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Abstract

We consider the problem of reconstructing the topology of a manifold $M$ from a finite sample of its points. For a submanifold of the Euclidean $n$-space Niyogi, Smale and Weinberger (2008) have shown that the $\varepsilon $-neighbourhood of a sufficiently dense sample is homotopy equivalent to $M$. We show a similar result for $C^1$-submanifolds of $\mathbb {H}^n$. The density bound we obtain turns out to be, in general case, less efficient. We construct an example showing that it is almost as good as it can be. For a closed, connected, $1$-dimensional submanifold there is a better bound.