Reconstruction of mapping spaces by inverse limits

TL;DR

The paper addresses reconstructing mapping spaces $Y^X$ for compact metric spaces via finite combinatorial models. It builds an inverse system from finite covers to produce an inverse limit $\\widetilde{Y^X}$ that captures the homotopy type of $Y^X$, with a map $p$ giving a strong deformation retract of $\\widetilde{Y^X}$. Key contributions include showing $Y^X$ is weakly equivalent to an inverse limit of compact polyhedra through McCORD-type correspondences and demonstrating that isotopies can be approximated by finite moves on associated finite $T_0$-spaces, linking shape-theoretic methods with practical finite models. The results provide explicit, computable finite models for mapping spaces, which has implications for shape theory and topological data analysis where complex mapping spaces are otherwise intractable.

Abstract

Extending the results of reconstruction of compact metric spaces by inverse limits, we show that if $(X, d), (Y, d)$ are compact metric spaces, then the mapping space $Y^X$ is homotopy equivalent to the inverse limit of an inverse system of finite $T_0$-spaces which depends only on the finite open covers of $X$ and $Y$. Applying our tools, we obtain that if $H$ is an isotopy of a compact metric space $(X, d)$, then $H_1H^{-1}_0$ can be approximated in terms of moves of a finite $T_0$-space.

Theorems & Definitions (27)

• Theorem 2.1
• Theorem 2.2
• Remark 3.1
• Remark 3.2
• Example 3.3
• Remark 3.4
• Example 3.5
• Lemma 3.6
• proof
• Lemma 3.7