhood retract. A generalized $n$-manifold is a locally compact $n$-dimensional ANR-space that is a $\mathbb Z$-homology $n$-manifold, meaning that for every $x\in X$,

$$H_k(X,X\setminus\{x\};\mathbb Z)=0 \quad\text{if } k<n,$$

and

$$H_n(X,X\setminus\{x\};\mathbb Z)=\mathbb Z.$$

Bryant's modified Bing–Borsuk conjecture. Every locally compact homogeneous ANR-space of dimension $n\geq 3$ is a generalized $n$-manifold. This modifies the Bing–Borsuk conjecture to allow generalized manifolds; it is motivated by known counterexamples to the original conjecture and remains unresolved.