# Metric Geometry of Finite Subset Spaces

May 2020

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

Mathematics

Leonid Kovalev

## Keywords

Lipschitz connectedness, Lipschitz retraction, Metric geometry, Subset space, Topological space

## Subject Categories

Physical Sciences and Mathematics

## Abstract

If X is a (topological) space, the nth 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--> X(n), (x_1,...,x_n)-->{x_1,...,x_n}. That is, a subset A of X(n) is open if and only if its preimage under q is open in the product space X^n.

Given a space X, let H(X) denote all homeomorphisms of X. For any subclass C of homeomorphisms in 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?".

Extensive work exists in the literature on the richness of the geometry of X(n). Nevertheless, despite the fact that the spaces X(n) considered in those investigations are metrizable (which is the case if and only if X is itself metrizable) the important role of metrics has been mostly ignored. Consequently, the existing results mostly elucidate topological aspects of the geometry of X(n).

The main goal of this thesis is to attempt to answer the above research question(s) for several geometric properties, with metrics playing a significant role (hence the title phrase "Metric Geometry of ..."). Some of the questions are relatively easy and will be answered completely. However, a question such as "If a normed space X is an absolute Lipschitz retract, does it follow that X(n) is also an absolute Lipschitz retract?" appears to require considerable effort and will be answered only partially. By the definition of an absolute Lipschitz retract, establishing the existence of Lipschitz retractions X(n) --> X(n-1) for all n>= 1 would be a partial positive answer to this question.

Among other things, we will prove the following. If X is a metrizable space, then so is X(n). If a metric space X is a snowflake, quasiconvex, or doubling then so is X(n). If two spaces X and Y are (Lipschitz) homotopy equivalent, then so are X(n) and Y(n). If X is a normed space (which is Lipschitz k-connected for all k>= 0), then X(n) is Lipschitz k-connected for all k>= 0. If X is a normed space, there exist (i) Holder retractions X(n)--> X(n-1), (ii) Lipschitz retractions X(n)--> X(1),X(2), and (iii) Lipschitz retractions X(n)--> X(n-1) when the dimension of X is finite or X is a Hilbert space.

Open Access

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