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arXiv · 2505.16907

Persistent homology of function spaces

Jonathan Block·Fedor Manin·Shmuel Weinberger
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Abstract

We can view the Lipschitz constant as a height function on the space of maps between two manifolds and ask (as Gromov did nearly 30 years ago) what its ``Morse landscape'' looks like: are there high peaks, deep valleys and mountain passes? A simple and relatively well-studied version of this question: given two points in the same component (homotopic maps), does a path between them (a homotopy) have to pass through maps of much higher Lipschitz constant? Now we also consider similar questions for higher-dimensional cycles in the space. We make this precise using the language of persistent homology and give some first results.

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Jonathan Block, Fedor Manin, Shmuel Weinberger. 2026-09-06. Persistent homology of function spaces. https://arxiv.org/abs/2505.16907

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