Examples of Ricci limit spaces with infinite holes

Let $n\geq 3$, $\lambda \in \mathbb{R}$, and $(X,h)$ be an $n$-dimensional smooth complete Riemannian manifold with ${\rm Ric}_h > \lambda$. In this paper, we construct, for each given $\epsilon >0$, a sequence of $(n+2)$-dimensional manifolds $(M_{i} ,g_i ) \stackrel{GH}{\longrightarrow} (X_\epsilon ,d_\epsilon )$ with ${\rm Ric}_{g_i} > \lambda$, such that $d_{GH} (X,X_\epsilon ) \leq \epsilon$, and $X_\epsilon$ is homeomorphic to the space obtained by removing an infinite number of balls from $X$. Hence $X_\epsilon$ has dense boundary with an infinite number of connected components. Moreover, $X_\epsilon$ has no open subset which is topologically a manifold. This generalizes Hupp-Naber-Wang's result (arxiv: 2308.03909) from $4$-dimensional case to the general case of dimension $n\geq 3$. Our construction differs from that of Hupp-Naber-Wang. In their approach, Hupp-Naber-Wang considered doing an infinite number of blow-ups on the local complex surface structure of $X$, thus relying on the $4$-dimensional condition. However, our method involves removing an infinite number of balls from $X$, allowing us to construct in the general case of dimensions greater than or equal to $3$. As a corollary, we provide a solution to an open problem posed by Naber in the $3$-dimensional case.