Hausdorff convergence of submanifolds in $\mathbb{S}^m$

Question

Let $\{X_i^n\}_{i\in \mathbb{N}}$ and $\{Y_i^n\}_{i\in \mathbb{N}}$ be sequences of connected closed submanifolds of $\mathbb{S}^{n+2}$, with $n> 5$. Suppose that $\{X_i^n\}_{i\in \mathbb{N}}$ (resp. $\{Y_i^n\}_{i\in \mathbb{N}}$) has $X_{\infty}\subset \mathbb{S}^{n+2}$ (resp. $Y_{\infty}\subset \mathbb{S}^{n+2}$) as its Hausdorff limit (with respect the Euclidean distance of $\mathbb{R}^{n+3}$). Suppose also that $X_{\infty}$ and $Y_{\infty}$ have Hausdorff dimension equal to $n$ (i.e., these sequences are noncollapsing sequences). Moreover, for any $i, j\in \mathbb{N}$, $X_i$ is homeomorphic to $X_j$ and $Y_i$ is homeomorphic to $Y_j$.

For each $i\in \mathbb{N}$, suppose that there is a diffeomorphism $h_i\colon \mathbb{S}^{n+2}\to \mathbb{S}^{n+2}$ such that $h_i(X_i)=Y_i$.

Question 1a. What is known about the relation between $X_{\infty}$ and $Y_{\infty}$? For example, are they homeomorphic, homotopy equivalent or homology equivalent?

Question 1b. The same Question 1a, when $X_{\infty}$ is smooth.

Answer 1

One cannot expect homology equivalence of limits. Here is a standard example for surfaces in $\mathbb R^3$ which can be easily adapted to your situation. Start from the unit sphere in $\mathbb R^3$ and attach a handle (still embedded in $\mathbb R^3$). Then look at the sequence with the handle getting smaller and disappearing in the limit. Compare it with the constant sequence where the handle is not disappearing.