---
abstract: 'The paper deals with the program of determining the complexity of various homeomorphism relations. The homeomorphism relation on compact Polish spaces is known to be reducible to an orbit equivalence relation of a continuous Polish group action (Kechris-Solecki). It is shown that this result extends to locally compact Polish spaces, but does not hold for spaces in which local compactness fails at only one point. In fact the result fails for those subsets of ${\mathbb{R}}^3$ which are unions of an open set and a point. In the end a list of open problems is given in this area of research.'
author:
- 'Vadim Kulikov[^1]'
bibliography:
- 'ref1.bib'
title: 'Classification and Non-classification of Homeomorphism Relations'
---

**MSC2010:** 03E15, 57N10, 57M25, 57N65.

#### Acknowledgments. {#acknowledgments. .unnumbered}

I am grateful to Professors Alexander Kechris and Su Gao for providing useful answers to my e-mails which helped and encouraged me to proceed with this work. During the time of the preparation I also had several valuable discussions on this topic with my Ph.D. supervisor Tapani Hyttinen and my friends and collegues Rami Luisto, Pekka Pankka and Marcin Sabok. Last but not least I would like to thank a dear to me woman (whose identity is concealed) for the infinite inspiration that she has given me during the time of this work.

This research was supported by the Austrian Science Fund (FWF) under project number P24654.

Introduction {#sec:Intro}
============

It is known that the homeomorphism relation on Polish spaces is $\Sigma^1_2$ [@Gao] and ${{\Sigma_1^1}}$-hard [@FerLouRos Thm 22]. On the other hand, it is known that restricted to compact spaces this homeomorphism relation is reducible to an orbit equivalence relation induced by continuous action of a Polish group which is known to be strictly below ${{\Sigma_1^1}}$-complete. This result is extended to locally compact spaces in Theorem \[thm:locCom\].

The main result of this paper is that this “nice” property of locally compact spaces breaks when just one point is added to them: The homeomorphism relation on the ${\sigma}$-compact spaces of the form $V\cup \{x\}$ where $x\in {\mathbb{R}}^3$ is fixed and $V\subset {\mathbb{R}}^3$ is open falls somewhere in between: it is ${{\Sigma_1^1}}$ and the equivalence relation known as $E_1$ is continuously reducible to it (Theorem \[thm:NonClass\]). This implies that this homeomorphism relation is not classifiable in a Borel way by any orbit equivalence relation arising from a Borel action of a Polish group.

The proof relies on known results in knot theory and low dimensional topology. We hope that these methods can be helpful in approaching Question \[open:Main\] and other questions listed in Section \[sec:Further\].

Sections \[sec:KnotTheory\] and \[sec:BkgrndDST\] are devoted to the required preliminaries. In Section \[sec:NonClass\] we prove the main non-classification result. In the final sections the research topic of classifying homeomorphism relations is looked at in more detail: In Section \[sec:Other\] it is reviewed what positive results there are in classification of homeomorphism relations and in Section \[sec:Further\] a list of open questions is given in the area.

Preliminaries in Topology and Knot Theory {#sec:KnotTheory}
=========================================

In this section we go through those definitions and lemmas in knot theory and topology that we need in the proofs later. We assume that the reader is familiar with the notion of the first homology group $H_1(X)$ of a topological space $X$. The standard definitions can be found for example in [@Hatcher].

We denote by ${\mathbb{R}}^n$ the $n$-dimensional Euclidean space and by ${\mathbb{S}}^n$ the one-point compactification of it, i.e. ${\mathbb{S}}^n={\mathbb{R}}^n\cup\{\infty\}$ and the neighborhoods of $\infty$ are the sets of the form $\{\infty\}\cup({\mathbb{R}}^n\setminus C)$ where $C$ is compact. By ${{\operatorname{int}}}A$ we denote the topological interior of $A$ and by $\bar A$ the closure.

Hausdorff Metric and Path Connected Subspaces
---------------------------------------------

\[def:HausdorffMetric\] Let $X$ be a compact metric space. The space of all non-empty compact subsets of $X$ is denoted by $K(X)$. We equip $K(X)$ with the Hausdorff-metric: An *${\varepsilon}$-collar* of a set $C\subset X$ is the set $$C_{\varepsilon}=\{x\mid d(x,C)< {\varepsilon}\}$$ and the Hausdorff-distance between two sets in $K(X)$ is determined by $$d_{K(X)}(C,C')=\max\{\inf\{{\varepsilon}\mid C\subset C'_{\varepsilon}\},\inf\{{\varepsilon}\mid C'\subset C_{\varepsilon}\}\}.$$

The following facts are standard to verify.

\[fact:Hausdorffmetric\] Let $X$ be a compact metric space. Then $K(X)$ is compact and if $(C_i)_{i\in{\mathbb{N}}}$ is a converging sequence in $K(X)$ and $C_*$ is its limit, then

1.  for every $x_*$ we have $x_*\in C_*$ if and only if there is a sequence $x_i$ converging to $x_*$ with $x_i\in C_i$ for all $i\in{\mathbb{N}}$. \[fact:Haus1\]

2.  if every $C_i$ is connected, then $C_*$ is connected. \[fact:Haus2\] \[fact:Haus3\]

\[def:DenselyPathConnected\] A subset $A\subset {\mathbb{R}}^n$ is *path metric* if the distance between two points is given by $$d_E(x,y)=\inf\{L({\gamma})\mid {\gamma}\subset A\text{ is a path joining }x\text{ and }y\}$$ where $d_E$ is the Euclidean distance and $L({\gamma})$ is the length of the path. Equivalently $A$ is path metric if and only if for every two points $x,y\in A$ and ${\varepsilon}>0$ there is a path ${\gamma}\subset A$ connecting $x$ to $y$ and $L({\gamma})<(1+{\varepsilon})d_E(x,y)$.

\[lem:HDPathMetric\] If the Hausdorff dimension of a closed $A\subset{\mathbb{R}}^n$ is less than $n-1$, then ${\mathbb{R}}^n\setminus A$ is path metric.

Let $D_0$ be the $(n-1)$-dimensional unit disc $$D_0=\{(x_1,\dots,x_{n-1})\mid x_1^2+\dots+x_{n-1}^2<1\}.$$ and let $C_0$ be the cylinder $$D_0\times [0,1]\subset {\mathbb{R}}^n.$$ For $x,y\in{\mathbb{R}}^n$ denote by $[x,y]$ the straight line segment connecting $x$ and $y$. Suppose $A_0\subset C_0$ and assume that for every $(x_1,\dots,x_{n-1})\in D_0$ the set $$A_0\cap [(x_1,\dots,x_{n-1},0),(x_1,\dots,x_{n-1},1)]$$ is non-empty. Then $A_0$ must have Hausdorff dimension at least $n-1$: $A_0$ can be projected onto $D_0$ with the Lipschitz map $$(x_1,\dots,x_{n-1},x_n)\mapsto (x_1,\dots,x_{n-1},0),$$ the latter has Hausdorff dimension $n-1$ and the Hausdorff dimension cannot increase in a Lipschitz map. Therefore we have the following claim:

If $A_0\subset C_0$ has Hausdorff dimension less than $n-1$, then there is $(x_1,\dots,x_{n-1})\in D_0$ such that $[(x_1,\dots,x_{n-1},0),(x_1,\dots,x_{n-1},1)]\cap A_0={\varnothing}.$

Let $x,y\in {\mathbb{R}}^n\setminus A$ and let ${\varepsilon}>0$. Since $A$ is closed there is ${\delta}<{\varepsilon}/2$ such that $\bar B(x,{\delta})\cap A=\bar B(y,{\delta})\cap A={\varnothing}$. Let $P_x$ and $P_y$ be $(n-1)$-dimensional affine hyperplanes passing through $x$ and $y$ respectively and which are orthogonal to $x-y$. Then there is an affine map $f\colon {\mathbb{R}}^n\to{\mathbb{R}}^n$ such that $f[P_x]={\mathbb{R}}^{n-1}\times\{0\}$, $f[P_y]={\mathbb{R}}^{n-1}\times \{1\}$ and $f[P_x\cap \bar B(x,{\delta})]=D_0\times\{0\}$. Since $\dim_H(A)<n-1$, also $\dim_H(f[A])<n-1$ (because $f$ is Lipschitz) and so by the claim above there is a line segment $s$ passing from $f[\bar B(x,{\delta})\cap P_x]$ to $f[\bar B(y,{\delta})\cap P_y]$ outside $f[A]$ which is orthogonal to ${\mathbb{R}}^{n-1}\times\{0\}$. By applying $f^{-1}$ to $s$, we obtain a straight line segment passing from $\bar B(x,\delta)\cap P_x$ to $\bar B(y,\delta)\cap P_y$ orthogonal to $P_x$. Now by connecting the endpoints of $f^{-1}[s]$ to $x$ and $y$ we obtain a path outside $A$ of length at most $d(x,y)+2{\delta}=d(x,y)+{\varepsilon}$ connecting these two points.

\[lemma:CauchyComponent\] Suppose $X,X'\subset {\mathbb{R}}^n$ are such that $X$ is a path metric space and there is a homeomorphism $h\colon X\to X'$. If $(x_i)_{i\in{\mathbb{N}}}$ is a Cauchy sequence in $X$ converging in ${\mathbb{R}}^n$ to some point $x\in {\mathbb{R}}^n\setminus X$, then all the accumulation points of $(h(x_i))_{i\in{\mathbb{N}}}$ lie in the same component of ${\mathbb{R}}^n\setminus X'$. In particular, if this component is a singleton, then $(h(x_i))_{i\in{\mathbb{N}}}$ is also a Cauchy sequence.

Let $(x_i)_{i\in{\mathbb{N}}}$ be as in the statement. Suppose for a contradiction that $y^1$ and $y^2$ are two points in two different components of ${\mathbb{R}}^n\setminus X'$ that are accumulation points of $(h(x_i))_{i\in{\mathbb{N}}}$ and let $(x^1_{i})_{i\in{\mathbb{N}}}$ and $(x^2_{i})_{i\in{\mathbb{N}}}$ be subsequences of $(x_i)_{i\in{\mathbb{N}}}$ such that $(h(x^1_i))_{i\in{\mathbb{N}}}$ and $(h(x^2_i))_{i\in{\mathbb{N}}}$ converge to $y^1$ and $y^2$ respectively. For $k\in\{1,2\}$ and $i\in{\mathbb{N}}$ denote $y^k_i=h(x^k_i)$.

For each $i\in {\mathbb{N}}$ let ${\gamma}_i$ be a path in $X$ connecting $x^1_i$ to $x^2_{i}$ such that $L({\gamma})<(1+2^{-i})d(x^1_i,x^2_{i})$. We think of the paths as compact subsets of ${\mathbb{S}}^n$. The sequence $\{{\gamma}_i\mid
  i\in{\mathbb{N}}\}$ converges in $K({\mathbb{S}}^n)$ to $\{x\}$. Consider the sequence $(h[\gamma_i])_{i\in{\mathbb{N}}}$. It is a sequence of compact subsets of ${\mathbb{S}}^n$, so it is a sequence of elements of $K({\mathbb{S}}^n)$. The latter is compact, so there is a converging subsequence: $(h[\gamma_{i(k)}])_{k\in{\mathbb{N}}}$. Denote by $\gamma$ the limit of that sequence. By Fact \[fact:Hausdorffmetric\].\[fact:Haus1\], we have $y^1,y^2\in {\gamma}$ since $y^1_i,y^2_i\in h[\gamma_i]$ for all $i\in{\mathbb{N}}$ and additionally, since every element in the sequence is connected, ${\gamma}$ is also connected by Fact \[fact:Hausdorffmetric\].\[fact:Haus2\].

Since $y^1$ and $y^2$ lie in different components of ${\mathbb{R}}^n\setminus X'$, there must be a point $z$ in ${\gamma}$ which is in $X'$. Now, by Fact \[fact:Hausdorffmetric\].\[fact:Haus1\] we can find a sequence $z_k\in h[{\gamma}_{i(k)}]$, $k\in{\mathbb{N}}$, such that $(z_k)_{k\in{\mathbb{N}}}$ converges to $z$. But $h^{-1}(z_k)$ lies in ${\gamma}_{i(k)}$ and so $(h^{-1}(z_k))_{k\in{\mathbb{N}}}$ converges to $x$. This is a contradiction, because $x\notin {\operatorname{dom}}h=X$, but $z\in {\operatorname{ran}}h= X'$.

Separation Theorems {#sec:Sep}
-------------------

Here we state, for the sake of completeness, two known results from finite dimensional topology that we will need.

\[thm:JordanBrouwer\] Let $h\colon {\mathbb{S}}^{n-1}\to {\mathbb{S}}^n$ be an embedding. Then ${\mathbb{S}}^n\setminus h[{\mathbb{S}}^{n-1}]$ consists of two open connected components.

(A Generalization of the Schönflies Theorem by M. Brown [@Brown])\[thm:Brown\] Let $h\colon {\mathbb{S}}^{n-1}\times [0,1]\to {\mathbb{S}}^n$ be an embedding. Then the closures of the complementary domains of $h[{\mathbb{S}}^{2}\times \{\frac{1}{2}\}]$ are topological $n$-cells, i.e. homeomorphic to closed balls. In particular there is a self-homeomorphism of ${\mathbb{S}}^n$ which takes $h[{\mathbb{S}}^{n-1}\times \{\frac{1}{2}\}]$ to the standard ${\mathbb{S}}^{n-1}$.

Knot Theory
-----------

We present the basics of knot theory here as neatly as possible and account only for the facts necessary for the present paper. Unless a specific reference is given below, the reader is referred to the classical textbooks on knot theory [@BurZie; @Kauf; @Mur] for the details and omitted proofs.

A *knot* is an embedding $K\colon {\mathbb{S}}^1\to {\mathbb{S}}^3$. We often identify a knot with its image, ${\operatorname{ran}}K$. This is in particular justified by the following equivalence relation on knots:

Two knots $K_0,K_1\colon {\mathbb{S}}^1\to {\mathbb{S}}^3$ are equivalent, if there is a homeomorphism $h\colon {\mathbb{S}}^3\to {\mathbb{S}}^3$ with $$K_0=h\circ K_1.$$ In literature this homeomorphism is often required to be orientation preserving in which case this equivalence relation coincides with the so-called *ambient isotopy*, but we do not require $h$ to be orientation preserving.

A knot is *trivial* if it is equivalent to the standard embedding ${\mathbb{S}}^1{\hookrightarrow}{\mathbb{S}}^3$. A knot is *tame* if it is equivalent to a smooth or a piecewise linear knot. As usual in knot theory, we consider only tame knots.

The following is a basic fact of knot theory:

There are infinitely many non-equivalent knots.

A not so basic fact is the following theorem:

([@GorLue])\[thm:GordonLuecke\] If two knots have homeomorphic complements, then they are equivalent.

Let $K$ be a knot in ${\mathbb{R}}^3$. A *Seifert surface* $S$ of $K$ is a compact orientable connected $2$-manifold with boundary $M\subset {\mathbb{R}}^3$ whose interior lies in ${\mathbb{R}}^3\setminus K$ and the boundary is exactly $K$.

\[fact:ExistsSeifertS\] For every open ball $B$ containing $K$ there exists a Seifert surface $S\subset B$ of $K$.

([@Rolfsen 5.D])\[fact:LinkingSeifert\] Let $K$ be a knot and $w$ a closed curve in ${\mathbb{R}}^3\setminus K$. The following are equivalent:

-   $w\cap S\ne{\varnothing}$ for every Seifert surface $S$ of $K$,

-   $w$ represents a non-trivial element in $H_1({\mathbb{R}}^3\setminus K)$.

\[fact:NonTrHomology\] For every knot $K$ we have $H_1({\mathbb{S}}^3\setminus K)\cong {\mathbb{Z}}$.

Preserving Knot Types
---------------------

The goal of this section is to prove Lemma \[lemma:FindPoints\] which says that if we carve out infinitely many knots from ${\mathbb{R}}^3$ in a certain way, then a self-homeomorphism of the left-over space will, in an approximate way, respect the knot-types of the carved knots.

\[def:Properties\] Let $(B_n)_{n\in{\mathbb{N}}}$ be a sequence of closed balls in ${\mathbb{R}}^3$, $(K_n)_{n\in{\mathbb{N}}}$ a sequence of knots, $Q\subset {\mathbb{R}}^3$ and $P\subset {\mathbb{R}}^3$. Here we list some properties for these sets which we will later refer to.

-   All the balls are disjoint from each other and are contained in a bounded region, i.e. there is $r$ such that ${\bigcup}_{n\in{\mathbb{N}}}B_n\subset B(0,r)$.

-   If $x$ is a limit of a sequence $(x_i)_{i\in{\mathbb{N}}}$ such that for all $i\in{\mathbb{N}}$ the point $x_i$ is in the ball $B_{n_i}$ and for all $i<j$, $n_i\ne n_j$, then $x$ is not in any of the balls. $Q$ is the set of such points $x$.

-   $P\supset Q$, every connected component of $P$ contains a point in $Q$ and for all $n$ there is ${\varepsilon}>0$ such that $P\cap (B_n)_{{\varepsilon}}={\varnothing}$. (Recall the definition of ${\varepsilon}$-collar, Definition \[def:HausdorffMetric\]).

-   $K_n\subset {{\operatorname{int}}}B_n$.

-   $X={\mathbb{R}}^3\setminus (P\cup{\bigcup}_{n\in{\mathbb{N}}}K_n)$ is path metric (Definition \[def:DenselyPathConnected\]).

\[lemma:FindPoints\] Suppose $(B_n)_{n\in{\mathbb{N}}}$, $(K_n)_{n\in{\mathbb{N}}}$, $Q$ and $P$ as well as $(B_n')_{n\in{\mathbb{N}}}$, $(K_n')_{n\in{\mathbb{N}}}$, $Q'$ and $P'$ satisfy the properties B1 – B5. Let $$X={\mathbb{S}}^3\setminus (P\cup{\bigcup}_{n\in{\mathbb{N}}}K_n)$$ and $$X'={\mathbb{S}}^3\setminus (P'\cup{\bigcup}_{n\in{\mathbb{N}}}K'_n).$$ Suppose further that $X$ and $X'$ are homeomorphic and $h$ is the homeomorphism. Then there is a bijection $\rho\colon{\mathbb{N}}\to{\mathbb{N}}$ such that for all $n\in{\mathbb{N}}$ we have that $K_n$ and $K_{\rho(n)}'$ have the same knot-type and for some $z\in B_n\setminus K_n$ we have $h(z)\in B_{\rho(n)}'\setminus K_{\rho(n)}'$.

Fix $n\in {\mathbb{N}}$. By the Jordan-Brouwer separation theorem (Theorem \[thm:JordanBrouwer\]) the complement of $h[\partial B_n]$ in ${\mathbb{S}}^3$ consists of two open connected components, say $Y_1$ and $Y_2$. In this case, however, we can prove even more, namely that $Y_1$ and $Y_2$ are homeomorphic to open balls and that there is a self-homeomorphism of ${\mathbb{S}}^3$ wich takes $h[\partial B_n]$ to ${\mathbb{S}}^2$. Let ${\varepsilon}$ be small enough so that $(\partial B_n)_{\varepsilon}\cap
  B_k={\varnothing}$ for all $k\ne n$, $(\partial B_n)_{\varepsilon}\cap P={\varnothing}$ and $(\partial B_n)_{{\varepsilon}}\cap K_n={\varnothing}$. This is possible by B2, B3 and B4. Let $$f\colon {\mathbb{S}}^2\times [0,1]\to (\partial B_n)_{{\varepsilon}}$$ be a homeomorphism such that $f[{\mathbb{S}}^2\times \{\frac{1}{2}\}]=\partial B_n$. We can think of $h\circ f$ as an embedding of ${\mathbb{S}}^2\times [0,1]$ into ${\mathbb{S}}^3$. Now apply the the generalized Schönflies theorem (Theorem \[thm:Brown\]) to $h\circ f$. Since $\partial B_n$ divides $X$ into two disjoint components as well as $h[\partial B_n]$ divides $X'$, $h$ takes them to one another. Assume without loss of generality that $h[{{\operatorname{int}}}B_n\setminus K_n]=Y_1\cap X'$.

\[claim:ConnectedThing\] The space $Y_1\setminus X'$ is connected.

For this we need a slight modification of the argument used to prove Lemma \[lemma:CauchyComponent\]. (Note that $B_n\setminus K_n$ is path metric.) Suppose there was two components $A$ and $B$ of $Y_1\setminus X'$ and let $(x_1,y_1,x_2,y_2,\dots)$ be a sequence such that $(x_i)_{i\in{\mathbb{N}}}$ converges (in ${\mathbb{S}}^3$) to a point in $A$ and $(y_i)_{i\in{\mathbb{N}}}$ converges to a point in $B$. Now $(h^{-1}(x_1),h^{-1}(y_1)\cdots)$ can only have accumulation points in $K_n$ (because the accumulation points cannot be in $X$). Pick Cauchy subsequences from both $(h^{-1}(x_i))_{i\in{\mathbb{N}}}$ and $(h^{-1}(y_i))_{i\in{\mathbb{N}}}$ and denote $(z_i)_{i\in{\mathbb{N}}}$ and $(w_i)_{i\in{\mathbb{N}}}$. Since $z=\lim_{i\to\infty}z_i$ and $w=\lim_{i\to\infty}w_i$ lie both in the knot, using the fact that $B_n\setminus K_n$ is path metric, it is possible to connect $z_i$ to $w_i$ by a curve ${\gamma}_i$ lying in $B_n\setminus K_n$ such that the sequence $({\gamma}_i)_{i\in{\mathbb{N}}}$ converges in $K({\mathbb{S}}^3)$ to a subset of $K_n$. Now pick (in $K({\mathbb{S}}^3)$) a converging subsequence $(\xi_{j})_{j\in{\mathbb{N}}}$ of $(h[{\gamma}_i])_{i\in{\mathbb{N}}}$. These are connected sets containing $h(z_i)$ and $h(w_i)$. Therefore the limit in $K({\mathbb{S}}^3)$ must intersect both $A$ and $B$ and since it is connected, it must contain a point $p$ in $Y_1\cap X'= h[{{\operatorname{int}}}B_n]$. By Fact \[fact:Hausdorffmetric\] there is a Cauchy sequence $(p_j)_{j\in{\mathbb{N}}}$ with $p_j\in \xi_{j}$ converging to $p$ but $((h^{-1}(p_j))_{j\in{\mathbb{N}}}$ does not have accumulation points in $B_n\setminus K_n$. This is a contradiction.

Thus, $Y_1\setminus X'$ is a connected component of ${\mathbb{S}}^3\setminus X'$ Note that this component must be in the interior of $\bar Y_1$, so it cannot be a subset of $P$, by B2, B3 and B4. Thus, it is $K_m'$ for some $m$. Since $h$ is a homeomorphism we have that $${{\operatorname{int}}}B_n\setminus K_n\approx Y_1\setminus K_m'.$$ Since $Y_1\approx {{\operatorname{int}}}B_n\approx {\mathbb{R}}^n$, we can conclude from Theorem \[thm:GordonLuecke\] that $K_n$ and $K_{m'}$ have the same knot-type.

By symmetry arguments using the fact that $h$ is a homeomorphism, this establishes a map $n\mapsto m$ which is actually bijective, so denote this bijection by $\rho$.

Let ${\gamma}\subset B_n\setminus K_n$ be a closed curve representing a non-trivial cycle in $H_1(B_n\setminus K_n)$ (such exists by Fact \[fact:NonTrHomology\]). Then $h[{\gamma}]$ will be a non-trivial cycle in $h[B_n]$. We would like to show that $h[{\gamma}]$ is also non-trivial in $({\mathbb{S}}^3\setminus Y_1)\cup h[B_n]$. But since we established a homeomorphism of ${\mathbb{S}}^3$ to itself taking $h[\partial B_n]$ to ${\mathbb{S}}^2$, we know that if ${\gamma}$ bounds a disk $D\subset ({\mathbb{S}}^3\setminus Y_1)\cup h[B_n]$, this disk can be isotoped to a disk $D'\subset Y_1$ keeping $Y_1\cap D$ fixed.

Let $S$ be a Seifert surface of $K_m'$ contained in $B_m'$ (see Fact \[fact:ExistsSeifertS\]). Then by Fact \[fact:LinkingSeifert\] there is a point $z'\in h[{\gamma}]\cap S$. Let $z=h^{-1}(z')$. This completes the proof, since $z'\in B_{m'}$.

Preliminaries in Descriptive Set Theory {#sec:BkgrndDST}
=======================================

A *Polish space* is a separable topological space which is homeomorphic to a complete metric space.

The most common examples of Polish spaces are ${\mathbb{R}}$, ${\mathbb{C}}$ and ${\mathbb{N}}^{\mathbb{N}}$ in the Tychonov product topology. Less common examples include the space of all homeomorphisms ${{\operatorname{Hom}}}(X)$ of a compact Polish space $X$ in the $\sup$-metric (see Fact \[fact:HomPolish\]) and the space of compact subsets of a compact space $X$ in the Hausdorff metric denoted $K(X)$ (see Fact \[fact:Hausdorffmetric\]).

([@Kechris])\[fact:Gdelta\] A subset of a Polish space is Polish in the subspace topology if and only if it is a $G_\delta$ subset.

([@Kechris Theorem 3.11 and Example 9B(8)]) \[fact:HomPolish\] For a compact Polish space $X$ equipped with the metric $d_X$, the space ${{\operatorname{Hom}}}(X)$ of homeomorphisms of $X$ in the $\sup$-metric, $\delta(h,g)=\sup\{d_X(h(x),g(x))\mid x\in X\}$ is a Polish space.

\[def:Reduction\] Suppose $E$ and $E'$ are equivalence relations on Borel subsets $A$ and $A'$ of Polish spaces $X$ and $X'$ respectively. The equivalence relation $E$ is *Borel reducible to* $E'$, denoted $E{\leqslant}_B E'$, if there is a Borel map $f\colon A\to A'$ such that $$\forall x,y\in A\big((x,y)\in E\iff (f(x),f(y))\in E'\big).$$ We say that an equivalence relation $E$ is *universal* among a set $X$ of equivalence relations, if $E\in X$ and for all $E'\in X$ we have $E'{\leqslant}_B E$.

A lot is known about the partial order ${\leqslant}_B$ on analytic equivalence relations which are defined on standard Borel spaces. A thorough treatment can be found in [@Gao]. Preface in [@Hjorth] gives a good glimpse of available applications. Here is an example of an equivalence relation which we will need:

\[def:E1\] Let $(2^{\mathbb{N}})^{\mathbb{N}}$ be the space of sequences of elements of $2^{\mathbb{N}}$ (the Cantor space). The topology on both $2^{\mathbb{N}}$ and $(2^{\mathbb{N}})^{{\mathbb{N}}}$ is given by the Tychonov product topology. Let $E_1$ be the equivalence relation given by: $$((r_n)_{n\in {\mathbb{N}}},(s_n)_{n\in {\mathbb{N}}})\in E_1\iff
  \exists m\forall k>m(r_k=s_k).$$

Another wide class of equivalence relations is given by Polish group actions:

Let $G$ be a Polish group acting in a Borel way on a Polish space $X$. Let $E^X_G$ be the equivalence relation where $x,y\in X$ are equivalent if and only if there exists $g\in G$ such that $y=gx$. This is called the *orbit equivalence relation* induced by this (Borel) action of a Polish group.

Many natural equivalence relations, in particular the isomorphism on countable structures (see the end of this section), can be viewed as orbit equivalence relations induced by Polish group actions. A proof of the following can be found in [@Gao Theorem 10.6.1].

\[thm:KecLou\](Kechris-Louveau [@KecLou]) Let $E$ be any orbit equivalence relation induced by a Borel action of a Polish group. Then $E_1\not{\leqslant}_B E$.

\[def:StrangeSpaces\] Let $X$ be a compact Polish space. For a fixed closed (and hence compact) subset $F\subset X$, let $$K^{F}(X)=\{A\in K(X)\mid F\subset A\}.$$ (See Definition \[def:HausdorffMetric\] for the definition of $K(X)$.) Then $K^{F}(X)$ is a closed subspace of $K(X)$ and so Polish itself by Fact \[fact:Gdelta\]. Let $$K^{F}_*(X)=\{(X\setminus A)\cup F\mid A\in K^F(X)\}.$$ The Polish topology on $K^{F}_*(X)$ is induced by the bijection $K^F(X)\to K^F_*(X)$ given by $A\mapsto (X\setminus A)\cup F$.

Let $F\subset X$ be closed. Then elements of $K^F_*(X)$ are of the form $U\cup F$ where $U$ is an open set disjoint from $F$. Therefore elements of this space are $\sigma$-compact $G_\delta$-subsets. Using Fact \[fact:Gdelta\] we obtain:

\[fact:TheyreAllPolish\] For a fixed closed $F\subset X$ and $X$ compact $K^F_*(X)$ consists of $\sigma$-compact Polish spaces.

\[def:Relations\] For a fixed closed $F\subset {\mathbb{S}}^3$, let $\approx^F$ be the homeomorphism relation on the space $K^F_*({\mathbb{S}}^3)$.

The main result of this paper (Theorem \[thm:NonClass\]) can be now stated: for a fixed $x\in {\mathbb{S}}^3$, $E_1{\leqslant}_B\ \approx^{\{x\}}$.

A countable model in a fixed vocabulary with universe ${\mathbb{N}}$ can be coded as an element of $2^{{\mathbb{N}}}$ in such a way that each $\eta\in 2^{\mathbb{N}}$ in fact represents some model. There are many nice ways to do this, see for example [@Gao]. Let $\cong$ be the equivalence relation of isomorphism. It is well known that given a vocabulary and any collection of countable models in this vocabulary whose set of codes is Borel, $\cong$ is reducible to $\cong_G$ where $\cong_G$ is the isomorphism of graphs, i.e. vocabulary consists of one binary symbol and the models are infinite graphs with domain ${\mathbb{N}}$. This equivalence relation is induced by the action of the infinite symmetric group $S_\infty$ (which is Polish in the standard product topology). A corollary to Theorem \[thm:NonClass\] which follows from Theorem \[thm:KecLou\] is that $\approx^{\{x\}}$ is not reducible to $\cong_G$, although this has been proved for the homeomorphism relation on compact spaces already by Hjorth [@Hjorth].

The original motivation of this research was the following, stronger, question:

\[open:Main\] Is $\approx^{{\varnothing}}$ reducible to $\cong_G$?

Note that $\approx^{{\varnothing}}$ is just the homeomorphism relation on open subsets of ${\mathbb{S}}^3$. See Section \[sec:Further\] for a discussion on this and other open questions.

Parametrization {#ssec:Par}
---------------

As was pointed out, the space $K^F_*(X)$ consists of ${\sigma}$-compact Polish spaces (Fact \[fact:TheyreAllPolish\]). However, not *all* ${\sigma}$-compact Polish spaces are found in $K^F_*(X)$. There are different ways to parametrize different classes of Polish spaces such as compact, locally compact, ${\sigma}$-compact, $n$-manifolds and so on. In this section we will present these different ways and show that essentially it does not matter which one we choose, all of them being essentially equivalent in some sense. Additionally in this section we introduce many new notations for various homeomorphism relations. A helpful list of notations can be found in Section \[sec:Conclusion\].

In [@HjoKec1] Hjorth and Kechris give a simple parametrization of all Polish spaces. Their parametrization, let us call it the *Hjorth-Kechris parametrization*, consists of two-fold sequences $\eta\in {\mathbb{R}}^{{\mathbb{N}}\times{\mathbb{N}}}$ which satisfy the requirements for a metric on ${\mathbb{N}}$. The set of such $\eta$ is easily seen to be Borel. Then the space $X(\eta)$ is obtained as a completion of this countable metric space. Another way to parametrize all Polish spaces is to view them as closed subsets of the Urysohn universal space $U$. Denote the space of all closed subsets of $U$ by $F(U)$. It can be equipped with a standard Borel structure which is inherited from $K(\bar U)$ where $\bar U$ is a compactification of $U$ (see [@Kechris Thm 12.6]). The Borel sets of $F(U)$ are generated by the sets of the form $$\{F\in F(U)\mid F\cap O\ne {\varnothing}\}$$ for some open $O\subset U$. This Borel structure is also generated by the *Fell topology* generated by the sets of the form

$$\label{eq:StandBor}
  \{F\in F(U)\mid F\cap K={\varnothing}\land F\cap O_1\ne{\varnothing}\land\dots\land F\cap O_n\ne{\varnothing}\},$$

where $K$ varies over $K(U)$ and $O_i$ are open sets in $U$ [@Kechris Exercise 12.7].

Let us show that these parametrizations are essentially equivalent. The universality property of $U$ is that given any finite metric space $H$ and $x\in H$, every isometric embedding of $H\setminus \{x\}$ into $U$ extends to an isometric embedding of $H$ into $U$. Thus, given a countable metric space as defined by $\eta$ as above, it can be isometrically embedded into $U$. The closure of the image will then be homeomorphic (and even isometric) to $X(\eta)$. We want to show that there are Borel reductions reducing the homeomorphism of Polish spaces in one parametrization to the other. To show this, let us define an “intermediate” parametrization. Let $U^{\mathbb{N}}$ be the set of all countable sequences in $U$. Each such sequence $\xi$ corresponds to the Polish space $Y(\xi)$ obtained as its closure taken in $U$. Let $f_1\colon U^{\mathbb{N}}\to {\mathbb{R}}^{{\mathbb{N}}\times{\mathbb{N}}}$ be defined by $f_1(\xi)=\eta$ where $\eta(n,m)=d_U(\xi(n),\xi(m))$. Obviously $X(\eta)$ and $Y(\xi)$ are isometric and $f_1$ is continuous. Let $f_2\colon U^{\mathbb{N}}\to F(U)$ be the map which takes $\xi$ to the closure of $\{\xi(n)\mid n\in{\mathbb{N}}\}$ in $U$.

There are Borel functions $g_1\colon {\operatorname{ran}}(f_1)\to U^{\mathbb{N}}$ and $g_2\colon F(U)\to U^{\mathbb{N}}$ such that $f_1\circ g_1={\operatorname{id}}$ and $f_2\circ g_2={\operatorname{id}}$.

For $g_2$ we will use [@Sri Cor 5.4] which says that if $f\colon X\to Y$ is a Borel function between Polish spaces $X$ and $Y$ such that $f[V]$ is open for all open $V\subset X$, $f^{-1}[V]$ is $F_{\sigma}$ for all open $V\subset Y$ and $f^{-1}\{y\}$ is $G_\delta$ for all $y\in Y$, then there is $g\colon Y\to X$ such that $f\circ g={\operatorname{id}}_Y$.

It is easy to see that the inverse image under $f_2$ of a set of the form is $F_{\sigma}$ in $U^{\mathbb{N}}$, so in particular $f_2$ is Borel. Additionally, given a closed set $C\in F(U)$, the inverse image of the singleton $f_2^{-1}\{C\}$ is $G_\delta$. To see this, let $Q=\{q_n\mid n\in{\mathbb{N}}\}$ be a dense countable subset of $C$. Then $f_2^{-1}\{C\}$ is the intersection of $\{\xi\mid {\operatorname{ran}}(\xi)\subset C\}$ and the sets $$O(k,m)=\{\xi\mid \exists n\in{\mathbb{N}}(\xi(n)\in B(q_k,1/m))\}.$$ The former is closed and the latter are open, so the intersection is $G_{\delta}$. Let $V\subset U^{\mathbb{N}}$ be a basic open set. It is of the form $$O_0\times\cdots\times O_n\times U^{{\mathbb{N}}\setminus \{0,\dots,n\}}.\label{eq:openset}$$

To see that $f_2[V]$ is open in $U^{\mathbb{N}}$ note that it can be represented in the form of with $K={\varnothing}$ and $O_i$ as in ; thus $f_2$ is an open map. By [@Sri Cor 5.4] there is a Borel $g_2\colon F(U)\to U^{\mathbb{N}}$ such that $f_2\circ g_2={\operatorname{id}}$.

Now consider $f_1$. It is continuous and the inverse image of a singleton is closed. By [@Kechris Thm 35.46] (or again by [@Sri Cor 5.4]) there is $g_1\colon {\operatorname{ran}}(f_1)\to U^{\mathbb{N}}$ such that $f_1\circ g_1={\operatorname{id}}$. Note that ${\operatorname{ran}}(f_1)$ is merely the Borel subset of ${\mathbb{R}}^{{\mathbb{N}}\times{\mathbb{N}}}$ on which the operation $\eta\mapsto X(\eta)$ is well defined and produces a Polish space.

Let $\approx_P$ be the equivalence relation on ${\operatorname{ran}}(f_1)\subset{\mathbb{R}}^{{\mathbb{N}}\times{\mathbb{N}}}$ where $\eta$ and $\eta'$ are equivalent if and only if $X(\eta)$ and $X(\eta')$ are homeomorphic, let $\approx_P'$ be the equivalence relation on $U^{\mathbb{N}}$ where two sequences $\xi$ and $\xi'$ are equivalent if and only if $Y(\xi)$ and $Y(\xi')$ are homeomorphic, and let $\approx''_P$ be the equivalence relation on $F(U)$ where two closed $C$ and $C'$ are equivalent if and only if they are homeomorphic.

Then $f_1$, $g_1$, $f_2$ and $g_2$ witness that these three equivalence relations, $\approx_P$, $\approx'_P$ and $\approx''_P$ are all Borel reducible to each other.

Hjorth and Kechris showed that the set of those $\eta$ for which $X(\eta)$ is compact and the set of those for which it is locally compact are both Borel subsets of ${\mathbb{R}}^{{\mathbb{N}}\times{\mathbb{N}}}$. Taking Borel inverse images under $f_1$ and $g_2$ we obtain the same conclusion for the other parametrizations.

In [@HjoKec1] it is shown that the set of complex $n$-manifolds is Borel. One has to replace “biholomorphic” by “homeomorphic” in order to relax from complex manifolds to (conventional) manifolds. But as also proved in [@HjoKec1], in the case of locally compact spaces, a function is defined to be a homeomorphism in a Borel way. Thus, the set of those $\eta$ for which $X(\eta)$ is an $n$-manifold is Borel. Using the functions $f_1$, $g_1$, $f_2$ and $g_2$ we finally obtain that the sets of $n$-manifolds in all the other parametrizations are also Borel.

Denote by $\approx_P$ the homeomorhism relation on all Polish spaces and by $\approx_{loc}$, $\approx_c$ and $\approx_n$ the same relation restricted to the sets of locally compact, compact Polish spaces and $n$-manifolds respectively. From what is shown above it follows that the chosen parametrization is irrelevant. Recall also the notation $\approx^{\{x\}}$ from Definition \[def:Relations\]. All of these equivalence relations are defined on Borel subsets of Polish spaces.

The following easily follows from [@Kechris Exercise (27.9)]:

The set of those $C\in F(U)$ which are ${\sigma}$-compact is ${{\Pi_1^1}}$-complete.

Because, as custom is, we require in Definition \[def:Reduction\] that the domains of equivalence relations are Borel subsets of Polish spaces, we do not talk directly about the homeomorphism relation restricted to the ${\sigma}$-compact spaces. However, by removing that requirement and relaxing from Borel sets to *relatively Borel* one could also talk about the Borel reducibility of $\approx_{\sigma}$, the homeomorphism relation on ${\sigma}$-compact spaces, to other equivalence relations. From our results it would follow in particular that $E_1{\leqslant}_B\ \approx_{\sigma}$ and that $\approx_{\sigma}$ is not classifiably by any equivalence relation induced by a Borel group action.

Yet another way to parametrize compact and locally compact spaces is to view them as subsets of the Hilbert cube $I^{\mathbb{N}}$ where $I$ is the unit interval. It is known that for every compact Polish space there is a homeomorphic copy as a subset of $I^{\mathbb{N}}$. For locally compact spaces we also obtain a parametrization: By [@Kechris Theorem 5.3], the one-point compactification of every locally compact Polish space is a compact Polish space. Now fix a point $x\in I^{\mathbb{N}}$ and for each $\xi\in (I^{\mathbb{N}}\setminus\{x\})^{\mathbb{N}}$ let $Z(\xi)$ be the space $\overline{\{\xi(n)\mid n\in{\mathbb{N}}\}}\setminus \{x\}$. If $P$ is any locally compact space, then let $\bar P=P\cup \{\infty\}$ be its one-point compactification. There is an embedding of $\bar P$ into $I^{\mathbb{N}}$ and since $I^{\mathbb{N}}$ is homogenous (see e.g. [@Fort]), there is an embedding such that $\infty$ is mapped to $x$. Thus, in the notation of \[def:StrangeSpaces\], $K^{\{x\}}(I^{\mathbb{N}})$ is a space parametrizing all locally compact spaces. By using the fact that $I^{\mathbb{N}}$ can be also isometrically embedded into the Urysohn space $U$, one can use the methods from above to conclude that this parametrization is in our sense equivalent to all the other parametrizations (i.e. the homeomorphism relation is Borel bireducible with the corresponding relation in other parametrizations and the relevant subsets such as $n$-manifolds are Borel subsets).

#### Summary.

When proving a classification or a non-classification result for any of $\approx_P$, $\approx^{\{x\}}$, $\approx_{loc}$, $\approx_{c}$, $\approx_n$ it is irrelevant which of the parametrizations is used. Additionally the sets of locally compact and compact spaces, of $n$-manifolds and of the spaces in $K^{\{x\}}_*({\mathbb{S}}^3)$ are Borel no matter which parametrization is used.

Non-classification of $\approx^{\{x\}}$ {#sec:NonClass}
=======================================

This section is devoted to proving the main result:

\[thm:NonClass\] The equivalence relation $E_1$ (Definition \[def:E1\]) is continuously reducible to the homeomorphism relation on $K^{\{x\}}_*({\mathbb{S}}^3)$ for any fixed $x\in{\mathbb{S}}^3$.

As before, we parametrize ${\mathbb{S}}^3$ as ${\mathbb{R}}^3\cup \{\infty\}$. Obviously the choice of $x$ does not matter. In our case $x=(1,1,\frac{1}{2})\in {\mathbb{R}}^3$ as will be seen below.

For every $n\in{\mathbb{N}}$, $k\in{\mathbb{N}}$ and $l\in \{0,1\}$, let $B_{n,k,l}\subset {\mathbb{R}}^3$ be a closed ball with the center at $(1-2^{-n},1-2^{-k},l)$ and radius $2^{-4(n+1)(k+1)}$. Define $Q$, $P'$ and $P$ as follows: $$\begin{aligned}
    Q&=&\{(1-2^{-n},1,l)\mid n\in{\mathbb{N}},l\in\{0,1\}\}
    \cup\{(1,1-2^{-k},l)\mid k\in{\mathbb{N}},l\in\{0,1\}\},\\
    P'&=&Q\cup {\bigcup}_{n\in{\mathbb{N}}}\{(1-2^{-n},1,t)\mid t\in [0,1]\}\\
    P&=&P'\cup \{(1,1,t)\mid t\in [0,1]\}\setminus \{(1,1,\frac{1}{2})\}.
  \end{aligned}$$

Thus, $(B_{n,k,l})$, $Q$ and $P$ satisfy the assumptions B1, B2 and B3 from Definition \[def:Properties\].

Let $\{P_{n,k,l}\mid n\in{\mathbb{N}},k\in{\mathbb{N}},l\in\{0,1\}\}$ be the set of all (mutually different) knot types indexed by the set ${\mathbb{N}}\times{\mathbb{N}}\times \{0,1\}$. Let $\bar r=(r_n)_{n\in{\mathbb{N}}}\in (2^{{\mathbb{N}}})^{{\mathbb{N}}}$ be a sequence of elements of $2^{\mathbb{N}}$. For each $(n,k,l)\in (2^{{\mathbb{N}}})^{{\mathbb{N}}}$, let $K^{\bar r}_{n,k,l}$ be a (piecewise linear) knot inside the interior of $B_{n,k,l}$. The knot-type of $K^{\bar r}_{n,k,l}$ is determined as follows:

-   If $n$ is odd, then it is $P_{n,k,l}$,

-   If $n$ is even and $r_{n/2}(k)=0$, then it is $P_{n,k,l}$,

-   If $n$ is even and $r_{n/2}(k)=1$, then it is $P_{n,k,1-l}$.

Let $R(\bar r)$ be ${\mathbb{S}}^3\setminus (P\cup{\bigcup}_{n,k,l}K^{\bar r}_{n,k,l}).$ Note that $R(\bar r)$ corresponds to $X$ in Definition \[def:Properties\] and properties B4 and B5 are now also satisfied (B5 follows easily from Lemma \[lem:HDPathMetric\] and the fact that ${\mathbb{S}}^3\setminus X$ is a countable union of piecewise linear curves and points). Notice also that $R(\bar r)\setminus \{(1,1,\frac{1}{2})\}$ is an open set, so $R(\bar r)\in K^{\{(1,1,\frac{1}{2})\}}_*({\mathbb{S}}^3)$. In the following three claims we will show that $F$ is a continuous reduction: $\bar r$ and $\bar r'$ are $E_1$-equivalent if and only if $R(\bar r)$ and $R(\bar r')$ are homeomorphic.

Suppose $\bar r$ and $\bar r'$ are $E_1$-equivalent. Then $R(\bar r)$ and $R(\bar r')$ are homeomorphic.

For every $(n,k)\in {\mathbb{N}}\times{\mathbb{N}}$ let $C_{n,k}$ be the convex hull of $B_{n,k,0}\cup B_{n,k,1}$, a “capsule” containing $B_{n,k,0}$ and $B_{n,k,1}$ disjoint from all other balls and from $P$. Denote for simplicity $X=R(\bar r)$ and $X'=R(\bar r')$. Now $C_{n,k}\cap X$ and $C_{n,k}\cap X'$ are homeomorphic because both are complements of two knots of types $P_{n,k,0}$ and $P_{n,k,1}$. If $n$ is odd or $n$ is even and $r_{n/2}(k)=r'_{n/2}(k)$ then identity on $C_{n,k}$ witnesses this. Otherwise there is a homeomorphism $g_{n,k}$ of ${\mathbb{S}}^3$ fixing ${\mathbb{S}}^3\setminus C_{n,k}$ and taking $C_{n,k}\cap X$ to $C_{n,k}\cap X'$. For each $(n,k)$, if $n$ is even and $r_{n/2}(k)\ne r'_{n/2}(k)$, let $h_{n,k}=g_{n,k}$. Otherwise let $h_{n,k}$ be the identity on ${\mathbb{S}}^3$. Let $\pi\colon {\mathbb{N}}\to {\mathbb{N}}\times{\mathbb{N}}$ be a bijection and define a sequence of functions $(t_m)_{m\in{\mathbb{N}}}$ by induction as follows: $$\begin{aligned}
      t_0&=&h_{\pi(0)}\\
      t_{m+1}&=&h_{\pi(m+1)}\circ t_m.
    \end{aligned}$$ We claim that for every $x\in R(\bar r)$ the limit $t(x)=\lim_{m\to\infty}t_m(x)$ exists and defines a homeomorphism $t$ from $R(\bar r)$ to $R(\bar r')$. Let us define a *support* of a homeomorphism $h$ to be the set ${\operatorname{sprt}}h=\{x\in{\operatorname{dom}}h\mid h(x)\ne x\}$. Now obviously for $m\ne
    m'$, the supports of $h_{\pi(m)}$ and $h_{\pi(m')}$ are disjoint, so the existence of the limit follows easily. In fact if $x\in
    C_{n,k}$ for some $n,k\in{\mathbb{N}}$, then $t(x)=h_{(n,k)}(x)$ and $t(x)=x$ otherwise. Same argument leads that $t$ is bijective. Let $(x,y,z)\in X$ and let us show that $t$ is continuous at $(x,y,z)$. If $y\ne 1$ and $x\ne 1$, then $(x,y,z)$ has a neighborhood intersecting only finitely many $C_{n,k}$, so $t$ is determined by a finite composition of continuous functions in this neighborhood. If $y=1$ and $x\notin \{1\}\cup\{1-2^{-n}\mid
    n\in{\mathbb{N}}\}$, then the same holds again and also if vice versa: If $x=1$ and $y\notin \{1\}\cup\{1-2^{-n}\mid n\in{\mathbb{N}}\}$. If $y=1$ and $x\in \{1-2^{-n}\mid n\in{\mathbb{N}}\}$, then $(x,y,z)\in X$ only if $z\notin [0,1]$ (by the definition of $P$) and in this case $(x,y,z)$ has again an open neighborhood intersecting only finitely many $C_{n,k}$. If $x=1$ and $y\in
    \{1\}\cup\{1-2^{-n}\mid n\in{\mathbb{N}}\}$, then every neighborhood intersects infinitely many $C_{n,k}$. Let $n_*$ be such that for all $n>n_*$ we have $r_{n}(k)=r'_{n}(k)$ which exists because $\bar r$ and $\bar r'$ are $E_1$-equivalent and let $U$ be a neighborhood of $(x,y,z)$ of radius $2^{-2n_*}$. Then $U$ intersects only those $C_{n,k}$ for which $n/2>n_*$ and so by the definition of $h_{n,k}$ it is identity on $C_{n,k}$ for all such $n$. Thus, $t_{m}$ is identity in $U$ for all $m$ and so $t$ is continuous. Now we should check that the inverse is also continuous. But with just a little care in the definition of $g_{n,k}$ we can assume that $g_{n,k}=g_{n,k}^{-1}$ and so $t=t^{-1}$. Thus by symmetry, $t^{-1}$ is also continuous.

Suppose $\bar r$ and $\bar r'$ are not $E_1$-equivalent. Then $R(\bar r)$ and $R(\bar r')$ are not homeomorphic.

Denote again $X=R(\bar r)$ and $X'=R(\bar r')$ and assume on contrary that there is a homeomorphism $h\colon X\to X'$. Since $\bar r$ and $\bar r'$ are not $E_1$-equivalent, there is a sequence $(n_i,k_i)_{i\in{\mathbb{N}}}$ such that $(n_i)_{i\in{\mathbb{N}}}$ is increasing and unbounded in ${\mathbb{N}}$ and for all $i$, $r_{n_i}(k_i)\ne r'_{n_i}(k_i)$.

Suppose first that $(k_i)_{i\in{\mathbb{N}}}$ is bounded in ${\mathbb{N}}$. Then there exists a subsequence $(n_{i(j)},k_{i(j)})_{j\in{\mathbb{N}}}$ such that $k_{i(j)}=k_*$ for all $j$ for some fixed $k_*$. By the construction each knot-type appears exactly once in either of the sets $$\{K^{\bar r}_{n,k,l}\mid (n,k,l)\in{\mathbb{N}}\times{\mathbb{N}}\times\{0,1\}\}$$ and $$\{K^{\bar r'}_{n,k,l}\mid (n,k,l)\in{\mathbb{N}}\times{\mathbb{N}}\times\{0,1\}\}.$$ For each $m\in{\mathbb{N}}$ define the point $x_{m}$ as follows: Let $x_{m}$ be the point in $B_{m,k_*,0}\setminus K^{\bar r}_{m,k_*,0}$ given by Lemma \[lemma:FindPoints\]. We know that if $m$ is odd, then $K^{\bar r}_{m,k_*,0}$ has the same knot-type as $K^{\bar r'}_{m,k_*,0}$ and if $m/2=n_{i(j)}$ for some $j$, then $K^{\bar r}_{m,k_*,0}$ has the same knot-type as $K^{\bar r'}_{m,k_*,1}$. Thus there are infinitely many $m$ such that $h(x_m)\in B_{m,k_*,0}$ and infinitely many $m$ such that $h(x_m)\in B_{m,k_*,1}$. Thus, both points $(1,1-2^{-k_*},0)$ and $(1,1-2^{-k_*},1)$ are accumulation points of $(h(x_m))_{m\in{\mathbb{N}}}$. But only $(1,1-2^{-k_*},0)$ is an accumulation point of $x_m$ which is a contradiction with Lemma \[lemma:CauchyComponent\], because both $\{(1,1-2^{-k_*},0)\}$ and $\{(1,1-2^{-k_*},1)\}$ are connected components of both ${\mathbb{S}}^3\setminus X$ and ${\mathbb{S}}^3\setminus X'$.

Suppose now that $(k_i)_{i\in {\mathbb{N}}}$ is unbounded in ${\mathbb{N}}$. Now pick a subsequence $(n_{i(j)},k_{i(j)})_{j\in{\mathbb{N}}}$ such that not only $n_{i(j)}$ is strictly increasing, but also $k_{i(j)}$ is. For all $j$, let $x_{2j}$ be the point in $B_{2n_{i(j)},k_{i(j)},0}$ given by Lemma \[lemma:FindPoints\]. By similar argumentation as above we know that $h(x_{2j})\in B_{2n_{i(j)},k_{i(j)},1}$. Now again for all $j$, define the point $x_{2j+1}$ to be a point in $B_{2j+1,k_{i(j)},0}$ given again by Lemma \[lemma:FindPoints\]. By the construction we know that $h(x_{2j+1})$ is in $B_{2j+1,k_{i(j)},0}$ too. Thus $(x_m)_{m\in{\mathbb{N}}}$ is now a Cauchy sequence converging to $(1,1,0)$ and $(h(x_m))_{m\in{\mathbb{N}}}$ is sequence with two accumulation points $(1,1,0)$ and $(1,1,1)$. The first of these points belongs to the connected component $\{(1,1,t)\mid 0{\leqslant}t<\frac{1}{2}\}$ of both ${\mathbb{S}}^3\setminus X$ and ${\mathbb{S}}^3\setminus X'$ (by the definition of $P$) and the second belongs to the other connected component $\{(1,1,t)\mid \frac{1}{2}<t{\leqslant}1\}$. Thus, we obtain a contradiction with Lemma \[lemma:CauchyComponent\] again.

$F$ is continuous.

The inverse image of an ${\varepsilon}$-neighborhood of $R(\bar r)$ consists of all $\bar r$ which are mapped inside the ${\varepsilon}$-collar of $R(\bar r)$ and in whose ${\varepsilon}$-collar $R(\bar r)$ is contained. It is evident that only finitely many of the knot-types are determined by the ${\varepsilon}$-collar, since the ${\varepsilon}$-collar of $Q$ (or $P$) “swallows” all but finitely many knots.

By Theorem \[thm:KecLou\] we have:

The homeomorphism relation $\approx^{\{x\}}$, is not Borel reducible to any orbit equivalence relation induced by a Borel action of a Polish group.

There is a collection $D$ of Polish spaces homeomorphic to subsets of ${\mathbb{S}}^3$ of the form $V\cup \{x\}$ where $V$ is open and $x\in {\mathbb{R}}^3$ such that all elements of $D$ have the same fundamental group, but are not classifiable up to homeomorphism by any equivalence relation arising from a Borel action of a Polish group.

The fundamental group of $R(\bar r)$ is the same for all $\bar r$ – the free product of the knot groups – as can be witnessed by the Seifert-van Kampen theorem by considering $R(\bar r)$ as the union of its open subsets $A_{n}={\mathbb{S}}^3\setminus (P\cup K_n\cup {\bigcup}_{k\ne n}B_k)$ (here we fall back to the easier enumeration of the balls by just one index used in Definition \[def:Properties\]).

Positive Classification Results {#sec:Other}
===============================

The results in this section are either known or follow easily from what is known. We give some of the proofs for the sake of completeness. Since the main result of the paper deals with $\sigma$-compact spaces, we begin this section with the following relatively simple observation:

The homeomorphism relation on any Borel collection of ${\sigma}$-compact spaces, such as $K^{\{x\}}_{*}$ is ${{\Sigma_1^1}}$.

Two ${\sigma}$-compact spaces $X$ and $X'$ are homeomorphic if and only if there exist sequences of compact sets $(C_n)_{n\in{\mathbb{N}}}$ and $(C_n')_{n\in{\mathbb{N}}}$ and homeomorphisms $h_n\colon C_n\to C_n'$ for each $n$ such that

1.  $C_n\subset C_{n+1}$ and $C'_n\subset C'_{n+1}$ for all $n$,

2.  $h_n\subset h_{n+1}$ for all $n$,

3.  $X={\bigcup}_{n\in{\mathbb{N}}}C_n$ and $X'={\bigcup}_{n\in{\mathbb{N}}}C'_n$.

To see this suppose $h\colon X\to X'$ is a homeomorphism. Since $X$ is ${\sigma}$-compact, there is a sequence of compact sets $(C_n)_{n\in{\mathbb{N}}}$ which satisfies the first parts of (1) and (3). Let $C'_n=h[C_n]$. Then $(C_n')_{n\in{\mathbb{N}}}$ and $(h_n)_{n\in{\mathbb{N}}}$ where $h_n=h{\!\restriction\!}C_n$ satisfy all the rest. On the other hand suppose that such sequences $(C_n)_{n\in{\mathbb{N}}}$, $(C_n')_{n\in{\mathbb{N}}}$ and $(h_n)_{n\in{\mathbb{N}}}$ exist. Then obviously ${\bigcup}_{n\in{\mathbb{N}}}h_n$ is a homeomorphism $X\to X'$.

Consider the space $F(U)$ defined in Section \[ssec:Par\] parametrizing all Polish spaces. Then, according to the above, the homeomorphism relation restricted to ${\sigma}$-compact spaces can be defined by saying that $C$ and $C'$ are equivalent if there exist sequences satisfying (1), (2) and (3). But these properties are all Borel properties. Also for $h_n$ to be a homeomorphism is Borel, because the domains $C_n$ are compact. This shows that the equivalence relation is ${{\Sigma_1^1}}$.

Now we turn to compact and locally compact spaces.

Let $I^{{\mathbb{N}}}$ be the Hilbert cube. A set of *infinite deficiency* $A\subset I^{{\mathbb{N}}}$ is a closed set whose projection onto infinitely many interval-coordinates is a singleton. A closed set $A$ is a *$Z$-set*, if for every open $U\subset I^{\mathbb{N}}$ with all homotopy groups vanishing, all the homotopy groups of $U\setminus A$ vanish too.

Anderson proved in [@Anderson] the following:

([@Anderson])\[Anderson\]

1.  If a set is of an infinite deficiency, then it is a $Z$-set.

2.  Each homeomorphism between two closed $Z$-subsets of $I^{\mathbb{N}}$ can be extended to a homeomorphism of $I^{\mathbb{N}}$ onto itself.

From this it is not difficult to obtain the following theorem:

\[thm:KechSol\] The homeomorphism relation on compact Polish spaces is continuously reducible to an orbit equivalence relation induced by a Polish group action.

Let $h_1\colon I^{\mathbb{N}}\to I^{\mathbb{N}}$ be an embedding defined as follows: $$h_1((x_i)_{i\in{\mathbb{N}}})=(y_i)_{i\in{\mathbb{N}}}$$ where for all $n$, $y_{2n}=x_n$ and $y_{2n+1}=0$. Let $X={\operatorname{ran}}(h_1)$. Then $X$ is homeomorphic to $I^{\mathbb{N}}$. Let $\approx^*_{c}$ be the equivalence relation on $K(I^{\mathbb{N}})$ where two compact sets $C$ and $C'$ are equivalent if there exists a homeomorphism $h\colon I^{\mathbb{N}}\to I^{\mathbb{N}}$ taking $C$ onto $C'$. By Fact \[fact:HomPolish\] this equivalence relation is induced by a Polish group action, and it is standard to verify that this action is continuous.

Let $\approx_{c}$ be the homeomorphism relation on $K(I^{\mathbb{N}})$ where $C$ and $C'$ are equivalent if they are homeomorphic. Thus, it is sufficient to find a reduction of this into $\approx^*_{c}$. For each $C\in K(I^{\mathbb{N}})$ let $F(C)=h_1[C]$. Now, if $C\approx_{c} C'$, then there is a homeomorphism between $F(C)$ and $F(C')$. But these are of infinite deficiency, so by Theorem \[Anderson\] there is a homeomorphism $h\colon I^{\mathbb{N}}\to I^{\mathbb{N}}$ taking $F(C)$ onto $F(C')$ and so $F(C)\approx^*_{c} F(C')$. If $C$ and $C'$ are not homeomorphic, then so are not $F(C)$ and $F(C')$, so no such homeomorphism can exist.

Arguments along the same lines give us a stronger result:

\[thm:locCom\] The homeomorphism relation on all locally compact Polish spaces is Borel reducible to an orbit equivalence relation induced by a continuous Polish group action.

Let ${{\operatorname{Hom}}}^{\{x\}}(I^{\mathbb{N}})$ be the subgroup of ${{\operatorname{Hom}}}(I^{\mathbb{N}})$ which consists of those homeomorphisms $h$ such that $h(x)=x$. As a closed subgroup of ${{\operatorname{Hom}}}(I^{\mathbb{N}})$ it is also Polish and acts continuously on $K^{\{x\}}(I^{\mathbb{N}})$ (see Definition \[def:StrangeSpaces\]).

As shown in Section \[ssec:Par\], the space of locally compact spaces can be parametrized as $K^{\{x\}}(I^{\mathbb{N}})$ each locally compact space being homeomorphic to $C\setminus \{x\}$ for some $C\in K^{\{x\}}(I^{\mathbb{N}})$. Applying the homeomorphism $h_1$ from the proof of Theorem \[thm:KechSol\], we may as well assume that this $C$ is of infinite deficiency. Now, if the two spaces $C\setminus \{x\}$ and $C'\setminus \{x\}$ are homeomorphic, the homeomorphism extends to their one-point compactifications and thus to $x$. Further, since $C$ and $C'$ are $Z$-sets, the homeomorphism extends to an element of ${{\operatorname{Hom}}}^{\{x\}}(I^{\mathbb{N}})$. On the other hand, it is obvious that if $C\setminus \{x\}$ and $C'\setminus \{x\}$ are not homeomorphic, then no such element of ${{\operatorname{Hom}}}^{\{x\}}(I^{\mathbb{N}})$ can exist.

By combining these results with Theorem \[thm:NonClass\] we can conclude that “not locally compact at one point” is in a sense the strongest requirement for Polish spaces to be non-classifiable by such an orbit equivalence relation. This is also reflected in the following Corollary:

\[cor:locsig\] Then $\approx^{\{x\}}\ \not{\leqslant}_B\ \approx_{loc}$.

We would like to apply Theorem \[thm:locCom\] to the homeomorphism on $n$-manifolds. It was discussed in Section \[ssec:Par\] that the set $M_n$ of $n$-manifolds is Borel as a subset of the space of all Polish spaces (in any of the parametrizations). As before, denote the homeomorphism relation on $M_n$ by $\approx_n$. Since manifolds are locally compact the inclusion into the locally compact spaces is a reduction $\approx_n\ {\leqslant}_B\ \approx_{loc}$. By applying Theorem \[thm:locCom\] we get the following:

$\approx_n$ is reducible to an orbit equivalence relation induced by a Polish group action.

More is known in the case $n=2$. There is a classification of $\approx_2$ by algebraic structures using cohomology groups by Goldman [@Goldman]. It is probably routine to verify that this gives a Borel reduction into the isomorphism on countable structures, but for now I leave it open in the form of a conjecture:

\[con:firstCon\] The classification in [@Goldman] is a Borel reduction into $\cong_G$, thus $\approx_2\ {\leqslant}_B\ \cong_G$.

If the conjecture holds, we obtain a consequence which follows from Theorem \[thm:graphstomanifolds\] below:

\[conj:2to3\] $\approx_2\ {\leqslant}_B\ \approx_3$.

The converse to Conjecture \[con:firstCon\] is known to hold: $\cong_G\ {\leqslant}_B\ \approx_2$. In fact $\cong_G\ {\leqslant}_B \ \approx_n$ for all $n{\geqslant}2$. We sketch two proofs of this fact – one is based on results by Camerlo and Gao and extension theorems from topology – the other one, for $n=3$, is based on the methods used in this paper, just to illustrate how these methods can be used.

\[thm:graphstomanifolds\] For all $n{\geqslant}2$ we have $\cong_G\ {\leqslant}_B\ \approx_n$.

I would like to thank Clinton Conley who came up with this proof at `mathoverflow.net`. It is sufficient to find a reduction into the homeomorphism relation on open subsets of ${\mathbb{S}}^n$ or ${\mathbb{R}}^n$. As shown in [@CamGao], $\cong_G$ is Borel reducible to the homeomorphism relation on $K(2^{\mathbb{N}})$. On one hand it is known that every homeomorphism of a totally disconnected compact subset of the plane extends to the whole plane ([@Moise Ch. 13, Thm 7]). On the other hand, by an application of Lemma \[lemma:CauchyComponent\], every homeomorphism of ${\mathbb{R}}^2\setminus C$ where $C$ is compact and totally disconnected, induces a homeomorphism of $C$. Thus, we can define a reduction from the homeomorphism relation on $K(2^{\mathbb{N}})$ to $\approx_2$: let $f\colon 2^{\mathbb{N}}\to {\mathbb{R}}^2$ be the standard embedding (the Cantor set) and with $C\subset 2^{\mathbb{N}}$ associate the open set ${\mathbb{R}}^2\setminus f[C]$. Of course these homeomorphisms extend to ${\mathbb{R}}^n$ for every $n>2$ as well, so in fact we have $\cong_G\ {\leqslant}_B\ \approx_n$ for all $n$.

Again, we consider only open subsets of ${\mathbb{R}}^3$. It was proved by H. Friedman and L. Stanley in [@FrSt] that $\cong_G$ is reducible to the isomorphism relation on countable linear orders.

Given a countable linear order $L$ with domain $\{x_n\mid n\in{\mathbb{N}}\}$, construct first a set of disjoint open intervals $U_n\subset [0,1]$ such that ${\bigcup}_{n\in{\mathbb{N}}}U_n$ is open and dense in $[0,1]$, $\sup U_n{\leqslant}\inf U_m$ if and only if $x_n<_Lx_m$ and if $x_m$ is an immediate successor of $x_n$ then $\sup U_n=\inf U_m$. Then, by considering $[0,1]$ as a subset of ${\mathbb{R}}^3$ in a canonical way, replace each open interval with a copy of the chain depicted on Figure \[fig:VadimsSuperLink\]. Let $C(L)$ be the closure of the union of all these chains in ${\mathbb{R}}^3$ By using methods similar to those above, one can show that two linear orders $L$ and $L'$ are isomorphic if and only if the complements of $C(L)$ and $C(L')$ are homeomorphic.

![The singular link.[]{data-label="fig:VadimsSuperLink"}](Singular_link_trefoil.mps){width="70.00000%"}

The idea is that the knot-types fix the orientation within the chain, and the set $Q$ – in this case, the set $[0,1]\setminus {\bigcup}_{n\in{\mathbb{N}}}U_n$ – is totally disconnected and the homeomorphism of the complement extends to it. Moreover it extends to it in an order preserving way and also preserves end-points of the chains. On the other hand all these chains are similar to one another, so any isomorphism of $L$ can be realized as a homeomorphism of the complement of $C(L)$.

Further Research {#sec:Further}
================

Let $O_n({\mathbb{S}}^n)$ be the space of all open subsets of ${\mathbb{S}}^n$ and let $\approx^o_n$ be the homeomorphism relation on this space. As before, $\approx_n$ is the homeomorphism relation on general non-compact $n$-manifolds without boundary. As before, let $\approx_{P}$ be the homeomorphism relation on all Polish, spaces, $\approx_{loc}$ the one on locally compact, $\approx_{c}$ the one on compact Polish spaces and $\approx^{\{x\}}$ as in Definition \[def:Relations\]. An open-ended research direction is to establish the places of these and other topological equivalence relations in the hierarchy of analytic equivalence relations. Positive and negative, new and old results have been reviewed in this paper.

We already stated the main open question:

Is $\approx_3\ {\leqslant}_B\ \cong_G$? If not, is it universal among orbit equivalence relations induced by a Polish group action?

And further one can ask:

\[q:whichnm\] For which $n$ and $m$ do we have $\approx_n\ {\leqslant}_B\ \approx_m$? The same for $\approx_n^o$ and $\approx_{m}^o$.

For which $n\in{\mathbb{N}}$ and known equivalence relations $E$ do we have or $E{\leqslant}_B\ \approx_n$ and same for $\approx^o_n$?

What about open subsets of the separable Hilbert space $\ell_2$? By the results of Henderson [@Hend] this covers all reasonable concepts of infinite-dimensional manifolds.

\[q:S12\] What is the exact complexity of $\approx_P$? It is known that it is $\Sigma^1_2$ [@Gao] and ${{\Sigma_1^1}}$-hard [@FerLouRos].

What are the precise locations of $\approx_{loc}$ and $\approx_{c}$ and in particular are they bireducible? Hjorth has shown using turbulence theory [@Hjorth] that $\cong_G\ <_B\ \approx_c$ (notice strict inequality) and by the results above, $\approx_c$ as well as $\approx_{loc}$ are below the universal equivalence relation induced by the Polish group action.

The following question has been asked already in [@FaToTo2]:

Is $\approx_c$ universal among all equivalence relations that are reducible to an orbit equivalence relation induced by a Polish group action?

And of course the conjectures from the end of the previous section:

\[conj:2to3dubl\] The classification in [@Goldman] is a Borel reduction into $\cong_G$, thus $\approx_2\ {\leqslant}_B\ \cong_G$. In particular $\approx_2\ {\leqslant}_B\ \approx_3$.

Concerning Question \[q:whichnm\] and Conjectures \[conj:2to3\] and \[conj:2to3dubl\]: at first it might seem that it holds that $\approx_n\ {\leqslant}_B\ \approx_{n+1}$. However, the obvious candidate for a reduction $M\mapsto M\times{\mathbb{R}}$ does not work: as shown in [@McMillan] there are open subsets $O$ of ${\mathbb{R}}^3$ which are not homeomorphic with ${\mathbb{R}}^3$, yet $O\times {\mathbb{R}}\approx {\mathbb{R}}^4$. There are no such manifolds in dimension $2$ [@Daverman], but it is still unclear to the author whether the general map $M\mapsto M\times{\mathbb{R}}$ from $2$- to $3$-manifolds provides a reduction between the homeomorphism relations.

Conclusion {#sec:Conclusion}
==========

As a conclusion we provide a diagram of all the relevant equivalence relations and which relations are knownbetween them. We omit some obvious arrows that follow e.g. from transitivity. In the diagram we use the following notation:

$$\begin{aligned}
  \xymatrix{E\ar[r] & \ E'} & & E {\leqslant}_B E',\\
  \xymatrix{E\ar@{.>}[r]|? & E'} & & \text{Not known whether or not } E {\leqslant}_B E',\\
  \xymatrix{E\ar@{.>}[r]|C & E'} & & \text{Conjectured in this paper that } E {\leqslant}_B E',\\
  \xymatrix{E\ar@{-|}[r] &\ E'} & &  E \not{\leqslant}_B E',\\
  E_1 && \text{Definition \ref{def:E1},}\\
  E_0 && \text{Two sequences $\eta,\xi\in {\mathbb{N}}^{\mathbb{N}}$ are equivalent if $\exists n\forall (m>n)(\eta(m)=\xi(m))$,}\\
  E_{Gr} && \text{The universal equivalence relation induced by a Borel Polish group action,}\\
  E_{{{\Sigma_1^1}}} && \text{The universal ${{\Sigma_1^1}}$ equivalence relation,}\\
  \approx^{\{x\}}&& \text{See Definition~\ref{def:Relations},}\\
  \approx_{loc}&& \text{Homeomorphism on locally compact Polish spaces,}\\
  \approx_c&& \text{Homeomorphism on compact Polish spaces,}\\
  \approx_P&& \text{Homeomorphism on all Polish spaces,}\\
  \approx_n &&  \text{Homeomorphism on $n$-manifolds.}\\
  \cong_G &&  \text{Isomorphism on countable graphs.}\end{aligned}$$

$$\xymatrix{ 
    & \approx_P \ar@<1ex>@{.>}[d]|?& \\  
    & E_{{{\Sigma_1^1}}}\ar[u] \ar@{.>}[dl]|? &\\
    \approx^{\{x\}}\ar@<1ex>[ur] &\approx_{loc}\ar@{|->}[u]\ar@<1ex>@{.>}[d]|? \ar@{|->}[l]\ar@<1ex>[r]& E_{Gr}\ar@{|->}[ul]\ar@{.>}[l]|? \\
    & \approx_c \ar[u] & \approx_n \ar@<1ex>@{.>}[dl]|?\ar[u]\ar[ul]\ar@<1ex>@{.>}[d]|?&\approx_m\ar@{<.>}[l]|?\\
    E_1 \ar[uu]& \cong_G\ar@{|->}[u]\ar[ur]\ar@<-1ex>[r]& \approx_2\ar@{.>}[l]|C\ar@<.5ex>@{.>}[u]|C \ar@/^/[uul]\\
    & E_0  \ar@{|->}[ul]\ar@{|->}[u]\ar@{|->}[ur]&
  }$$

[^1]: Affiliation: Kurt Gödel Research Center, University of Vienna, vadim.kulikov@iki.fi, phone: 00436804461218
