Borel structurability by locally finite simplicial complexes

TL;DR

This paper demonstrates that countable Borel equivalence relations can be embedded into ones structurable by finite-dimensional contractible simplicial complexes with controlled vertex degrees, generalizing previous results for the case n=1.

Contribution

It extends Jackson-Kechris-Louveau's result to higher dimensions, showing embeddings with bounded vertex degrees for structurable equivalence relations.

Findings

Every countable Borel equivalence relation structurable by n-dimensional complexes embeds into one with bounded vertex degree.

The bound on vertex degree is explicitly given by a formula involving n.

The proof leverages classical Whitehead results on countable CW-complexes.

Abstract

We show that every countable Borel equivalence relation structurable by $n$-dimensional contractible simplicial complexes embeds into one which is structurable by such complexes with the further property that each vertex belongs to at most $M_n := 2^{n-1}(n^2+3n+2)-2$ edges; this generalizes a result of Jackson-Kechris-Louveau in the case $n = 1$. The proof is based on that of a classical result of Whitehead on countable CW-complexes.

Equations

$$\{(x_0,y_0)\le \cdots \le (x_n,y_n)\}\in S\times T⟺\{x_0\le \cdots \le x_n\}\in S\wedge \{y_0\le \cdots \le y_n\}\in T.$$

$$f(S):=\{f(s)\mid s\in S\},$$

$${\mathcal{T}}_0(T)$$

$${\mathcal{T}}_n(T)$$

$${\mathcal{T}}_n(T)=(p_1(T)\times N)\cup (p_2(T)\times N\times \{0\})\cup \cdots \cup (p_n(T)\times N\times \{0{\}}^{n\text{mathchar}451})\cup (p_n(T)\times \{0{\}}^n)$$

$$T_n\subseteq S_n\times N^n,$$

$$T_2\times N$$

$$T_{n+1}^{\prime }:=(T_n\times N)\cup {\bigcup }_{s\in S^{(n+1)}}({\mathcal{T}}_n(T_n\text{mathchar}124(s\times {\mathbb{N}}^n))\times \{c_{n+1}(s)\}).$$

$$T_{n+1}^{\prime }\text{mathchar}124(t\times {\mathbb{N}}^{n+1})$$

$$=(\underset{A}{\underbrace{T_n\text{mathchar}124p_n^{\text{mathchar}451}(t)\times N}})\cup {\bigcup }_{s\in S^{(n+1)}}(\underset{B_s}{\underbrace{{\mathcal{T}}_n(T_n\text{mathchar}124p_n^{\text{mathchar}451}(s\cap t))\times \{c_{n+1}(s)\}}})$$

$$A\cap B_s$$

$$=(T_n\text{mathchar}124((s\cap t)\times {\mathbb{N}}^n)\cap {\mathcal{T}}_n(T_n\text{mathchar}124((s\cap t)\times {\mathbb{N}}^n)))\times \{c_{n+1}(s)\}$$

$$=T_n\text{mathchar}124((s\cap t)\times {\mathbb{N}}^n)\times \{c_{n+1}(s)\}$$

$$=T_n\text{mathchar}124p_n^{\text{mathchar}451}(s\cap t)\times \{c_{n+1}(s)\}$$

$$T_{n+1}:=T_{n+1}^{\prime }\cup \{s\times \{0{\}}^n\times \{c_{n+1}(s)\}\mid s\in S^{(n+1)}\}.$$

$$T_{n+1}\text{mathchar}124p_{n+1}^{\text{mathchar}451}(s)$$

$$T_{n+1}\text{mathchar}124p_{n+1}^{\text{mathchar}451}(s)=T_{n+1}^{\prime }\text{mathchar}124p_{n+1}^{\text{mathchar}451}(s)\cup (S\text{mathchar}124s\times \{0{\}}^n\times \{c_{n+1}(s)\}),$$

$$K_1:=3$$

$$K_{n+1}:=2K_n+(n+2)2^n$$

$$K_n=2^{n\text{mathchar}452}(n^2+3n+2).$$

$$M_n:=2(K_n\text{mathchar}451)=2^{n\text{mathchar}451}(n^2+3n+2)\text{mathchar}452$$

$$T_n^{\ast }:={\bigcup }_{k\in \mathbb{N}}{T}_{n,k}$$

$$i_n:T_n\cong T_n\times \{0\}\subseteq T_n\times N\subseteq T_{n+1}^{\prime }\subseteq T_{n+1}.$$

$$T_0$$

$$i^n:X\times {\mathbb{N}}^n\cong X\times {\mathbb{N}}^n\times \{0{\}}^{\infty }\subseteq X\times {\mathbb{N}}^{\infty };$$

$$Y$$

$$T$$

$${\tilde{T}}_n:={\bigcup }_{m\le n}(i^m(T_m)\times N\text{mathchar}124\{m,m+1\}).$$

$${\tilde{T}}_0$$

$$(x,i_1,\dots ,i_n)F(x^{\prime },i_1^{\prime },\dots ,i_n^{\prime })⟺xE{x}^{\prime },$$

Full text

1 Introduction

A countable Borel equivalence relation $E$ on a standard Borel space $X$ is a Borel equivalence relation $E\subseteq X^{2}$ for which each equivalence class is countable. The class of treeable countable Borel equivalence relations, for which there is a Borel way to put a tree (acyclic connected graph) on each equivalence class, has been studied extensively by many authors, especially in relation to ergodic theory; see e.g., [Ada], [Ga1], [JKL], [KM], [HK], [Hjo]. It is a basic result, due to Jackson-Kechris-Louveau [JKL, 3.10], that every treeable equivalence relation embeds into one treeable by trees in which each vertex has degree at most 3. The purpose of this paper is to present a generalization of this result to higher dimensions.

Recall that a simplicial complex on a set $X$ is a collection $S$ of finite nonempty subsets of $X$ which contains all singletons and is closed under nonempty subsets. A simplicial complex $S$ has a geometric realization $\lvert S\rvert$, which is a topological space formed by gluing together Euclidean simplices according to $S$ (see Section 2 for the precise definition); $S$ is contractible if $\lvert S\rvert$ is. Given a distinguished class $\mathcal{K}$ of simplicial complexes (e.g., the contractible ones) and a countable Borel equivalence relation $(X,E)$, a (Borel) structuring of $E$ by simplicial complexes in $\mathcal{K}$ is, informally (see Section 2), a Borel assignment of a simplicial complex $S_{C}\in\mathcal{K}$ on each equivalence class $C\in X/E$. If such a structuring exists, we say that $E$ is structurable by complexes in $\mathcal{K}$. We are interested here mainly in $\mathcal{K}=$ $n$-dimensional contractible simplicial complexes; when $n=1$, we recover the notion of treeability. The study of equivalence relations structurable by $n$-dimensional contractible simplicial complexes was initiated by Gaboriau [Ga2], who proved (among other things) that for $n=1,2,3,\dotsc$ these classes of countable Borel equivalence relations form a strictly increasing hierarchy under $\subseteq$.

Recall also the notion of a Borel embedding $f:E\to F$ between countable Borel equivalence relations $(X,E)$ and $(Y,F)$, which is an injective Borel map $f:X\to Y$ such that $x\mathrel{E}x^{\prime}\iff f(x)\mathrel{F}f(x^{\prime})$ for all $x,x^{\prime}\in X$.

Theorem 1**.**

Let $n\geq 1$, and let $(X,E)$ be a countable Borel equivalence relation structurable by $n$-dimensional contractible simplicial complexes. Then $E$ Borel embeds into a countable Borel equivalence relation $(Y,F)$ structurable by $n$-dimensional contractible simplicial complexes in which each vertex belongs to at most (or even exactly) $M_{n}:=2^{n\mathchar 45\relax 1}(n^{2}+3n+2)\mathchar 45\relax 2$ edges.

In particular, every $E$ structurable by $n$-dimensional contractible simplicial complexes Borel embeds into an $F$ structurable by locally finite such complexes, where a simplicial complex is locally finite if each vertex is contained in finitely many edges (or equivalently finitely many simplices). The constant $M_{n}$ above is not optimal: for $n=1$ we have $M_{1}=4$, whereas by the aforementioned result of Jackson-Kechris-Louveau we may take $M_{1}=3$ instead, which is optimal; for $n=2$ we have $M_{2}=22$, whereas by a construction different from the one below we are able to get $M_{2}=10$. We do not know what the optimal $M_{n}$ is for $n\mathchar 62\relax 1$; however, the result of Gaboriau mentioned above implies that the optimal $M_{n}$ is at least $n+1$.

The referee has pointed out that by an easy argument, one may strengthen “at most” to “exactly” in Theorem 1 (as well as in the following reformulations).

We may reformulate Theorem 1 in terms of compressible countable Borel equivalence relations, which are those admitting no invariant probability Borel measure (see e.g., [DJK] for various equivalent definitions of compressibility):

Corollary 2**.**

Let $n\geq 1$, and let $(X,E)$ be a compressible countable Borel equivalence relation structurable by $n$-dimensional contractible simplicial complexes. Then $E$ is structurable by $n$-dimensional contractible simplicial complexes in which each vertex belongs to at most (or even exactly) $M_{n}$ edges.

Note that by the theory of cost (see [Ga1], [KM]), Corollary 2 cannot be true of non-compressible equivalence relations, i.e., there cannot be a uniform bound $M_{n}$ on the number of edges containing each vertex.

Theorem 1 fits into a general framework for classifying countable Borel equivalence relations according to the (first-order) structures one may assign in a Borel way to each equivalence class; see [JKL], [Mks], [CK]. As with most such results, the “underlying” result is that there is a procedure for turning every structure of the kind we are starting with ($n$-dimensional contractible simplicial complexes) into a structure of the kind we want ($n$-dimensional contractible simplicial complexes satisfying the additional condition), which is “uniform” enough that it may be performed simultaneously on all equivalence classes in a Borel way. We state this as follows. We say that a simplicial complex is locally countable if each vertex is contained in countably many edges (or equivalently countably many simplices).

Theorem 3**.**

There is a procedure for turning a locally countable simplicial complex $(X,S)$ into a locally finite simplicial complex $(Y,T)$, such that

• (i)

$T$* is homotopy equivalent to $S$;*

• (ii)

if $S$ is $n$-dimensional, then $T$ can be chosen to be $n$-dimensional and with each vertex in at most (or even exactly) $M_{n}$ edges.

Furthermore, given a countable Borel equivalence relation $(X,E)$ and a structuring $S$ of $E$ by simplicial complexes, this procedure may be performed simultaneously (in a Borel way) on all $E$-classes, yielding a countable Borel equivalence relation $(Y,F)$ with a structuring $T$ by simplicial complexes and a Borel embedding $f:E\to F$ such that applying the above procedure to the complex $S_{[x]_{E}}$ on an $E$-class $[x]_{E}$ yields the complex $T_{[f(x)]_{F}}$ on the corresponding $F$-class $[f(x)]_{F}$.

The theorem in this form also yields the following (easy) corollary:

Corollary 4**.**

Every countable Borel equivalence relation $(X,E)$ embeds into a countable Borel equivalence relation $(Y,F)$ structurable by locally finite contractible simplicial complexes.

Again, this may be reformulated as

Corollary 5**.**

Every compressible countable Borel equivalence relation $(X,E)$ is structurable by locally finite contractible simplicial complexes.

The proof of Theorem 3 is based on a classical theorem of Whitehead on CW-complexes [Wh, Theorem 13], which states that every locally countable CW-complex is homotopy equivalent to a locally finite CW-complex of the same dimension. While the statement of this theorem is useless for Theorem 3 (every contractible complex is homotopy equivalent to a point, but one cannot replace every class of a non-smooth equivalence relation with a point), its proof may be adapted to our setting, with the help of some lemmas from descriptive set theory.

We review some definitions and standard lemmas in Section 2, then give the proofs of the above results in Section 3; the proofs are structured so that it should be possible to read the combinatorial/homotopy-theoretic argument without the descriptive set theory, and vice-versa. In Section 4 we list some other properties of treeable equivalence relations which we do not currently know how to generalize to higher dimensions.

Acknowledgments. We would like to thank Alexander Kechris, Damien Gaboriau, and the anonymous referee for providing some comments on drafts of this paper.

2 Preliminaries

We begin by reviewing some notions related to simplicial complexes; see e.g., [Spa].

A simplicial complex on a set $X$ is a set $S$ of finite nonempty subsets of $X$ such that $\{x\}\in S$ for all $x\in X$ and every nonempty subset of an element of $S$ is in $S$. The elements $s\in S$ are called simplices. The dimension $\dim(s)$ of $s\in S$ is $\mathchar 124\relax s\mathchar 124\relax\mathchar 45\relax 1$; if $\dim(s)=n$, we call $s$ an $n$-simplex. We let $S^{(n)}:=\{s\in S\mid\dim(s)=n\}$ be the $n$-simplices, and call $S$ $n$-dimensional if $S^{(m)}=\varnothing$ for $m\mathchar 62\relax n$. (To avoid confusion, we will sometimes call a simplicial complex with an $n$-simplex containing all other simplices a standard $n$-simplex.)

A subcomplex of $(X,S)$ is a simplicial complex $(Y,T)$ such that $Y\subseteq X$ and $T\subseteq S$. For a simplicial complex $(X,S)$ and a subset $Y\subseteq X$, the induced subcomplex on $Y$ is $S\mathchar 124\relax Y:=\{s\in S\mid s\subseteq Y\}$. A simplicial map $f:S\to T$ between complexes $(X,S)$ and $(Y,T)$ is a map $f:X\to Y$ such that $f(s)\in T$ for all $s\in S$.

The geometric realization of a simplicial complex $(X,S)$ is the topological space $\mathchar 124\relax S\mathchar 124\relax$ formed by gluing together standard Euclidean $n$-simplices $\Delta^{n}$ for each $s\in S^{(n)}$, according to the subset relation. Explicitly, $\mathchar 124\relax S\mathchar 124\relax$ can be defined as the set $\bigcup_{s\in S}\mathchar 124\relax s\mathchar 124\relax_{S}\subseteq[0,1]^{X}$, where $\mathchar 124\relax s\mathchar 124\relax_{S}:=\{(a_{x})_{x\in X}\mid\sum_{x\in X}a_{x}=1,\,\forall x\not\in s\,(a_{x}=0)\}$ is (thought of as) the set of formal convex combinations of elements of $X$ supported on $s$, equipped with the topology where a subset of $\mathchar 124\relax S\mathchar 124\relax$ is open iff its intersection with each $\mathchar 124\relax s\mathchar 124\relax_{S}$ is open in the Euclidean topology on $\mathchar 124\relax s\mathchar 124\relax_{S}$. We say that $S$ is contractible if $\mathchar 124\relax S\mathchar 124\relax$ is. Likewise, a simplicial map $f:S\to T$ induces a continuous map $\mathchar 124\relax f\mathchar 124\relax:\mathchar 124\relax S\mathchar 124\relax\to\mathchar 124\relax T\mathchar 124\relax$ in the obvious way; we say that $f$ is a homotopy equivalence if $\mathchar 124\relax f\mathchar 124\relax$ is.

We also need the more refined notion of an ordered simplicial complex, which is a simplicial complex $S$ on a poset $X$ such that every simplex $s\in S$ is a chain $\{x_{0}\mathchar 60\relax\dotsb\mathchar 60\relax x_{n}\}$ in $X$. The product of ordered simplicial complexes $(X,S)$ and $(Y,T)$ is the complex $(X\times Y,S\times T)$ where $X\times Y$ is the usual product poset and

[TABLE]

It is standard that $\mathchar 124\relax S\times T\mathchar 124\relax$ is canonically homeomorphic to $\mathchar 124\relax S\mathchar 124\relax\times\mathchar 124\relax T\mathchar 124\relax$ with the CW-product topology (which coincides with the product topology if $S,T$ are locally countable).

In order to prove contractibility/homotopy equivalence, we use the following standard results from homotopy theory.

Lemma 6**.**

Let $S,T$ be simplicial complexes which are the unions of subcomplexes $S=\bigcup_{i\in I}S_{i}$ and $T=\bigcup_{i\in I}T_{i}$ over the same index set $I$, and let $f:S\to T$ be a simplicial map such that $f(S_{i})\subseteq T_{i}$ for each $i$. If for each finite family of indices $i_{1},\dotsc,i_{n}\in I$, the restriction $f:S_{i_{1}}\cap\dotsb\cap S_{i_{n}}\to T_{i_{1}}\cap\dotsb\cap T_{i_{n}}$ is a homotopy equivalence, then $f:S\to T$ is a homotopy equivalence.

Proof.

See e.g., [Hat, 4K.2]. ∎

Corollary 7**.**

Let $S$ be a simplicial complex which is the union of subcomplexes $U,V\subseteq S$. If the inclusion $U\cap V\to U$ is a homotopy equivalence, then so is the inclusion $V\to S$. In particular, if $U$, $V$, and $U\cap V$ are contractible, then so is $S$.

Proof.

Apply Lemma 6 to the inclusion from $V=(U\cap V)\cup V$ into $S=U\cup V$. ∎

Corollary 8**.**

Let $S=\bigcup_{i\in I}S_{i}$ and $T=\bigcup_{i\in I}T_{i}$ be simplicial complexes which are directed unions of subcomplexes (over the same directed poset), and let $f:S\to T$ be a simplicial map such that $f(S_{i})\subseteq T_{i}$ for each $i$. If each restriction $f\mathchar 124\relax S_{i}:S_{i}\to T_{i}$ is a homotopy equivalence, then so is $f$.

In particular, if $S_{i}$ is contractible for each $i$, then (taking $T=T_{i}=$ a point) $S$ is contractible.

Proof.

In the case where $I$ is a well-ordered set, this is immediate from Lemma 6; the two places below where we use this result both follow from this case. (To deduce the general form of the result, one can appeal to Iwamura’s lemma from order theory which reduces an arbitrary directed union to iterated well-ordered unions; see e.g., [Mky].) ∎

We say that a simplicial map $f:S\to T$ is a trivial pseudofibration if for each $t\in T$, the subcomplex $S\mathchar 124\relax f^{\mathchar 45\relax 1}(t)\subseteq S$ is contractible.

Corollary 9**.**

A trivial pseudofibration is a homotopy equivalence.

Proof.

Apply Lemma 6 to $S=\bigcup_{t\in T}S\mathchar 124\relax f^{\mathchar 45\relax 1}(t)$ and $T=\bigcup_{t\in T}T\mathchar 124\relax t$. ∎

Finally, we come to the notion of Borel structurability. Let $(X,E)$ be a countable Borel equivalence relation. We say that a simplicial complex $S$ on $X$ is Borel if for each $n$ the $(n+1)$-ary relation “$\{x_{0},\dotsc,x_{n}\}\in S$” is Borel, or equivalently $S$ is Borel as a subset of the standard Borel space of finite subsets of $X$. A Borel simplicial complex $S$ on $X$ is a Borel structuring of $E$ by simplicial complexes if in addition each simplex $s\in S$ is contained in a single $E$-class; such an $S$ represents the “Borel assignment” $C\mapsto S_{C}:=S\mathchar 124\relax C$ of the (countable) complex $S_{C}$ to each $E$-class $C\in X/E$. More generally, for a class $\mathcal{K}$ of simplicial complexes (e.g., the contractible ones), $S$ is a structuring of $E$ by complexes in $\mathcal{K}$ if $S_{C}\in\mathcal{K}$ for each $C\in X/E$; if such a structuring exists, we say that $E$ is structurable by complexes in $\mathcal{K}$.

3 Proofs

3.1 Some lemmas

Let $N=\{\{i\},\{i,i+1\}\mid i\in\mathbb{N}\}$ denote the ordered simplicial complex on $\mathbb{N}=\{0\mathchar 60\relax 1\mathchar 60\relax 2\mathchar 60\relax\dotsc\}$ with an edge between $i,i+1$ for each $i$, whose geometric realization is a ray.

For a simplicial complex $(X,S)$, a set $Y$, and a map $f:X\to Y$, define the image complex

[TABLE]

which is a simplicial complex on $f(X)$; we write $f(X,S)$ for $(f(X),f(S))$. If $(X,S)$ is an ordered simplicial complex, $Y$ is a poset, and $f$ is monotone, then $(f(X),f(S))$ is also ordered.

Let $X$ be a poset and $T$ be an ordered simplicial complex on $X\times\mathbb{N}^{n}$, for some $n\in\mathbb{N}$. We define the telescope $\mathcal{T}_{n}(T)$, an ordered simplicial complex on $X\times\mathbb{N}^{n}$, by induction on $n$ as follows:

[TABLE]

where $p_{i}:X\times\mathbb{N}^{n}\to X\times\mathbb{N}^{n\mathchar 45\relax i}$ is the projection onto all but the last $i$ factors. Explicitly, we have

[TABLE]

(the last term $p_{n}(T)\times\{0\}^{n}$ is redundant unless $n=0$). Here are some simple properties of $\mathcal{T}_{n}(T)$:

Lemma 10**.**

1. (a)

$T\subseteq\mathcal{T}_{n}(T)$. 2. (b)

The projection $p_{n}:\mathcal{T}_{n}(T)\to p_{n}(T)$ is a homotopy equivalence (with homotopy inverse the inclusion $p_{n}(T)\cong p_{n}(T)\times\{0\}^{n}\subseteq\mathcal{T}_{n}(T)$). 3. (c)

For a subset $Z\subseteq X$, we have $\mathcal{T}_{n}(T)\mathchar 124\relax(Z\times\mathbb{N}^{n})=\mathcal{T}_{n}(T\mathchar 124\relax(Z\times\mathbb{N}^{n}))$. 4. (d)

If $T$ is (at most) $k$-dimensional, then $\mathcal{T}_{n}(T)$ is (at most) $(k+1)$-dimensional.

Proof.

(a), (c), and (d) are straightforward. For $n\geq 1$, it is easily seen that $\mathchar 124\relax\mathcal{T}_{n}(T)\mathchar 124\relax$ deformation retracts onto $\mathchar 124\relax\mathcal{T}_{n\mathchar 45\relax 1}(p_{1}(T))\times\{0\}\mathchar 124\relax\cong\mathchar 124\relax\mathcal{T}_{n\mathchar 45\relax 1}(p_{1}(T))\mathchar 124\relax$; a simple induction then yields (b). ∎

We need one more (straightforward) lemma:

Lemma 11**.**

A trivial pseudofibration $f:S\to T$ is surjective on simplices.

Proof.

Let $t\in T$. Put $S^{\prime}:=\{s\in S\mid f(s)\subsetneq t\}=S\mathchar 124\relax f^{\mathchar 45\relax 1}(t)\setminus\{s\in S\mid f(s)=t\}$. Since $f$ is a trivial pseudofibration, for every $t^{\prime}\subsetneq t$, $S^{\prime}\mathchar 124\relax f^{\mathchar 45\relax 1}(t^{\prime})=S\mathchar 124\relax f^{\mathchar 45\relax 1}(t^{\prime})$ is contractible; thus $f:S^{\prime}\to T\mathchar 124\relax t\setminus\{t\}$ is a homotopy equivalence. But $T\mathchar 124\relax t\setminus\{t\}$ is the boundary of the simplex $t$, hence not contractible; thus for $S\mathchar 124\relax f^{\mathchar 45\relax 1}(t)$ to be contractible, there must be $s\in S$ with $f(s)=t$. ∎

3.2 The main construction

We now give the main construction in the proof of Theorem 3. Let $(X,S)$ be a locally countable simplicial complex, which we may assume to be ordered by taking any linear order on $X$. By local countability, for each $n$ we may find a function $c_{n}:S^{(n)}\to\mathbb{N}$ which colors the intersection graph on the $n$-simplices $S^{(n)}$, which means that for $s,t\in S^{(n)}$ with $s\neq t$ and $s\cap t\neq\varnothing$ we have $c_{n}(s)\neq c_{n}(t)$. The idea is that for each $n$, we will multiply the complex by the ray $N$ and then attach each $n$-simplex $s\in S^{(n)}$ at position $c_{n}(s)$ along the ray, so that distinct simplices have non-overlapping boundaries.

Let $S_{n}:=\bigcup_{m\leq n}S^{(m)}=\{s\in S\mid\dim(s)\leq n\}$, the $n$-skeleton of $S$. We will inductively define ordered simplicial complexes $T_{n}$ on $X\times\mathbb{N}^{n}$ and for $n\geq 1$, $T_{n}^{\prime}$ on $X\times\mathbb{N}^{n}$ such that

[TABLE]

fitting into the following commutative diagram of monotone simplicial maps:

[TABLE]

The horizontal maps are the inclusions, while the vertical/diagonal maps are the projections $p_{i}:X\times\mathbb{N}^{n}\to X\times\mathbb{N}^{n\mathchar 45\relax i}$ onto all but the last $i$ factors as before; furthermore each vertical/diagonal map will be a trivial pseudofibration between the respective complexes.

Start with $T_{0}:=S_{0}$. Given $T_{n}$ such that $p_{n}:T_{n}\to S_{n}$ is a trivial pseudofibration, put

[TABLE]

Clearly this is an ordered simplicial complex on $X\times\mathbb{N}^{n+1}$.

Claim**.**

$p_{n+1}:(X\times\mathbb{N}^{n+1},T_{n+1}^{\prime})\to(X,S_{n})$* is a trivial pseudofibration.*

Proof.

Let $t\in S_{n}$; we must check that $T_{n+1}^{\prime}\mathchar 124\relax p_{n+1}^{\mathchar 45\relax 1}(t)=T_{n+1}^{\prime}\mathchar 124\relax(t\times\mathbb{N}^{n+1})$ is contractible. We have

[TABLE]

(using Lemma 10(c)); let $A,B_{s}$ be as shown. The subcomplex $A$ is contractible since $p_{n}:T_{n}\to S_{n}$ is a trivial pseudofibration by the induction hypothesis whence $T_{n}\mathchar 124\relax p_{n}^{\mathchar 45\relax 1}(t)$ is contractible. For each $s\in S^{(n+1)}$ such that $s\cap t\neq\varnothing$ (otherwise $B_{s}$ is empty), the subcomplex $B_{s}$ is contractible since the telescope $\mathcal{T}_{n}(T_{n}\mathchar 124\relax p_{n}^{\mathchar 45\relax 1}(s\cap t))$ is homotopy equivalent (by Lemma 10(b)) to the projection $p_{n}(T_{n}\mathchar 124\relax p_{n}^{\mathchar 45\relax 1}(s\cap t))=p_{n}(T_{n})\mathchar 124\relax(s\cap t)=S_{n}\mathchar 124\relax(s\cap t)$ which is a standard simplex; and also $A\cap B_{s}$ is contractible since

[TABLE]

(the second equality since the telescope is a complex on $(s\cap t)\times\mathbb{N}^{n}$, the third equality by Lemma 10(a)), which is contractible because again $p_{n}$ is a trivial pseudofibration. For two distinct $s,s^{\prime}\in S^{(n+1)}$, we have $B_{s}\cap B_{s^{\prime}}=\varnothing$: either $c_{n+1}(s)\neq c_{n+1}(s^{\prime})$ in which case clearly $B_{s}\cap B_{s^{\prime}}=\varnothing$, or $c_{n+1}(s)=c_{n+1}(s^{\prime})$ whence by the coloring property of $c_{n+1}$ we have $s\cap s^{\prime}=\varnothing$. Now by repeated use of Corollary 7, we get that $A\cup B_{s_{1}}\cup\dotsb\cup B_{s_{m}}$ is contractible for every finite collection of $s_{1},\dotsc,s_{m}\in S^{(n+1)}$, whence by Corollary 8, $T_{n+1}^{\prime}\mathchar 124\relax(t\times\mathbb{N}^{n+1})$ is contractible. ∎

Now put

[TABLE]

Claim**.**

$T_{n+1}$* is an ordered simplicial complex on $X\times\mathbb{N}^{n+1}$.*

Proof.

The only thing that needs to be checked is that for each $s\in S^{(n+1)}$, a nonempty subset $s^{\prime}\times\{0\}^{n}\times\{c_{n+1}(s)\}$ of $s\times\{0\}^{n}\times\{c_{n+1}(s)\}$ is still in $T_{n+1}$. We may assume $s^{\prime}\subsetneq s$. Then $s^{\prime}\in S_{n}$, so since $p_{n}:T_{n}\to S_{n}$ is a trivial pseudofibration, hence surjective on simplices, we have $s^{\prime}\in p_{n}(T_{n}\mathchar 124\relax(s\times\mathbb{N}^{n}))$, whence $s^{\prime}\times\{0\}^{n}\times\{c_{n+1}(s)\}\in p_{n}(T_{n}\mathchar 124\relax(s\times\mathbb{N}^{n}))\times\{0\}^{n}\times\{c_{n+1}(s)\}\subseteq\mathcal{T}_{n}(T_{n}\mathchar 124\relax(s\times\mathbb{N}^{n}))\times\{c_{n+1}(s)\}\subseteq T_{n+1}^{\prime}\subseteq T_{n+1}$. ∎

Claim**.**

$p_{n+1}:(X\times\mathbb{N}^{n+1},T_{n+1})\to(X,S_{n+1})$* is a trivial pseudofibration.*

Proof.

Let $s\in S_{n+1}$; we must check that $T_{n+1}\mathchar 124\relax p_{n+1}^{\mathchar 45\relax 1}(s)$ is contractible. If $s\in S_{n}$ then clearly $T_{n+1}\mathchar 124\relax p_{n+1}^{\mathchar 45\relax 1}(s)=T_{n+1}^{\prime}\mathchar 124\relax p_{n+1}^{\mathchar 45\relax 1}(s)$ so this follows from the previous claim that $p_{n+1}:T_{n+1}^{\prime}\to S_{n}$ is a trivial pseudofibration. So we may assume that $s\in S^{(n+1)}$, in which case

[TABLE]

Since $p_{n+1}:T_{n+1}^{\prime}\to S_{n}$ is a trivial pseudofibration, so is the restriction $p_{n+1}:T_{n+1}^{\prime}\mathchar 124\relax p_{n+1}^{\mathchar 45\relax 1}(s)\to S_{n}\mathchar 124\relax s$; but this restriction has one-sided inverse the inclusion $S_{n}\mathchar 124\relax s\cong S_{n}\mathchar 124\relax s\times\{0\}^{n}\times\{c_{n+1}(s)\}\subseteq\mathcal{T}_{n}(T_{n}\mathchar 124\relax(s\times\mathbb{N}^{n}))\times\{c_{n+1}(s)\}\subseteq T_{n+1}^{\prime}\mathchar 124\relax p_{n+1}^{\mathchar 45\relax 1}(s)$, which is therefore a homotopy equivalence. Now applying Corollary 7 to

[TABLE]

where the two subcomplexes on the right-hand side have intersection $S_{n}\mathchar 124\relax s\times\{0\}^{n}\times\{c_{n+1}(s)\}$, yields that the inclusion $S\mathchar 124\relax s\times\{0\}^{n}\times\{c_{n+1}(s)\}\subseteq T_{n+1}\mathchar 124\relax p_{n+1}^{\mathchar 45\relax 1}(s)$ is a homotopy equivalence; but $S\mathchar 124\relax s$ is a standard simplex, hence contractible, whence $T_{n+1}\mathchar 124\relax p_{n+1}^{\mathchar 45\relax 1}(s)$ is contractible. ∎

This completes the definition of the complexes $T_{n},T_{n}^{\prime}$ and the verification that $p_{n}:T_{n}\to S_{n}$ is a homotopy equivalence for each $n$. Note that from the definition and Lemma 10(d), it is clear that each $T_{n}$ is $n$-dimensional.

3.3 The constant bound

We next bound the number of edges containing a point in $T_{n}$. To do so, we will define for each $n\geq 1$ a constant $K_{n}$ such that for each $y\in X\times\mathbb{N}^{n}$ there are at most $K_{n}$ distinct $y^{\prime}\in X\times\mathbb{N}^{n}$ with $y\leq y^{\prime}$ and $\{y,y^{\prime}\}\in T_{n}$, and also the same holds with $y^{\prime}\leq y$.

For $n=1$, we have $T_{1}^{\prime}=T_{0}\times N=S_{0}\times N$, while $T_{1}=T_{1}^{\prime}\cup\{s\times\{c_{1}(s)\}\mid s\in S^{(1)}\}$. Thus

[TABLE]

works: for $t=\{y\leq y^{\prime}\}\in T_{1}$, either $t\in T_{1}^{\prime}$, in which case we have $y=(x,i)$ and $y^{\prime}\in\{(x,i),(x,i+1)\}$ for some $(x,i)\in X\times\mathbb{N}$, or $t=s\times\{c_{1}(s)\}$ for some $s\in S^{(1)}$, in which case $y=(x,c_{1}(s))$ and $y^{\prime}=(x^{\prime},c_{1}(s))$ for some $s=\{x\mathchar 60\relax x^{\prime}\}\in S^{(1)}$, which is uniquely determined by $y$ by the coloring property of $c_{1}$; and similarly for $y^{\prime}\leq y$.

Now suppose for $n\geq 1$ that we are given $K_{n}$; we find $K_{n+1}$ by a similar argument. Let $t=\{y\leq y^{\prime}\}\in T_{n+1}$. Since $n+1\geq 2$, $T_{n+1}$ adds no [math]- or $1$-simplices to $T_{n+1}^{\prime}$, so $t\in T_{n+1}^{\prime}$. If $t\in T_{n}\times N$, then we have $y=(z,i)$ and $y^{\prime}=(z^{\prime},i^{\prime})$ for some $\{z\leq z^{\prime}\}\in T_{n}$ and $\{i\leq i^{\prime}\}\in N$, i.e., $i^{\prime}\in\{i,i+1\}$; there are thus $\leq 2K_{n}$ choices for $y^{\prime}$ given $y$ in this case. Otherwise, we have $t\in\mathcal{T}_{n}(T_{n}\mathchar 124\relax(s\times\mathbb{N}^{n}))\times\{c_{n+1}(s)\}\subseteq S\mathchar 124\relax s\times N^{n}\times\{c_{n+1}(s)\}$ for some $s\in S^{(n+1)}$, whence $y=(x,i_{1},\dotsc,i_{n},c_{n+1}(s))$ and $y^{\prime}=(x^{\prime},i_{1}^{\prime},\dotsc,i_{n}^{\prime},c_{n+1}(s))$ where $x,x^{\prime}\in s$ and each $i_{j}^{\prime}\in\{i_{j},i_{j}+1\}$; by the coloring property of $c_{n+1}(s)$, $s$ is uniquely determined by $y$, hence there are at most $\mathchar 124\relax s\mathchar 124\relax=n+2$ choices for $x^{\prime}$ and so at most $(n+2)2^{n}$ choices for $y^{\prime}$ given $y$. In total, there are thus at most

[TABLE]

choices for $y^{\prime}\geq y$; similarly for $y^{\prime}\leq y$.

Solving this recurrence yields

[TABLE]

So, for each $n\geq 1$ and $y\in X\times\mathbb{N}^{n}$, there are at most $2(K_{n}\mathchar 45\relax 1)$ distinct edges $\{y\mathchar 60\relax y^{\prime}\}$ or $\{y^{\prime}\mathchar 60\relax y\}$ in $T_{n}$; that is, there are at most

[TABLE]

edges in $T_{n}$ containing $y$. When $S=S_{n}$ is $n$-dimensional, truncating the above inductive construction at $T_{n}$ and taking $T:=T_{n}$ proves the combinatorial part of Theorem 3 (with the weaker condition “at most $M_{n}$” in (ii)) in this case.

3.4 Growing edges

Still in the $n$-dimensional case, in order to modify $T_{n}$ so that each vertex is contained in exactly $M_{n}$ edges, we use the following simple construction. Put $T_{n,0}:=T_{n}$. Given $T_{n,k}$, let $T_{n,k+1}$ be $T_{n,k}$ together with, for each vertex $y$ of $T_{n}$ with fewer than $M_{n}$ edges, a new vertex $y^{\prime}$ and an edge $\{y,y^{\prime}\}$. Then clearly

[TABLE]

is still $n$-dimensional and has each vertex contained in exactly $M_{n}$ edges. Also, clearly $T_{n,k+1}$ deformation retracts onto $T_{n,k}$; thus (by Corollary 8) the inclusion $T_{n}=T_{n,0}\subseteq T_{n}^{*}$ is a homotopy equivalence. So we may replace $T_{n}$ with $T_{n}^{*}$ to get the stronger form of Theorem 3(ii).

3.5 The infinite-dimensional case

Next we handle the case where $S$ is infinite-dimensional. Let $i_{n}:(X\times\mathbb{N}^{n},T_{n})\hookrightarrow(X\times\mathbb{N}^{n+1},T_{n+1})$ be the composite

[TABLE]

From the above diagram ($*$), we get a commutative diagram

[TABLE]

We would like to let $T$ be the direct limit of the top row of this diagram, but that might not be locally finite. Instead, we take the mapping telescope of the top row, which can be defined explicitly as follows.

Let $\mathbb{N}^{\infty}$ be the direct limit of $\mathbb{N}\cong\mathbb{N}\times\{0\}\subseteq\mathbb{N}^{2}\cong\mathbb{N}^{2}\times\{0\}\subseteq\mathbb{N}^{3}\subseteq\dotsb$; explicitly, $\mathbb{N}^{\infty}$ can be taken as the subset of $\mathbb{N}^{\mathbb{N}}$ consisting of the eventually zero sequences. Then $X\times\mathbb{N}^{\infty}$ is the direct limit of the sequence $X\times\mathbb{N}^{0}\xrightarrow{i_{0}}X\times\mathbb{N}^{1}\xrightarrow{i_{1}}\dotsb$, with injections

[TABLE]

and so the direct limit of the top row of ($\dagger$) can be taken explicitly as the ordered simplicial complex $\bigcup_{n\in\mathbb{N}}i^{n}(T_{n})$ on $X\times\mathbb{N}^{\infty}$.

The mapping telescope of the top row of ($\dagger$) is the complex $(Y,T)$ where

[TABLE]

For each $n$, let

[TABLE]

It is easy to see that the projection $p_{1}:X\times\mathbb{N}^{\infty}\times\mathbb{N}\to X\times\mathbb{N}^{\infty}$ restricts to simplicial maps $\widetilde{T}_{n}\to i^{n}(T_{n})$ for each $n$, yielding a commutative diagram

[TABLE]

in which the horizontal maps are inclusions and the vertical maps are homotopy equivalences by the usual argument: the (geometric realization of the) first cylinder $i^{0}(T_{0})\times N\mathchar 124\relax\{0,1\}$ in $\widetilde{T}_{n}$ deformation retracts onto its base $i^{0}(T_{0})\times\{1\}$, which is contained in the second cylinder $i^{1}(T_{1})\times N\mathchar 124\relax\{1,2\}$, which deformation retracts onto its base $i^{1}(T_{1})\times\{2\}$, etc. Since, as noted above, the bottom row of ($\ddagger$) may be identified with the top row of ($\dagger$), combining the two diagrams and applying Corollary 8 yields that $T=\bigcup_{n}\widetilde{T}_{n}$ is homotopy equivalent to $S=\bigcup_{n}S_{n}$ (via the restriction of the projection $X\times\mathbb{N}^{\infty}\times\mathbb{N}\to X$).

Since, clearly, each $T_{n}$ being locally finite implies that $T$ is locally finite, this proves the combinatorial part of Theorem 3 in the infinite-dimensional case.

3.6 The Borel case

Finally, suppose we start with a Borel structuring $S$ of a countable Borel equivalence relation $(X,E)$ by simplicial complexes. Recall that this means $S$ is a simplicial complex on $X$ with simplices contained in $E$-classes and such that $S$ is Borel in the standard Borel space of finite subsets of $X$. We may then simply apply the above construction to the locally countable simplicial complex $(X,S)$, while observing that each step is Borel. To do so, we first pick a Borel linear order on $X$ to turn $(X,S)$ into an ordered simplicial complex, and then pick the coloring functions $c_{n}:S^{(n)}\to\mathbb{N}$ to be Borel (in fact restrictions of a single $c:S\to\mathbb{N}$) using the following standard lemma:

Lemma 12** (Kechris-Miller [KM, 7.3]).**

Let $(X,E)$ be a countable Borel equivalence relation, and let $[E]^{\mathchar 60\relax\infty}$ be the standard Borel space of finite subsets of $X$ which are contained in some $E$-class. Then there is a Borel $\mathbb{N}$-coloring of the intersection graph on $[E]^{\mathchar 60\relax\infty}$, i.e., a Borel map $c:[E]^{\mathchar 60\relax\infty}\to\mathbb{N}$ such that if $A,B\in[E]^{\mathchar 60\relax\infty}$ with $A\neq B$ and $A\cap B\neq\varnothing$ then $c(A)\neq c(B)$.

It is now straightforward to check that the definitions of $T_{n},T_{n}^{\prime}$ are Borel; in the definition of $T_{n+1}^{\prime}$, note that the union over $s\in S^{(n+1)}$ is disjoint, by the coloring property of $c_{n+1}$. In the $n$-dimensional case, we end up with an ordered Borel simplicial complex $(X\times\mathbb{N}^{n},T_{n})$ such that the projection $p_{n}:X\times\mathbb{N}^{n}\to X$ is a homotopy equivalence $T_{n}\to S_{n}=S$. Defining the countable Borel equivalence relation $F$ on $Y:=X\times\mathbb{N}^{n}$ by

[TABLE]

we get that $T:=T_{n}$ is a Borel structuring of $(Y,F)$; and we have a Borel embedding $f:(X,E)\to(Y,F)$ given by $f(x):=(x,0,\dotsc,0)$ such that $S\mathchar 124\relax[x]_{E}$ is homotopy equivalent to $T\mathchar 124\relax[f(x)]_{F}$ (via the map $p_{n}\mathchar 124\relax([x]_{E}\times\mathbb{N}^{n})=p_{n}\mathchar 124\relax[f(x)]_{F}:T\mathchar 124\relax[f(x)]_{F}\to S\mathchar 124\relax[x]_{E}$) for each $x\in X$.

For the stronger condition that each vertex is contained in exactly $M_{n}$ edges, it is straightforward that the definition of $T_{n}^{*}$ above can be taken to be a Borel simplicial complex on a standard Borel space $Y^{*}\supseteq Y$; letting $F^{*}\supseteq F$ be the obvious equivalence relation on $Y^{*}$ (so that each newly added edge in $T_{n}^{*}$ lies in one $F^{*}$-class), $T_{n}^{*}$ is a Borel structuring of $(Y^{*},F^{*})$ such that the composite $(X,E)\xrightarrow{f}(Y,F)\subseteq(Y^{*},F^{*})$ is a homotopy equivalence on each class. So we may replace $(Y,F,T_{n})$ by $(Y^{*},F^{*},T_{n}^{*})$.

Similarly, in the infinite-dimensional case, it is straightforward that the definition of the mapping telescope $T$ on $Y\subseteq X\times\mathbb{N}^{\infty}\times\mathbb{N}$ is Borel; so the same definitions of $F,f$ as in the finite-dimensional case work (note that $(x,0,\dotsc,0)\in Y$ for all $x\in X$). This completes the proof of Theorem 3, which implies Theorem 1.

To prove Corollary 2, apply Theorem 1 to get $(Y,F)$ with structuring $T$ and an embedding $f:(X,E)\to(Y,F)$; since $E$ is compressible, $f$ may be modified so that its image is $F$-invariant (see [DJK, 2.3]), whence we get the desired structuring of $E$ by restricting $T$.

To prove Corollary 4, let $S$ be the trivial structuring of $E$ given by $\{x_{0},\dotsc,x_{n}\}\in S\iff x_{0}\mathrel{E}\dotsb\mathrel{E}x_{n}$; this is obviously contractible on each $E$-class, so by Theorem 3 $E$ Borel embeds into some $F$ structurable by locally finite contractible complexes. As before, this implies Corollary 5.

3.7 Some remarks

In the dimension $n=1$ case, the construction of $T_{1}$ above can be seen as a slight variant of the proof of Jackson-Kechris-Louveau [JKL, 3.10]. Thus the general case of our construction can be seen as a generalization of their proof to higher dimensions.

As mentioned in the Introduction, our construction is based on the proof of Whitehead [Wh, Theorem 13] that every countable CW-complex is homotopy equivalent to a locally finite complex of the same dimension. That proof uses the same idea of “spreading out” cells along a ray to make their boundaries disjoint, but uses more abstract tools from homotopy theory in place of our explicit “telescope” construction $\mathcal{T}_{n}$. While it should be possible to give a more direct combinatorial transcription of Whitehead’s proof, using (for example) simplicial sets, it does not seem that such an approach would yield a uniform bound $M_{n}$ on the number of edges containing a vertex in the $n$-dimensional case.

4 Problems

There are several other nice properties of treeable countable Borel equivalence relations, for which we do not know if they generalize to higher dimensions. Each of the following is known to be true in the case $n=1$; see [JKL, 3.3, 3.12, 3.17].

Problem 1**.**

Let $E,F$ be countable Borel equivalence relations such that $E$ Borel embeds into $F$. If $F$ is structurable by $n$-dimensional contractible simplicial complexes, then must $E$ be also?

Problem 2**.**

Let $E$ be a countable Borel equivalence relation. If $E$ is structurable by $n$-dimensional contractible simplicial complexes, then is $E$ necessarily structurable by $n$-dimensional locally finite contractible simplicial complexes? (As noted in the Introduction, there cannot be a uniform bound on the number of edges containing each vertex.)

Problem 3**.**

Is there a single countably infinite $n$-dimensional contractible simplicial complex $S_{n}$, such that every countable Borel equivalence relation $E$ structurable by $n$-dimensional contractible simplicial complexes Borel embeds into an $F$ structurable by isomorphic copies of $S_{n}$?

Problem 4**.**

Is there a countable group $\Gamma_{n}$ with an $n$-dimensional Eilenberg-MacLane complex $K(\Gamma_{n},1)$, such that every countable Borel equivalence relation $E$ structurable by $n$-dimensional contractible simplicial complexes Borel embeds into the orbit equivalence relation of a free Borel action of $\Gamma_{n}$?