Topological dimension of the Gromov-Hausdorff and Gromov-Prokhorov spaces
Hiroki Nakajima, Takamitsu Yamauchi, Nicolò Zava
TL;DR
The work addresses the problem of determining the topological dimension of the Gromov-Hausdorff space $\mathcal{M}$ and the Gromov-Prokhorov/box-distance space $\mathcal{X}$ and their finite subspaces. It develops a finite-dimensional-parameter representation via quotients $R_n/\sim_{GH}$ and $(R_n\times \Delta_{n-1}^\circ)/\sim_b$ to obtain exact formulas for $\dim \mathcal{M}_{\le n}$ and $\dim \mathcal{X}_{\le n}$, and demonstrates that the Hilbert cube embeds into $\mathcal{X}$ (and hence into $\mathcal{M}$), implying strong infinite dimensionality of the full spaces, with the finite-point strata dense in the whole spaces. The paper also provides explicit, self-contained constructions of the embeddings into $\mathcal{M}$ and $\mathcal{X}$ that underpin the dimensional results, offering insights into embeddability and the geometric structure of these comparison spaces. These results clarify the dimensional complexity of spaces of metric and metric-measure spaces, with potential implications for computational topology and related embedding problems.
Abstract
The Gromov-Hausdorff distance is a dissimilarity metric capturing how far two spaces are from being isometric. The Gromov-Prokhorov distance is a similar notion for metric measure spaces. In this paper, we study the topological dimension of the Gromov-Hausdorff and Gromov-Prokhorov spaces. We show that the dimension of the space of isometry classes of metric spaces with at most $n$ points endowed with the Gromov-Hausdorff distance is $\frac{n(n-1)}{2}$, and that of mm-isomorphism classes of metric measure spaces whose support consists of $n$ points is $\frac{(n+2)(n-1)}{2}$. Hence, the spaces of all isometry classes of finite metric spaces and of all mm-isomorphism classes of finite metric measure spaces are strongly countable dimensional. If, instead, the cardinalities are not limited, the spaces are strongly infinite-dimensional.