A family of \(4\)-manifolds with nonnegative Ricci curvature and prescribed asymptotic cone

In this paper, we show that for any finite subgroup \(\Gamma < O(4)\) acting freely on \(\mathbb{S}^3\), there exists a \(4\)-dimensional complete Riemannian manifold \((M,g)\) with \({\rm Ric}_g \geq 0 \), such that the asymptotic cone of \((M,g)\) is \(C(\mathbb{S}_\delta^3 /\Gamma )\) for some \(\delta = \delta (\Gamma ) >0\). This answers a question of Bruè-Pigati-Semola [arXiv:2405.03839] about the topological obstructions of \(4\)-dimensional non-collapsed tangent cones. Combining this result with a recent work of Bruè-Pigati-Semola [arXiv:2405.03839], one can classify the \(4\)-dimensional non-collapsed tangent cone in the topological sense.