A family of $4$-manifolds with nonnegative Ricci curvature and prescribed asymptotic cone

In this paper, we show that for any finite subgroup Γ < O(4) acting freely on S³, there exists a $4$-dimensional complete Riemannian manifold $(M,g)$ with Ricg ≥ 0, such that the asymptotic cone of $(M,g)$ is C(S_δ³ /Γ ) for some δ = δ (Γ ) >0. This answers a question of Bru\`e-Pigati-Semola [arXiv:2405.03839] about the topological obstructions of $4$-dimensional non-collapsed tangent cones. Combining this result with a recent work of Bru\`e-Pigati-Semola [arXiv:2405.03839], one can classify the $4$-dimensional non-collapsed tangent cone in the topological sense.