Positive Bakry-Émery Ricci Curvature on Homotopy Spheres

Wei and Wylie asked whether a complete weighted manifold with nonnegative Bakry--Émery Ricci curvature and bounded potential must admit a Riemannian metric with nonnegative Ricci curvature. We answer this question negatively in every dimension \(8k+1\) and \(8k+2\), where \(k\geq1\). Our main geometric result is that every smooth homotopy sphere of dimension at least seven admits a weighted core metric \((g,e^{-f})\) with \(\Ric_g+\Hess_g f>0\). On the other hand, in dimensions \(8k+1\) and \(8k+2\), where \(k\geq1\), we prove that a homotopy sphere with nonzero \(α\)-invariant admits no Riemannian metric with \(\Ric\geq0\). Since homotopy spheres with nonzero \(α\)-invariant exist in every dimension \(8k+1\) and \(8k+2\), and since compactness makes the potential bounded, these manifolds provide the required counterexamples.