A note on the homotopical characterization of \(\mathbb{R}^ n\) (Q1318619)

Using results of [\textit{J. Dydak}, Lect. Notes Math. 870, 48-72 (1981; Zbl 0491.55006)], it is proven that the one-point compactification of a stable at infinity, homologically trivial topological manifold \(M\) is a generalized homology sphere. In addition, if \(M\) is \(1-LC\) at infinity, \(M^ +\) is a topological homology sphere by results of [\textit{J. L. Bryant} and \textit{R. C. Lacher}, Math. Proc. Camb. Philos. Soc. 83, 403- 413 (1978; Zbl 0373.57002)]. As a consequence it is shown that Siebenmann's characterization of the Euclidean space \(\mathbb{R}^ n\) is equivalent to the Poincaré Conjecture. These facts have been independently noticed by D. Repovš in dimension 3.