Metric Geometry of Finite Subset Spaces

TL;DR

This paper investigates the metric geometry of finite subset spaces of topological spaces, exploring how geometric properties of the original space relate to those of its finite subset spaces, with implications for manifold structures and invariants.

Contribution

It provides a detailed analysis of how geometric properties are preserved or altered in finite subset spaces, extending understanding of their topological and metric structures.

Findings

Finite subset spaces can have richer geometric structures than the original space.

Properties like orientability may change when passing to finite subset spaces.

The study offers criteria for when geometric properties are preserved in finite subset spaces.

Abstract

If $X$ is a (topological) space, the $n$th finite subset space of $X$, denoted by $X(n)$, consists of $n$-point subsets of $X$ (i.e., nonempty subsets of cardinality at most $n$) with the quotient topology induced by the unordering map $q:X^n\to X(n)$, $(x_1,\cdots,x_n)\mapsto\{x_1,\cdots,x_n\}$. That is, a set $A\subset X(n)$ is open if and only if its preimage $q^{-1}(A)$ is open in the product space $X^n$. Given a space $X$, let $H(X)$ denote all homeomorphisms of $X$. For any class of homeomorphisms $C\subset H(X)$, the $C$-geometry of $X$ refers to the description of $X$ up to homeomorphisms in $C$. Therefore, the topology of $X$ is the $H(X)$-geometry of $X$. By a ($C$-) geometric property of $X$ we will mean a property of $X$ that is preserved by homeomorphisms of $X$ (in $C$). Metric geometry of a space $X$ refers to the study of geometry of $X$ in terms of notions of metrics (e.g., distance, or length of a path, between points) on $X$. In such a study, we call a space $X$ metrizable if $X$ is homeomorphic to a metric space. Naturally, $X(n)$ always inherits some aspect of every geometric property of $X$ or $X^n$. Thus, the geometry of $X(n)$ is in general richer than that of $X$ or $X^n$. For example, it is known that if $X$ is an orientable manifold, then (unlike $X^n$) $X(n)$ for $n>1$ can be an orientable manifold, a non-orientable manifold, or a non-manifold. In studying geometry of $X(n)$, a central research question is "If $X$ has geometric property $P$, does it follow that $X(n)$ also has property $P$?". A related question is "If $X$ and $Y$ have a geometric relation $R$, does it follow that $X(n)$ and $Y(n)$ also have the relation $R$?". (Truncated)