#  $d$-representability as an embedding problem

Moshe J. White
Institute of Mathematics,
Hebrew University, Jerusalem, Israel Research supported in part by the Israel
Science Foundation (Grant No. 2669/21) and in part by the ERC Advanced Grant
(Grant No. 834735)

###### Abstract

An abstract simplicial complex is said to be $d$-representable if it records
the intersection patterns of a collection of convex sets in $\mathbb{R}^{d}$.
In this paper, we show that $d$-representability of a simplicial complex is
equivalent to the existence of a map with certain properties, from a closely
related simplicial complex into $\mathbb{R}^{d}$. This equivalence suggests a
framework for proving (and disproving) $d$-representability of simplicial
complexes using topological methods such as applications of the Borsuk–Ulam
theorem, which we begin to explore.

##  1 Introduction

The _nerve_ of any finite collection of sets $C_{1},\ldots,C_{n}$, is a
simplicial complex which records all subcollections with non-empty
intersection: the faces are precisely those
$I\subseteq\left\\{1,\ldots,n\right\\}$ for which $\bigcap_{i\in
I}C_{i}\neq\emptyset$. A simplicial complex $\mathsf{K}$ is said to be _$d$
-representable_ if it is the nerve of a collection of convex sets in
$\mathbb{R}^{d}$.

The general problem of classifying $d$-representable complexes is wide open,
but many partial results are known. Perhaps the earliest result on
$d$-representability is Helly’s classical theorem from 1923. There are
additional topological restrictions: a $d$-representable complex cannot have
non-trivial homology in any dimension $m\geq d$, and the same holds for any
induced subcomplex. A complex with these homological properties is said to be
_$d$ -Leray_. Another important property, combinatorial in nature, was
introduced by Wenger [12] in 1975. In an _elementary $d$-collapsing_ of a
simplicial complex $\mathsf{K}$, we take a face $F$ of $\mathsf{K}$ which has
dimension $<d$ and is contained in a unique maximal face of $\mathsf{K}$, and
delete from $\mathsf{K}$ the face $F$ and every face containing it. If there
is a sequence of elementary $d$-collapsings from $\mathsf{K}$ to $\emptyset$,
we say that $\mathsf{K}$ is _$d$ -collapsible_. Wegner proved that every
$d$-representable complex is $d$-collapsible, and every $d$-collapsible
complex is $d$-Leray. This result was extended by Eckhoff [3], who introduced
_strong $d$-collapsibility_, a property implied by $d$-representability and
implying $d$-collapsibility.
For further reading on representability and related properties, the reader is
referred to a survey by Tancer [11].

Therefore, if we discover that a certain complex $\mathsf{K}$ is, say, not
$d$-collapsible, we may deduce that $\mathsf{K}$ is not $d$-representable.
Another approach for disproving $d$-representability appeared in Wegner’s PhD
thesis from 1967, and was later extended by Tancer [10]. In the simplest case,
they prove that a certain $\mathsf{K}$ (the barycentric subdivision of
$K_{5}$, the complete graph on $5$ vertices) is not $2$-representable. This is
done by showing that if $\mathsf{K}$ is the nerve of convex sets in
$\mathbb{R}^{2}$, the vertices and edges of $K_{5}$ can be careful mapped
within the convex sets, in a manner implying that $K_{5}$ is a planar graph.

Our setting can be seen as a generalization of this result. For every
$\mathsf{K}$ we define the dual simplicial complex $\mathsf{K}^{\prime}$, as
the nerve of the facets of $\mathsf{K}$. Explicitly:

######  Definition 1.1.

Let $\mathsf{m}_{\mathsf{K}}$ denote the set of inclusion-maximal faces of a
simplicial complex $\mathsf{K}$. The dual of $\mathsf{K}$ is defined as:

$\mathsf{K}^{\prime}=\left\\{\alpha\subseteq\mathsf{m}_{\mathsf{K}}\ :\
\bigcap\alpha\neq\emptyset\right\\}.$

In Theorem 3.3, we prove that $\mathsf{K}$ is $d$-representable if and only if
there exists a map from $\|\mathsf{K}^{\prime}\|$ (the geometric realization
of $\mathsf{K}^{\prime}$) to $\mathbb{R}^{d}$, satisfying certain properties.
As in Tancer’s proof, this is done by mapping the faces of
$\mathsf{K}^{\prime}$ within corresponding intersections of those convex sets
in $\mathbb{R}^{d}$ which represent $\mathsf{K}$, whenever such sets exist. We
also mention that a precursor of the idea to construct a map
$\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{d}$ by placing points among certain
intersections of a collection of sets in $\mathbb{R}^{d}$, appeared in an
earlier work of Matoušek [6].

This method can be used to obtain either representability or non-
representability results, by either proving (as in Theorem 3.12) or disproving
(as in Theorem 3.13) the existence of the suitable map. These theorems may by
applied regardless of the method by which we prove or disprove the existence
of the required map $\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{d}$. Proceeding
with this goal, we will mostly focus on a topological method described in a
book by Matoušek [7], on the application of the Borsuk–Ulam theorem. This
leads to the introduction of a new class of simplicial complexes, which we
name _$d$ -Matoušek_ complexes:

######  Definition 1.2.

Let $\mathsf{K}$ be a simplicial complex. We define $\widehat{\mathsf{K}}$ as
the topological space:

$\widehat{\mathsf{K}}=\left\\{(x,y)\in\|\mathsf{K}^{\prime}\|^{2}\ :\
\left(\bigcap\ \textnormal{supp}(x)\right)\cup\left(\bigcap\
\textnormal{supp}(y)\right)\notin\mathsf{K}\right\\}.$

######  Definition 1.3.

Let $\mathsf{K}$ be a simplicial complex, and let $d\in\mathbb{N}$. We say
that $\mathsf{K}$ is $d$-Matoušek, if there exists a continuous map
$f:\widehat{\mathsf{K}}\to S^{d-1}$ satisfying ${f(x,y)=-f(y,x)}$ for every
${(x,y)\in\widehat{\mathsf{K}}}$.

In the terminology of $\mathbb{Z}_{2}$-spaces, the configuration space
$\widehat{\mathsf{K}}$ has an involution given by swapping coordinates, and
$\mathsf{K}$ is $d$-Matoušek if the $\mathbb{Z}_{2}$-index of
$\widehat{\mathsf{K}}$ is at most $d-1$.

In broad terms, the configuration space $\widehat{\mathsf{K}}$ consists of
pairs of points which cannot be mapped to the same point in $\mathbb{R}^{d}$
without “ruining” the representation of $\mathsf{K}$. Therefore every
$d$-representable is also $d$-Matoušek (Proposition 3.7). We begin to explore
the $d$-Matoušek property, and how it relates to $d$-representability.

It is natural to start by restricting the general problem to low dimension,
and there are three ways to do this: by fixing the dimension of the complex
$\mathsf{K}$, or the dimension of the dual complex $\mathsf{K}^{\prime}$, or
by fixing $d$ (the dimension of the space in which we look for representing
convex sets). For $d=1$ representability was fully classified in 1962 by
Lekkerkerker and Boland [5]. We apply their results in the context of Matoušek
complexes, and reclassify $1$-representable complexes as $1$-Matoušek
complexes (Theorem 4.4). For simplicial complexes $\mathsf{K}$ such that
$\dim\mathsf{K}^{\prime}=1$, we again obtain that $d$-Matoušek and
$d$-representability are identical (Theorem 4.7).

However, we describe a complex $\mathsf{K}$ such that $\dim\mathsf{K}=1$,
which is $2$-Matoušek but not $2$-representable (see Example 4.13).

The structure of the remainder of this paper more or less mimics the
introduction: in Section 2 we briefly summarize existing notation,
definitions, and results, for simplicial complexes and the
$\mathbb{Z}_{2}$-index. In Section 3 we introduce new concepts, and prove the
main general results for which they are useful. In Section 4 we explore
results for $d$-Matoušek complexes, and we conclude in Section 5 by outlining
some unanswered questions.

###  1.1 Acknowledgments

This work was completed as part of the PhD thesis of the author. I would like
to thank my advisor Gil Kalai for his continued guidance and support, without
which this work would not be possible. I would also like to thank Pavle
Blagojević for useful discussions.

##  2 Preliminaries

In this section we choose notations and give a brief and incomplete summary of
previously existing results required for our setup.

Let $V$ be a finite set. A collection $\mathsf{K}\subseteq 2^{V}$ is called a
_simplicial complex_ if it is closed under downward inclusion: whenever
$F\in\mathsf{K}$ and $G\subseteq F$, we have $G\in\mathsf{K}$. The elements of
$\mathsf{K}$ are called _faces_ and inclusion maximal faces are called
_facets_. A _vertex_ is any $i\in V$, and we assume that
$\left\\{i\right\\}\in\mathsf{K}$ for every $i\in V$. We also denote the
vertex set of $\mathsf{K}$ by $V(\mathsf{K})$.
To ease notation, we mostly use numbers and lowercase Latin letters
($1,2,\ldots,i,j,n,x$ etc.) to denote single elements, uppercase Latin letters
($I,J,F,G,V$ etc.) for sets of elements such as faces, and Greek lowercase
letters ($\alpha,\beta,\gamma$ etc.) for collections of sets of elements.
Simplicial complexes will mostly be denoted by sans-serif capital letters
($\mathsf{L},\mathsf{K},\mathsf{G}$ etc.). We also often abbreviate notation
for sets described by a list of elements, for example writing $123$ instead of
$\left\\{1,2,3\right\\}$ when possible. For every $n\in\mathbb{N}$, let
$[n]=\left\\{1,\ldots,n\right\\}$.
Simplicial complexes are _isomorphic_ if we can obtain one from the other by
renaming the vertices (and extending to faces).
The _dimension_ of a face $F\in\mathsf{K}$ equals $|F|-1$. The dimension of a
simplicial complex $\mathsf{K}$ is the maximal dimension among all faces of
$\mathsf{K}$.
For every $d\in\mathbb{N}$, the _$d$ -dimensional simplex_ is
$\Delta_{d}=2^{[d+1]}$.
The _$k$ -skeleton_ of a simplicial complex $\mathsf{K}$ consists of all faces
of dimension $\leq k$, and is denoted by $\mathsf{K}^{(k)}$.
We consider graphs as $1$-dimensional simplicial complexes: we say that a
simplicial complex $\mathsf{G}\subseteq 2^{V}$ is a graph if all faces of
$\mathsf{G}$ have dimension $\leq 1$. The edges $E=E(\mathsf{G})$ are the
$1$-dimensional faces of $\mathsf{G}$.

A simplicial complex $\mathsf{K}\subseteq 2^{V}$ is _$d$ -representable_, if
there exist sets $\left\\{C_{i}\right\\}_{i\in V}$, such that each
$C_{i}\subseteq\mathbb{R}^{d}$ is convex, and

$\mathsf{K}=\left\\{I\subseteq V\ :\ \bigcap_{i\in
I}C_{i}\neq\emptyset\right\\}.$

Let $\left\\{e_{v}\right\\}_{v\in V}$ denote the standard basis of
$\mathbb{R}^{V}$. For every face $F\in\mathsf{K}$, the (geometric) simplex
corresponding to $F$ is

$\|F\|=\textnormal{conv}\left\\{e_{v}\ :\ v\in F\right\\}$

where conv denotes the convex hull. The geometric realization of $\mathsf{K}$
is the topological space

$\|\mathsf{K}\|=\bigcup_{F\in\mathsf{K}}\|F\|.$

For a point $x\in\|\mathsf{K}\|$, the support of $x$, denoted
$\textnormal{supp}(x)$, is the minimal face $F\in\mathsf{K}$ such that
$x\in\|F\|$.

For any simplicial complex $\mathsf{K}$, the _barycentric subdivision_ of
$\mathsf{K}$ is a new simplicial complex denoted by
$\textnormal{sd}(\mathsf{K})$. The vertices of $\textnormal{sd}(\mathsf{K})$
are $\mathsf{K}\setminus\emptyset$ (non-empty faces of $\mathsf{K}$), and
faces in $\textnormal{sd}(\mathsf{K})$ are chains of faces in $\mathsf{K}$. In
other words, a collection of faces
$\left\\{F_{1},\ldots,F_{n}\right\\}\subseteq\mathsf{K}$ is a face in
$\textnormal{sd}(\mathsf{K})$, if $\emptyset\subsetneq
F_{1}\subsetneq\cdots\subsetneq F_{n}$.
It is straightforward to verify that $\|\mathsf{K}\|$ and
$\|\textnormal{sd}(\mathsf{K})\|$ are canonically homeomorphic, where the
homeomorphism $\|\textnormal{sd}(\mathsf{K})\|\to\|\mathsf{K}\|$ is defined on
vertices by

$e_{F}\ \mapsto\ \frac{1}{|F|}\sum_{i\in F}e_{i}\ ,$

and extending linearly to simplices. We also note the easily verifiable fact:

######  Lemma 2.1.

Let $\mathsf{K}$ be a simplicial complex, and let $x\in\|\mathsf{K}\|$ be a
point supported by some face $F\in\mathsf{K}$. Then the point in
$\|\textnormal{sd}(\mathsf{K})\|$ corresponding to $x$ is supported by
$\left\\{F_{1},\ldots,F_{n}\right\\}$, where $\emptyset\subsetneq
F_{1}\subsetneq\cdots\subsetneq F_{n}=F$ is a chain of faces in $\mathsf{K}$.

We will use the category of _$\mathbb{Z}_{2}$ -spaces_: the objects (called
_$\mathbb{Z}_{2}$ -spaces_) are topological spaces with a continuous
$\mathbb{Z}_{2}$-action, and morphisms (called _equivariant maps_) are
continuous and commute with the $\mathbb{Z}_{2}$-action. The action of the
non-trivial element in $\mathbb{Z}_{2}$ on the $\mathbb{Z}_{2}$-space is
called the _involution_.

Important $\mathbb{Z}_{2}$-spaces that we use include:

  * •

The $d$-dimensional sphere $S^{d}$ (where $d$ is a non-negative integer).

  * •

For any topological space $X$, the deleted product of $X$ is

${X}^{2}_{\Delta}=\left\\{(x,y)\in X^{2}:x\neq y\right\\}.$

  * •

For any simplicial complex $\mathsf{K}$, the deleted product of $K$ is the
topological space:

$\|{\mathsf{K}}^{2}_{\Delta}\|=\left\\{(x,y)\in\|\mathsf{K}\|^{2}:\textnormal{supp}(x)\cap\textnormal{supp}(y)=\emptyset\right\\}.$

The involution on $S^{d}$ is the antipodal map ($x\mapsto-x$). For the deleted
products (both of a topological space and of a simplicial complex), the
involution is the coordinate swap, $(x,y)\mapsto(y,x)$. We will often omit
repeatedly specifying the involution when it should be clear from the context.
With these definitions, the Borsuk–Ulam theorem from 1933 simply asserts:

######  Theorem 2.2 (Borsuk–Ulam).

Let $d\in\mathbb{N}$. There is no equivariant map $f:S^{d}\to S^{d-1}$.

To use the Borsuk–Ulam theorem in a setting other than spheres, we recall the
definition of the $\mathbb{Z}_{2}$-index:

######  Definition 2.3 ([7], Definition 5.3.1).

For any $\mathbb{Z}_{2}$-space $A$, we define the $\mathbb{Z}_{2}$-index of
$A$ as

$\textnormal{ind}_{\mathbb{Z}_{2}}(A)=\min\\{d\in\\{0,1,2,\ldots\\}\ |\
\exists\textnormal{ an equivariant }f:A\to S^{d}\\}$

This allows us to use spheres as a measure of the ‘size’ of a
$\mathbb{Z}_{2}$-space, by the following immediate observations:

######  Proposition 2.4 ([7], Proposition 5.3.2).

  1. 1.

If $A,B$ are $\mathbb{Z}_{2}$-spaces and there exists an equivariant $f:A\to
B$, we must have
$\textnormal{ind}_{\mathbb{Z}_{2}}(A)\leq\textnormal{ind}_{\mathbb{Z}_{2}}(B)$
(any equivariant map $B\to S^{d}$ can be precomposed with $f$).

  2. 2.

For every non-negative integer $d$, we have
$\textnormal{ind}_{\mathbb{Z}_{2}}(S^{d})=d$ (follows from Borsuk–Ulam).

So ‘large’ $\mathbb{Z}_{2}$-spaces (as measured by the $\mathbb{Z}_{2}$-index)
can’t be mapped to ‘smaller’ ones. This gives us a general framework for
proving non-embeddability results: suppose we want to show that there is no
continuous injective map $f:X\to Y$, for some pair of topological spaces $X$
and $Y$. Consider the deleted products ${X}^{2}_{\Delta}$ and
${Y}^{2}_{\Delta}$ (the former is often called the _configuration space_ , and
the latter is the _target space_ of the problem). If an injective continuous
map $f:X\to Y$ exists, we may apply $f$ to each coordinate of the points in
${X}^{2}_{\Delta}$, to obtain a map ${X}^{2}_{\Delta}\to{Y}^{2}_{\Delta}$,
which is clearly equivariant (an equivariant map
${X}^{2}_{\Delta}\to{Y}^{2}_{\Delta}$ is called a _test map_). Therefore if we
prove that
$\textnormal{ind}_{\mathbb{Z}_{2}}({X}^{2}_{\Delta})>\textnormal{ind}_{\mathbb{Z}_{2}}({Y}^{2}_{\Delta})$,
no test map exists, and any continuous $f:X\to Y$ must not be injective.

We note our choice to work with deleted products of simplicial complexes in
this paper, rather than deleted joins. It is true that deleted joins have the
advantage of being simplicial complexes themselves, and are often easier to
work with (consider for example the deleted join and deleted product of
$K_{5}$). However, the deleted product has the advantage of having dimension
lower by $1$ than its deleted join counterpart, resulting in a proof of
Theorem 4.4 which is easier to visualize.

##  3 Main setup and results

###  3.1 Dual complexes and faithful maps

Our first goal is to define faithful maps, and prove that they are closely
related to representability. Recall Definition 1.1: for every simplicial
complex $\mathsf{K}$ we let $\mathsf{m}_{\mathsf{K}}$ denote the facets of
$\mathsf{K}$, and defined the dual of $\mathsf{K}$ as the nerve of
$\mathsf{m}_{\mathsf{K}}$, denoted $\mathsf{K}^{\prime}$. We now specify some
faces in $\mathsf{K}^{\prime}$:

######  Definition 3.1.

Let $\mathsf{K}$ be a simplicial complex. If $i$ is a vertex of $\mathsf{K}$,
let $\gamma_{i}$ denote the set of $\mathsf{K}$-facets containing $i$:

$\gamma_{i}=\left\\{J\in\mathsf{m}_{\mathsf{K}}:i\in J\right\\}.$

If $F$ is a face of $\mathsf{K}$, we define:

$\gamma_{F}=\left\\{J\in\mathsf{m}_{\mathsf{K}}:F\subseteq J\right\\}.$

Note that $\gamma_{F}$ and $\gamma_{i}$ are always faces of
$\mathsf{K}^{\prime}$. Every facet of $\mathsf{K}^{\prime}$ is of the form
$\gamma_{j}$ for some vertex $j$.

The motivation behind the dual complex $\mathsf{K}^{\prime}$ is to create the
simplest collection of convex sets with $\mathsf{K}$ as their nerve. Clearly
$\mathsf{K}$ is the nerve of the collection $\left\\{\gamma_{i}\right\\}_{i\in
V}$, so the images of $\left\\{\|\gamma_{i}\|\right\\}_{i\in V}$ under any map
from $\|\mathsf{K}^{\prime}\|$ which doesn’t introduce additional
intersections among the images of these simplices, will still have
$\mathsf{K}$ as the nerve. We call such a map _faithful_ :

######  Definition 3.2.

Let $\mathsf{K}\subseteq 2^{V}$ be a simplicial complex. A _faithful_ map (for
$\mathsf{K}$), is a continuous $f:\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{d}$,
such that for every $I\subseteq V$ which satisfies

$\bigcap_{i\in I}f\left(\|\gamma_{i}\|\right)\neq\emptyset,$

we have $I\in\mathsf{K}$ (equivalently $\underset{i\in
I}{\cap}\gamma_{i}\neq\emptyset$).

######  Theorem 3.3.

Let $\mathsf{K}$ be a simplicial complex. Then $\mathsf{K}$ is
$d$-representable if and only if there exists a map
$f:\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{d}$ which is linear and faithful.

###### Proof.

W.l.o.g. assume $\mathsf{K}\subseteq 2^{[n]}$ (we may label the vertices of
$\mathsf{K}$ as $\left\\{1,\ldots,n\right\\}$). If
$f:\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{d}$ is an linear and faithful map,
for every $i\in[n]$ let

$C_{i}=f\left(\|\gamma_{i}\|\right).$

The sets $C_{1},\ldots,C_{n}\subseteq\mathbb{R}^{d}$ are convex because $f$ is
linear, and we claim that $\mathsf{K}$ is their nerve. Indeed, every face
$I\in\mathsf{K}$ is a subset of some facet $J\in\mathsf{m}_{\mathsf{K}}$. Thus
for every $i\in I$ we have $J\in\gamma_{i}$, so $\underset{i\in
I}{\cap}\|\gamma_{i}\|\neq\emptyset$ and therefore $\underset{i\in
I}{\cap}C_{i}\neq\emptyset$ as well. If $I\notin\mathsf{K}$ then
$\underset{i\in I}{\cap}C_{i}=\emptyset$ due to $f$ being faithful.
Conversely, suppose that $\mathsf{K}$ is the nerve of the convex sets
$C_{1},\ldots,C_{n}\subseteq\mathbb{R}^{d}$. For every facet
$J\in\mathsf{m}_{\mathsf{K}}$, choose some point $p_{J}\in\underset{i\in
J}{\cap}C_{i}$ (this intersection is non-empty as $J$ is in the nerve).
Construct $f:\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{d}$ by taking $e_{J}\mapsto
p_{J}$ and extending linearly. To prove that $f$ is faithful note that for
every $i\in[n]$ we have

$f\left(\|\gamma_{i}\|\right)=\textnormal{conv}{\left\\{p_{J}:J\in\gamma_{i}\right\\}}\subseteq
C_{i}.$

So whenever $I\subseteq[n]$ and $\underset{i\in
I}{\cap}f\left(\|\gamma_{i}\|\right)$ is non-empty, the intersection
$\underset{i\in I}{\cap}C_{i}$ is also non-empty, thus $I\in\mathsf{K}$. ∎

An example illustrating the proof is given in Figure 3.1 below:

1234567$C_{1}$$C_{2}$$C_{3}$$C_{4}$$C_{5}$$C_{6}$$C_{7}$12131425635456737

Figure 1

From left to right, we have a simplicial complex $\mathsf{K}$, seven convex
sets in the plane which have $\mathsf{K}$ as the nerve, and the image of
$\|\mathsf{K}^{\prime}\|$ under a linear faithful map within the original
convex sets.

It is also worth noting that $\|\mathsf{K}\|$ and $\|\mathsf{K}^{\prime}\|$
are homotopy equivalent (due to the nerve theorem, see e.g. [7, Theorem
4.4.4]). While we don’t need the to prove this equivalence, we will prove the
following:

######  Lemma 3.4.

Let $\mathsf{K}$ be a simplicial complex. There exists a map
$\iota_{\mathsf{K}}:\|\mathsf{K}\|\to\|\mathsf{K}^{\prime}\|$, such that for
every $x\in\|\mathsf{K}\|$, we have

$\textnormal{supp}(x)\subseteq\bigcap\textnormal{supp}\left(\iota_{\mathsf{K}}(x)\right).$

###### Proof.

Using the canonical homeomorphism between $\|\mathsf{K}\|$ and
$\|\textnormal{sd}(\mathsf{K})\|$, we may in fact define a linear map
$g:\|\textnormal{sd}(\mathsf{K})\|\to\|\mathsf{K}^{\prime}\|$. For every
$\emptyset\neq F\in\mathsf{K}$, let $g(e_{F})\in\|\mathsf{K}^{\prime}\|$ be
some point with $\textnormal{supp}(g(e_{F}))=\gamma_{F}$ (we can choose
$g(e_{F})$ to be the center of the simplex $\|\gamma_{F}\|$). This indeed
extends to a linear
$g:\|\textnormal{sd}(\mathsf{K})\|\to\|\mathsf{K}^{\prime}\|$: if
$x\in\|\textnormal{sd}(\mathsf{K})\|$ is supported by
$\left\\{F_{1},\ldots,F_{n}\right\\}$ such that $\emptyset\subsetneq
F_{1}\subsetneq\cdots\subsetneq F_{n}$, we have
$\gamma_{F_{n}}\subseteq\ldots\subseteq\gamma_{F_{1}}$, therefore $g(x)$ is
well defined in $\|\mathsf{K}^{\prime}\|$, and

$\textnormal{supp}\left(g(x)\right)\subseteq\gamma_{F_{1}}.$ (3.1)

By Lemma 2.1, if $x\in\|\mathsf{K}\|$ is supported by $F\in\mathsf{K}$, then
the corresponding point in $\|\textnormal{sd}(\mathsf{K})\|$ is supported by
some $\left\\{F_{1},\ldots,F_{n}\right\\}$, where $\emptyset\subsetneq
F_{1}\subsetneq\cdots\subsetneq F_{n}=F$. So (3.1) becomes

$\textnormal{supp}\left(\iota_{\mathsf{K}}(x)\right)\subseteq\gamma_{F_{1}}$

and therefore

$\textnormal{supp}(x)=F_{n}\subseteq\bigcap\gamma_{F_{n}}\subseteq\ldots\subseteq\bigcap\gamma_{F_{1}}\subseteq\bigcap\textnormal{supp}\left(\iota_{\mathsf{K}}(x)\right).$

∎

###  3.2 Configuration spaces and $d$-Matoušek complexes

Recall Definitions 1.2-1.3: given a simplicial complex $\mathsf{K}$, we let

$\widehat{\mathsf{K}}=\left\\{(x,y)\in\|\mathsf{K}^{\prime}\|^{2}\ :\
\left(\bigcap\ \textnormal{supp}(x)\right)\cup\left(\bigcap\
\textnormal{supp}(y)\right)\notin\mathsf{K}\right\\},$

and say that $\mathsf{K}$ is $d$-Matoušek if there exists a continuous map
$f:\widehat{\mathsf{K}}\to S^{d-1}$ which is equivariant (that is,
$f(y,x)=-f(x,y)$ for every $(x,y)\in\widehat{\mathsf{K}}$). Equivalently,
$\textnormal{ind}_{\mathbb{Z}_{2}}(\widehat{\mathsf{K}})\leq d-1$, where the
involution on $\widehat{\mathsf{K}}$ is $(x,y)\mapsto(y,x)$. We now prove that
$\widehat{\mathsf{K}}$ is the configuration space consisting of pairs that
can’t be identified under a faithful map:

######  Lemma 3.5.

Let $\mathsf{K}$ be a simplicial complex, and suppose that
$f:\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{d}$ has $f(x)=f(y)$ for some
$(x,y)\in\widehat{\mathsf{K}}$. Then $f$ is not faithful.

###### Proof.

Let $F=\bigcap\textnormal{supp}(x)$ and $G=\bigcap\textnormal{supp}(y)$. Note
that $i\in F$ implies $\textnormal{supp}(x)\subseteq\gamma_{i}$ (for every
$i\in V(\mathsf{K})$). Therefore $\textnormal{supp}(x)\subseteq\bigcap_{i\in
F}\gamma_{i}$ and similarly $\textnormal{supp}(y)\subseteq\bigcap_{j\in
G}\gamma_{j}$. We now deduce

$\emptyset\neq f\left(\|\textnormal{supp}(x)\|\right)\cap
f\left(\|\textnormal{supp}(y)\|\right)\subseteq f\left(\bigcap_{i\in
F}\|\gamma_{i}\|\right)\cap f\left(\bigcap_{j\in
G}\|\gamma_{j}\|\right)\subseteq\bigcap_{v\in F\cup
G}f\left(\|\gamma_{v}\|\right),$

but $(x,y)\in\widehat{\mathsf{K}}$ implies $\left(\cap\
\textnormal{supp}(x)\right)\cup\left(\cap\ \textnormal{supp}(y)\right)=F\cup
G\notin\mathsf{K}$, so $f$ is not faithful. ∎

This implies that whenever a faithful map
$\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{d}$ exists (and in particular if
$\mathsf{K}$ is $d$-representable), $\mathsf{K}$ is $d$-Matoušek:

######  Proposition 3.6.

Let $d\in\mathbb{N}$, and let $\mathsf{K}$ be a simplicial complex such that
there exists a faithful map $\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{d}$. Then
$\mathsf{K}$ is $d$-Matoušek.

###### Proof.

If $f:\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{d}$ is faithful, define
$\widehat{f}:\widehat{\mathsf{K}}\to S^{d-1}$, by

$\widehat{f}(x,y)=\frac{f(x)-f(y)}{\left|f(x)-f(y)\right|}$

for every $(x,y)\in\widehat{\mathsf{K}}$ (where $\left|\cdot\right|$ denotes
the norm in $\mathbb{R}^{d}$). We know that $\widehat{f}$ is well-defined from
Lemma 3.5, and clearly $\widehat{f}$ is equivariant. ∎

Theorem 3.3 now implies:

######  Proposition 3.7.

For any $d\in\mathbb{N}$, every $d$-representable simplicial complex is
$d$-Matoušek.

######  Remark 3.8.

Note that if $\mathsf{K}$ is $d$-Matoušek, and $\mathsf{L}$ is an induced
subcomplex, that is $\mathsf{L}=\mathsf{K}\cap 2^{W}$ for some $W\subseteq
V(\mathsf{K})$, then $\mathsf{L}$ is also $d$-Matoušek. This is because every
maximal face of $\mathsf{L}$ is of the form $W\cap J$, where
$J\in\mathsf{m}_{\mathsf{K}}$ is some maximal face in $\mathsf{K}$. This gives
us a map $\mathsf{m}_{\mathsf{L}}\to\mathsf{m}_{\mathsf{K}}$, and it is
straightforward to verify that by extending to $\|\mathsf{L}^{\prime}\|$ and
then applying on each coordinate, we obtain an equivariant
$\widehat{\mathsf{L}}\to\widehat{\mathsf{K}}$.

###  3.3 Good covers

Note that in Proposition 3.6 we only required a faithful map
$\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{d}$, while $\mathsf{K}$ is also
$d$-representable iff there exists a faithful map which is also _linear_. This
implies that the collection of $d$-Matoušek complexes includes a larger class
of complexes which satisfy some topological generalizations of (convex)
representability. Indeed, this holds for every complex which is the nerve of a
good covers:

A collection of sets $U_{1},U_{2},\ldots,U_{n}\subseteq\mathbb{R}^{d}$ is a
_good cover_ , if every intersection of sets $\cap_{i\in I}U_{i}$ is either
empty or homeomorphic to an open $d$-dimensional ball. As in the convex case,
the nerve of these sets is the simplicial complex whose faces are sets
$I\subseteq[n]$ for which $\cap_{i\in I}U_{i}\neq\emptyset$.

######  Lemma 3.9.

Let $\mathsf{K}$ the nerve of a good cover in $\mathbb{R}^{d}$. Then there
exists a faithful map $f:\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{d}$.

###### Proof.

Let $\mathsf{K}$ be the nerve of a good cover
$U_{1},\ldots,U_{n}\subseteq\mathbb{R}^{d}$, and we will construct a faithful
$f:\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{d}$.
First, choose some $p_{J}\in\cap_{i\in J}U_{i}$ for every
$J\in\mathsf{m}_{\mathsf{K}}$, and let $f(e_{J})=p_{J}$. Continue constructing
$f$ face by face, inductively on the dimension of the faces $n$: suppose that
$f$ has already been defined on every simplex $\|\alpha\|$ such that
$\alpha\in\mathsf{K}^{\prime}$ is a face of dimension $<n$. Furthermore,
assume our $f$ satisfies

$f(\|\alpha\|)\subseteq\bigcap_{i\in\cap\alpha}U_{i}$ (3.2)

for every such $\alpha$. Our next task is to extend the domain of $f$ to some
simplex $\|\sigma\|$, where $\sigma\in\mathsf{K}^{\prime}$ is a face of
dimension $n>0$. We have already defined $f|_{\|\alpha\|}$ for every
$\alpha\subsetneq\sigma$, and as $\alpha\subseteq\sigma$ implies
$\cap\sigma\subseteq\cap\alpha$, we deduce

$f(\|\alpha\|)\subseteq\bigcap_{i\in\cap\alpha}U_{i}\subseteq\bigcap_{i\in\cap\sigma}U_{i}.$

This tells us that $f(\partial\|\sigma\|)\subseteq\cap_{i\in\cap\sigma}U_{i}$,
and since $\cap_{i\in\cap\sigma}U_{i}$ is contractible, we may extend $f$ to
$\|\sigma\|$ within $\cap_{i\in\cap\sigma}U_{i}$ as required.
We finally obtain a map $f:\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{d}$,
satisfying (3.2) for every $\alpha\in\mathsf{K}^{\prime}$. In particular,
$f(\|\gamma_{i}\|)\subseteq U_{i}$ for every $i=1,2,\ldots,n$, and we conclude
that $f$ is faithful: if $I\subseteq[n]$ and $\cap_{i\in
I}f(\|\gamma_{i}\|)\neq\emptyset$ then also $\cap_{i\in I}U_{i}\neq\emptyset$,
thus $I\in\mathsf{K}$. ∎

######  Corollary 3.10.

Let $\mathsf{K}$ be a simplicial complex which is the nerve of a good cover in
$\mathbb{R}^{d}$. Then $\mathsf{K}$ is $d$-Matoušek.

######  Remark 3.11.

The contractibility of all non-empty intersections is a sufficient condition
allowing us to construct a faithful map for the nerve, but it isn’t a
necessary one. Even in the proof of Lemma 3.9, we only needed that
$\cap_{i\in\cap\sigma}U_{i}$ is contractible for faces
$\sigma\in\mathsf{K}^{\prime}$ of dimension $n>0$. If $\mathsf{K}\subseteq
2^{V}$ is the nerve of a collection $\left\\{U_{i}\right\\}_{i\in V}$ in
$\mathbb{R}^{d}$, such that $\cap_{i\in F}U_{i}$ is contractible for every
non-maximal face $F\in\mathsf{K}$, the Lemma still applies with the same
proof.

The converse of Lemma 3.9 (and of Corollary 3.10) doesn’t hold, as shown in
Example 4.13.

###  3.4 Proving and disproving $d$-representability

As Theorem 3.3 implies, there is a close connection between
$d$-representability of $\mathsf{K}$ and our ability to linearly embed (or
“approximately” embed) $\mathsf{K}^{\prime}$ in $\mathbb{R}^{d}$. Indeed, we
have

######  Theorem 3.12.

Let $\mathsf{K}$ be a simplicial complex, such that there exists a continuous
injective map $f:\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{d}$. Then $\mathsf{K}$
is the nerve of a good cover in $\mathbb{R}^{d}$. If $f$ is also linear,
$\mathsf{K}$ is $d$-representable. In particular, every $\mathsf{K}$ is
$\left(2\dim\mathsf{K}^{\prime}+1\right)$-representable.

###### Proof.

If $f$ is injective, for every collection
$\alpha_{1},\ldots,\alpha_{n}\in\mathsf{K}^{\prime}$ we have

$\bigcap_{i=1}^{n}f\left(\|\alpha_{i}\|\right)=f\left(\bigcap_{i=1}^{n}\|\alpha_{i}\|\right).$

This clearly implies that $f$ is faithful, so the linear case follows from
Theorem 3.3. Furthermore, every intersection among the sets
$\left\\{f\left(\|\gamma_{i}\|\right)\right\\}_{i\in V(\mathsf{K})}$ is either
empty or homeomorphic to a closed ball of dimension $\leq d$. For some
sufficiently small $\varepsilon>0$, we may replace the sets
$\left\\{f\left(\|\gamma_{i}\|\right)\right\\}_{i\in V(\mathsf{K})}$ with an
$\varepsilon$-neighborhood of them without introducing new intersections, thus
obtaining our required good cover.
We conclude by proving that whenever $\mathsf{L}$ is a simplicial complex and
$d=2\dim\mathsf{L}+1$, there exists a linear injective map
$f:\|\mathsf{L}\|\to\mathbb{R}^{d}$. Indeed, every linear $f$ taking the
vertices of $\mathsf{L}$ to points in general position will suffice (for
instance, we may map the vertices of $\mathsf{L}$ to distinct points on the
moment curve, $\left\\{\left(t,t^{2},\ldots,t^{d}\right)\ :\
t\in\mathbb{R}\right\\}$), see [7, Theorem 1.6.1]. ∎

Of course the converse doesn’t hold - some faithful maps are not injective
(see Figure 3.1). It is therefore insufficient (in the general case) to prove
that $\|\mathsf{K}^{\prime}\|$ doesn’t embed in $\mathbb{R}^{d}$, if our goal
is to prove that $\mathsf{K}$ is not $d$-representable. We can however prove:

######  Theorem 3.13.

Let $\mathsf{L}$ be a simplicial complex with the property: for every
continuous ${\|\mathsf{L}\|\to\mathbb{R}^{d}}$ there exist in $\mathsf{L}$ two
disjoint faces whose images in $\mathbb{R}^{d}$ intersect.
Let $\mathsf{K}$ be a simplicial complex, and suppose that there exists a
continuous ${\varphi:\|\mathsf{L}\|\to\|\mathsf{K}^{\prime}\|}$, which extends
to $\|{\mathsf{L}}^{2}_{\Delta}\|\to\widehat{\mathsf{K}}$ (by applying
$\varphi$ on each coordinate). Then $\mathsf{K}$ is not $d$-representable (and
in fact not even the nerve of a good cover in $\mathbb{R}^{d}$).

###### Proof.

Assume by contradiction that $\mathsf{K}$ is $d$-representable or the nerve of
a good cover. In particular there exists a faithful
$f:\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{d}$ (by Theorem 3.3 or Lemma 3.9). Of
course $f\circ\varphi:\|\mathsf{L}\|\to\mathbb{R}^{d}$ is continuous, so there
exist $a,b\in\|\mathsf{L}\|$ supported by disjoint faces, such that
$f(\varphi(a))=f(\varphi(b))$. But now
$(a,b)\in\|{\mathsf{L}}^{2}_{\Delta}\|$, implying
$\left(\varphi(a),\varphi(b)\right)\in\widehat{\mathsf{K}}$ and contradicting
Lemma 3.5. ∎

######  Remark 3.14.

Theorem 3.13 is applicable in a wide variety of cases. In some cases,
$\widehat{\mathsf{K}}$ is precisely the deleted product of
$\mathsf{L}=\mathsf{K}^{\prime}$, so $\varphi$ is just the identity map (for
example when $\mathsf{K}^{\prime}$ is a graph, see Theorem 4.7). In Lemma 3.15
we prove that every $\mathsf{L}$ has some $\mathsf{K}$ for which the theorem
applies, so every such non-embeddability result translates to a non-
representability result for some $\mathsf{K}$.
In another example due to Tancer [10], $\mathsf{L}$ is the $d$-skeleton of the
$\left(2d+2\right)$-dimensional simplex, and $\mathsf{K}$ is the barycentric
subdivision of $\mathsf{L}$. The Van Kampen-Flores Theorem states that any
continuous $\|\mathsf{L}\|\to\mathbb{R}^{2d}$ has some two disjoint
$\mathsf{L}$-faces whose images intersect. Tancer’s proof that $\mathsf{K}$ is
not $2d$-representable is essentially equivalent to an application of Theorem
3.13 with an implicit construction of
$\varphi:\|\mathsf{L}\|\to\|\mathsf{K}^{\prime}\|$.

We now want to show that the value $\left(2\dim\mathsf{K}^{\prime}+1\right)$
in Theorem 3.12 is the best possible, in other words there exist complexes
$\mathsf{K}$ such that $\mathsf{K}^{\prime}$ has dimension $d$ but
$\mathsf{K}$ is not $2d$-representable. We know (from Van Kampen-Flores) of a
suitable $\mathsf{L}$, so to apply Theorem 3.13 we prove:

######  Lemma 3.15.

Let $\mathsf{L}$ be a simplicial complex. Then there exists a simplicial
complex $\mathsf{K}$ and an isomorphism
$\varphi:\mathsf{L}\to\mathsf{K}^{\prime}$, which extends (by applying
$\varphi$ on each coordinate) to an equivariant
$\|{\mathsf{L}}^{2}_{\Delta}\|\to\widehat{\mathsf{K}}$.

###### Proof.

Assume that $V(\mathsf{L})=\left[n\right]$. Let $\mathsf{K}$ be the nerve of
$\mathsf{L}$. Explicitly,

$\mathsf{K}=\left\\{\alpha\subseteq\mathsf{L}\ :\
\bigcap\alpha\neq\emptyset\right\\}.$

For every $i\in\left[n\right]$ denote

$\gamma_{i}=\left\\{F\in\mathsf{L}\ :\ i\in F\right\\},$

and for every $I\subseteq\left[n\right]$ denote

$\mathcal{C}_{I}=\left\\{\gamma_{i}\ :\ i\in I\right\\}.$

Clearly
$\mathsf{m}_{\mathsf{K}}=\left\\{\gamma_{1},\ldots,\gamma_{n}\right\\}$. Our
isomorphism is obtained by applying $i\mapsto\gamma_{i}$ to map the vertices.
To prove that this induces an isomorphism $\mathsf{L}\to\mathsf{K}^{\prime}$,
we must show that every $I\subseteq\left[n\right]$ has $I\in\mathsf{L}$ if and
only if $\mathcal{C}_{I}\in\mathsf{K}^{\prime}$.

Indeed, if $I\in\mathsf{L}$ then $I\in\gamma_{i}$ for every $i\in I$, thus
$\bigcap_{i\in I}\gamma_{i}\neq\emptyset$ proving
$\mathcal{C}_{I}\in\mathsf{K}^{\prime}$. On the other hand, if
$\mathcal{C}_{I}\in\mathsf{K}^{\prime}$ then
$\cap\mathcal{C}_{I}\neq\emptyset$, so let $F\in\bigcap_{i\in I}\gamma_{i}$.
In particular $F\in\mathsf{L}$. Furthermore, $F\in\gamma_{i}$ implies $i\in
F$, and this holds for every $i\in I$, proving $I\subseteq F$ and therefore
$I\in\mathsf{L}$.

We prove the remaining claim on the configuration spaces by showing that for
any $F,G\in\mathsf{L}$, we have

$F\cap
G\neq\emptyset\quad\iff\quad\left(\bigcap\mathcal{C}_{F}\right)\cup\left(\bigcap\mathcal{C}_{G}\right)\in\mathsf{K}\
.$

Suppose that $x\in F\cap G$. Then $\bigcap\mathcal{C}_{F}=\bigcap_{i\in
F}\gamma_{i}\subseteq\gamma_{x}$, and similarly
$\bigcap\mathcal{C}_{G}\subseteq\gamma_{x}$, thus
$\left(\bigcap\mathcal{C}_{F}\right)\cup\left(\bigcap\mathcal{C}_{G}\right)\subseteq\gamma_{x}\in\mathsf{K}$.
On the other hand, suppose that
$\alpha:=\left(\bigcap\mathcal{C}_{F}\right)\cup\left(\bigcap\mathcal{C}_{G}\right)\in\mathsf{K}$.
For every $i\in F$ we have $F\in\gamma_{i}$, thus
$F\in\left(\bigcap\mathcal{C}_{F}\right)$ and $F\in\alpha$. Similarly
$G\in\alpha$. Note that $\alpha\in\mathsf{K}$ implies
$\bigcap\alpha\neq\emptyset$, and we deduce
$\emptyset\neq\bigcap\alpha\subseteq F\cap G$. ∎

######  Corollary 3.16.

For every $d\in\mathbb{N}$, there exist simplicial complexes $\mathsf{K}$ such
that $\dim\mathsf{K}^{\prime}=d$, and $\mathsf{K}$ is not $2d$-representable.

###### Proof.

Let $\mathsf{L}$ be the $d$-skeleton of the $\left(2d+2\right)$-dimensional
simplex, and apply Lemma 3.15 on $\mathsf{L}$ to obtain a simplicial complex
$\mathsf{K}$ and corresponding
$\varphi:\|\mathsf{L}\|\to\|\mathsf{K}^{\prime}\|$. The Van Kampen-Flores
Theorem (see [7, Theorem 5.1.1]) states that any continuous
$\|\mathsf{L}\|\to\mathbb{R}^{2d}$ has some two disjoint $\mathsf{L}$-faces
whose images intersect. We deduce from Theorem 3.13 that $\mathsf{K}$ is not
$2d$-representable. ∎

##  4 Results on $d$-Matoušek complexes

We now turn our attention to the $d$-Matoušek property, and begin to explore
some cases where it fully captures $d$-representability, and others where it
fails to do so.

In the classification of $1$-representable complexes, Lekkerkerker and Boland
[5] introduced the notion of _asteroidal triples_ in graphs, and proved that
$1$-skeletons of a $1$-representable complexes are precisely graphs which have
no asteroidal triple and no induced cycle of length $\geq 4$. Before proving
that every $1$-representable complex is $1$-Matoušek, we extend the notion of
an asteroidal triple.

###  4.1 Asteroidal maps

Recall that $\Delta_{d+1}^{(d)}$ denotes the $d$-dimensional skeleton of a
$(d+1)$-dimensional simplex.

######  Definition 4.1.

Let $\mathsf{K}$ be a simplicial complex, and let $d\in\mathbb{N}$. A _$d$
-asteroidal_ map into $\mathsf{K}$ is a continuous
$f:\|\Delta_{d+1}^{(d)}\|\to\|\mathsf{K}\|$, such that for every
$x,y\in\|\Delta_{d+1}^{(d)}\|$ with
$\textnormal{supp}(x)\cap\textnormal{supp}(y)=\emptyset$, we have

$\textnormal{supp}\left(f(x)\right)\cup\textnormal{supp}\left(f(y)\right)\notin\mathsf{K}\
.$

To see how this generalizes the notion of asteroidal triples (when $d=1$),
note that any $3$ vertices form an asteroidal triple (as defined in [5]) if
they are the image of the $3$ vertices of $\Delta_{2}^{(1)}$ under a
$1$-asteroidal map. By allowing the vertices of $\Delta_{2}^{(1)}$ to be
mapped to general points in $\|\mathsf{K}\|$ (not necessarily vertices), we
find that a $1$-asteroidal map into $\mathsf{K}$ exists whenever $\mathsf{K}$
is not $1$-representable.

Additionally, we obtain a $d$-dimensional analogue of asteroidal triples, and
a property which implies that a complex is not $d$-representable: we can
define $d+2$ points in $\|\mathsf{K}\|$ to be in _asteroidal position_ if they
are the image of the vertices of $\Delta_{d+1}^{(d)}$ under a $d$-asteroidal
map. We now prove that any complex with $d+2$ points in asteroidal position is
not $d$-Matoušek (and therefore by Proposition 3.7 not $d$-representable):

######  Proposition 4.2.

Let $f:\|\Delta_{d+1}^{(d)}\|\to\|\mathsf{K}\|$ be a $d$-asteroidal map, for
some simplicial complex $\mathsf{K}$ and $d\in\mathbb{N}$. Then $\mathsf{K}$
is not $d$-Matoušek.

###### Proof.

Denote $\mathsf{L}=\Delta_{d+1}^{(d)}$, and define
$\varphi:\iota_{\mathsf{K}}\circ f:\|\mathsf{L}\|\to\|\mathsf{K}^{\prime}\|$,
where $\iota_{\mathsf{K}}:\|\mathsf{K}\|\to\|\mathsf{K}^{\prime}\|$ is the map
from Lemma 3.4, which satisfies
$\textnormal{supp}(z)\subseteq\bigcap\textnormal{supp}\left(\iota_{\mathsf{K}}(z)\right)$
for every $z\in\|\mathsf{K}\|$. This implies that for every
$x,y\in\|\mathsf{L}\|$ with disjoint supports, we have

$\mathsf{K}\not\ni\textnormal{supp}(f(x))\cup\textnormal{supp}(f(y))\subseteq\left(\bigcap\textnormal{supp}(\varphi(x))\right)\cup\left(\bigcap\textnormal{supp}(\varphi(y))\right),$

so by applying $\varphi$ on each coordinate, we obtain an equivariant map
$\|{\mathsf{L}}^{2}_{\Delta}\|\to\widehat{\mathsf{K}}$. Our claim now follows
from Proposition 2.4 and the fact that

$\|{\mathsf{L}}^{2}_{\Delta}\|=\|{\left(\Delta_{d+1}\right)}^{2}_{\Delta}\|\cong
S^{d}$

as a $\mathbb{Z}_{2}$-space. ∎

######  Remark 4.3.

Note that we may also prove that an asteroidal map into $\mathsf{K}$ implies
that $\mathsf{K}$ is not $d$-representable by using the more general statement
given in Theorem 3.13. We have our complex $\mathsf{L}=\Delta_{d+1}^{(d)}$ and
a continuous $\varphi=\iota_{\mathsf{K}}\circ
f:\|\mathsf{L}\|\to\|\mathsf{K}^{\prime}\|$. The fact that every continuous
$\|\mathsf{L}\|\to\mathbb{R}^{d}$ identifies some pair from disjoint faces is
simply the topological version of Radon’s theorem [2].

###  4.2 $1$-representable complexes

In this subsection we prove:

######  Theorem 4.4.

A simplicial complex is $1$-representable if and only if it is $1$-Matoušek.

This gives us a topological description of $1$-representable complexes. To
illustrate, consider Figure 2 below:

123456123142536(25,36)(25,123)(25,14)(123,14)(36,14)(36,123)(36,25)(123,25)(14,25)(14,123)(14,36)(123,36)

Figure 2

From left to right: a simplicial complex $\mathsf{K}$ which is not
$1$-representable despite being contractible, the dual complex
$\mathsf{K}^{\prime}$, and the configuration space $\widehat{\mathsf{K}}$. The
equivariant copy of $S^{1}$ within $\widehat{\mathsf{K}}$ shows that
$\mathsf{K}$ is not $1$-Matoušek. Our proof implies that this holds in
general: for any non $1$-representable complex $\mathsf{K}$, we have an
equivariant $S^{1}\to\widehat{\mathsf{K}}$.

Note we don’t give a new proof of [5], but rely on their main result:

######  Theorem 4.5 (Lekkerkerker, Boland).

Let $\mathsf{K}\subseteq 2^{V}$ be a simplicial complex, and let $\mathsf{G}$
be the $1$-skeleton of $\mathsf{K}$. Then $\mathsf{K}$ is $1$-representable if
and only if the following three conditions are satisfied:

  1. 1.

$\mathsf{K}$ is the clique complex of $\mathsf{G}$: whenever $I\subseteq V$
and $\left\\{u,v\right\\}\in\mathsf{G}$ for all $u,v\in I$, we have
$I\in\mathsf{K}$.

  2. 2.

$\mathsf{G}$ has no induced cycle of length $n\geq 4$.

  3. 3.

There is no $1$-asteroidal map $\|\Delta_{2}^{(1)}\|\to\|\mathsf{G}\|$ which
maps vertices to vertices.

###### Proof.

The necessity of the 1st condition is due to Helly’s theorem. The remaining
conditions are equivalent to [5, Theorem 3], noting that any $3$ vertices in
$\mathsf{G}$ form an asteroidal triple (as defined in [5]) if they are the
image of the $3$ vertices of $\Delta_{2}^{(1)}$ under some $1$-asteroidal map.
∎

Next, we show that the entire theorem can be rephrased in terms of asteroidal
maps:

######  Proposition 4.6.

Let $\mathsf{K}\subseteq 2^{V}$ be a simplicial complex which is not
$1$-representable. Then there exists a $1$-asteroidal map
$\|\Delta_{2}^{(1)}\|\to\|\mathsf{K}\|$.

###### Proof.

Note that $\|\Delta_{2}^{(1)}\|$ is just a triangle (without the interior), so
constructing a map $\|\Delta_{2}^{(1)}\|\to\|\mathsf{K}\|$ requires specifying
$3$ points in $\|\mathsf{K}\|$ (one for each vertex), and $3$ paths in
$\|\mathsf{K}\|$ (one for each edge, connecting every pair of vertices). We
will specify vertices and paths in $\textnormal{sd}(\mathsf{K})$. That is, we
will find $F_{1},F_{2},F_{3}\in\mathsf{K}$, and
$\alpha_{12},\alpha_{13},\alpha_{23}\subseteq\mathsf{K}$, such that each
$\alpha_{jk}$ is a path in $\textnormal{sd}(\mathsf{K})$ from $F_{j}$ to
$F_{k}$.
In other words,
$\alpha_{jk}=\left\\{F_{jk}^{0},F_{jk}^{1},\ldots,F_{jk}^{n}\right\\}$, such
that $F_{jk}^{0}=F_{j}$, $F_{jk}^{n}=F_{k}$, and for every $m=1,\ldots,n$
either $F_{jk}^{m-1}\subseteq F_{jk}^{m}$ or $F_{jk}^{m}\subseteq
F_{jk}^{m-1}$.
A point in $\|\textnormal{sd}(\mathsf{K})\|$ supported by
$\left\\{F,G\right\\}$ corresponds to a point in $\|\mathsf{K}\|$ supported by
$F\cup G$. Thus our specified
$F_{1},F_{2},F_{3},\alpha_{12},\alpha_{13},\alpha_{23}$ correspond to a
$1$-asteroidal map into $\mathsf{K}$, if whenever $i=1,2,3$ and
$F\in\alpha_{jk}$ such that $i\neq j,k$, we have

$F\cup F_{i}\notin\mathsf{K}.$

Since $\mathsf{K}$ is not $1$-representable, from Theorem 4.5 we have at least
one of the following:

  1. 1.

There exists $I\subseteq V$ such that $\left\\{u,v\right\\}\in\mathsf{K}$ for
all $u,v\in I$, but $I\notin\mathsf{K}$.

  2. 2.

$\mathsf{K}$ has an induced cycle of length $n\geq 4$.

  3. 3.

There exists a $1$-asteroidal map $\|\Delta_{2}^{(1)}\|\to\|\mathsf{K}\|$
which maps vertices to vertices.

If (1) holds, let $I\subseteq V$ be an inclusion-minimal set satisfying the
condition. Clearly we have $|I|\geq 3$, so let $v_{1},v_{2},v_{3}\in I$ be
distinct vertices. Now define $F_{i}=I\setminus\left\\{v_{i}\right\\}$ for
$i=1,2,3$, and $\alpha_{jk}=\left\\{F_{j},F_{j}\cap F_{k},F_{k}\right\\}$ for
$jk=23,13,12$. For every $F\in\alpha_{jk}$ we have $v_{i}\in F$ (where $i\neq
j,k$), therefore $F\cup F_{i}=I\notin\mathsf{K}$ and the requirement for
asteroidality is fulfilled.
If (2) holds, let $(v_{0},v_{1},v_{2},\ldots,v_{n}=v_{0})$ be an induced cycle
in $\mathsf{K}$ of length $n\geq 4$: the vertices $v_{1},\ldots,v_{n}\in V$
are distinct, $\left\\{v_{i-1},v_{i}\right\\}\in\mathsf{K}$ for
$i=1,2,\ldots,n$, and there are no other faces in $\mathsf{K}$ among
$\left\\{v_{1},\ldots,v_{n}\right\\}$. Let

$F_{1}=\left\\{v_{0},v_{1}\right\\}\quad,\quad
F_{2}=\left\\{v_{1},v_{2}\right\\}\quad,\quad
F_{3}=\left\\{v_{2},v_{3}\right\\},$

and let

$\begin{split}\alpha_{12}&=\left\\{\left\\{v_{0},v_{1}\right\\},\left\\{v_{1}\right\\},\left\\{v_{1},v_{2}\right\\}\right\\},\\\
\alpha_{23}&=\left\\{\left\\{v_{1},v_{2}\right\\},\left\\{v_{2}\right\\},\left\\{v_{2},v_{3}\right\\}\right\\},\\\
\alpha_{13}&=\left\\{\left\\{v_{n},v_{1}\right\\},\left\\{v_{n}\right\\},\left\\{v_{n-1},v_{n}\right\\},\left\\{v_{n-1}\right\\},\left\\{v_{n-2},v_{n-1}\right\\},\ldots,\left\\{v_{3}\right\\},\left\\{v_{2},v_{3}\right\\}\right\\}.\end{split}$

Note that no face $F$ in $\alpha_{jk}$ is a subset of $F_{i}$ (where $i\neq
j,k$). Therefore $F\cup F_{i}$ contains $3$ or $4$ vertices, and in particular
$\left\\{v_{x},v_{y}\right\\}\subseteq F\cup F_{i}$ for some non-consecutive
$x,y$ (here we must require $n\geq 4$). Since our cycle is induced, we must
have $\left\\{v_{x},v_{y}\right\\}\notin\mathsf{K}$, therefore $F\cup
F_{i}\notin\mathsf{K}$, proving that the requirement for asteroidality is
fulfilled.
If (3) holds there is nothing more to prove, so our proof is complete. ∎

We can now deduce Theorem 4.4:

###### Proof of Theorem 4.4.

Every $1$-representable complex is $1$-Matoušek (from Proposition 3.7), so it
remains to show that if $\mathsf{K}$ is not $1$-representable, it isn’t
$1$-Matoušek either. This immediately follows from Propositions 4.2 and 4.6. ∎

###  4.3 Representability of graphs and co-graphs

If $\mathsf{K}$ is a complex such that $\dim\mathsf{K}^{\prime}=1$ (i.e.
$\mathsf{K}^{\prime}$ is a graph), we informally say that $\mathsf{K}$ a co-
graph. In these cases, we find that $d$-Matoušek and $d$-representability are
identical:

######  Theorem 4.7.

Let $\mathsf{K}$ be a simplicial complex, such that
$\dim\mathsf{K}^{\prime}=1$. Then for every $d\in\mathbb{N}$, $\mathsf{K}$ is
$d$-representable if and only if it is $d$-Matoušek.

The first case not covered by previously stated results is when $d=2$, and we
will show that $2$-representability and $2$-Matoušek are both equivalent to
the planarity of $\dim\mathsf{K}^{\prime}$. We break this down into a number
of lemmas. First, we prove that our configuration space is simply the deleted
product of $\mathsf{K}^{\prime}$:

######  Lemma 4.8.

Let $\mathsf{K}$ be a simplicial complex such that
$\dim\mathsf{K}^{\prime}=1$, i.e. $\mathsf{G}:=\mathsf{K}^{\prime}$ is a
graph. Then $\widehat{\mathsf{K}}=\|{\mathsf{G}}^{2}_{\Delta}\|$.

###### Proof.

We will prove that for any non-empty faces $\alpha,\beta\in\mathsf{G}$ we have

$\alpha\cap\beta\neq\emptyset\ \iff\
\left(\bigcap\alpha\right)\cup\left(\bigcap\beta\right)\in\mathsf{K}.$

This is straightforward: if $J\in\alpha\cap\beta$, then $(\cap\alpha)\subseteq
J$ and also $(\cap\beta)\subseteq J$, so $(\cap\alpha)\cup(\cap\beta)\subseteq
J\in\mathsf{K}$ implying $(\cap\alpha)\cup(\cap\beta)\in\mathsf{K}$.
On the other hand, suppose $(\cap\alpha)\cup(\cap\beta)\in\mathsf{K}$. Then
$(\cap\alpha)\cup(\cap\beta)\subseteq J$ for some $\mathsf{K}$-maximal face
$J$. In particular $\cap\alpha\subseteq J$. Since $\mathsf{G}$ is a graph,
$\alpha$ is either a singleton or an edge. If $\alpha=\left\\{J_{1}\right\\}$,
then $\cap\alpha\subseteq J$ implies $J_{1}\subseteq J$. But
$J_{1}\in\mathsf{m}_{\mathsf{K}}$ is a maximal face in $\mathsf{K}$, so
$J=J_{1}$. If $\alpha=\left\\{J_{1},J_{2}\right\\}$ for $J_{1}\neq J_{2}$,
then $\emptyset\neq J_{1}\cap J_{2}\subseteq J$ (recall that
$\alpha\in\mathsf{G}=\mathsf{K}^{\prime}$ thus $\cap\alpha\neq\emptyset$),
therefore $\emptyset\neq J_{1}\cap J_{2}=J\cap J_{1}\cap J_{2}$ implying
$\left\\{J,J_{1},J_{2}\right\\}\in\mathsf{K}^{\prime}=\mathsf{G}$. But
$\mathsf{G}$ is a graph, so in fact $\left\\{J,J_{1},J_{2}\right\\}$ has at
most $2$ distinct elements, and $J=J_{1}$ or $J=J_{2}$. In every case, we have
$J\in\alpha$. The same argument applies for $\beta$, therefore $J\in\beta$ and
$\alpha\cap\beta\neq\emptyset$ as required. ∎

This allows us to prove our theorem by relying on the following criterion for
graph planarity from [8, “A graph planarity criterion” p. 597]:

######  Lemma 4.9 (Sarkaria).

Let $\mathsf{G}$ be a graph. Then $\mathsf{G}$ is planar iff
$\textnormal{ind}_{\mathbb{Z}_{2}}(\|{\mathsf{G}}^{2}_{\Delta}\|)\leq 1$.

###### Proof of Theorem 4.7.

By Theorem 3.12, every $\mathsf{K}$ with $\dim\mathsf{K}^{\prime}=1$ is
$3$-representable (and thus also $3$-Matoušek by Proposition 3.7). This
clearly implies the same for any $d\geq 3$. For $d=1$ we apply Theorem 4.4, so
the only remaining case is $d=2$. For this we denote
$\mathsf{G}=\mathsf{K}^{\prime}$, and prove that the following our equivalent:

  1. 1.

$\mathsf{G}$ is planar.

  2. 2.

$\mathsf{K}$ is $2$-representable.

  3. 3.

$\mathsf{K}$ is $2$-Matoušek.

To prove (1)$\implies$(2), note that any planar graph can by drawn in
$\mathbb{R}^{2}$ using straight line segments, so if $\mathsf{G}$ is planar we
obtain a linear injective map $\|\mathsf{G}\|\to\mathbb{R}^{2}$, and conclude
from Theorem 3.12 that $\mathsf{K}$ is $2$-representable. The implication
(2)$\implies$(3) is simply Proposition 3.7 for $d=2$. To prove
(3)$\implies$(1), by Lemma 4.8 we have
$\widehat{\mathsf{K}}=\|{\mathsf{G}}^{2}_{\Delta}\|$. If $\mathsf{K}$ is
$2$-Matoušek then $\textnormal{ind}_{\mathbb{Z}_{2}}(\widehat{\mathsf{K}})\leq
1$, and Lemma 4.9 implies that $\mathsf{G}$ is planar. ∎

In contrast to our previous results, for a simplicial complex $\mathsf{K}$
which is a a graph, being $2$-representable and being $2$-Matoušek are not
identical properties. Some previous results regarding dimension are relevant:
we know that $\mathsf{K}$ is $1$-representable iff it is $1$-Matoušek (from
Theorem 4.4). Every simplicial complex $\mathsf{K}$ is
$(2\dim\mathsf{K}+1)$-representable (see [11]), therefore $\mathsf{K}$ is
$d$-representable and $d$-Matoušek, for every $d\geq 3$. And of course every
$2$-representable complex is also $2$-Matoušek. But the converse fails: there
exist graphs $\mathsf{K}$ which are $2$-Matoušek but not $2$-representable:
see Example 4.13.

There are, however, non-trivial examples of graphs which are not $2$-Matoušek.
For example, let $\mathsf{K}$ be the barycentric subdivision of some non-
planar graph $\mathsf{G}$. Consider the canonical map
$\varphi:\|\mathsf{G}\|\to\|\mathsf{K}^{\prime}\|$ which is the composition of
the homeomorphism $\|\mathsf{G}\|\to\|\mathsf{K}\|$ and
$\iota_{\mathsf{K}}:\|\mathsf{K}\|\to\|\mathsf{K}^{\prime}\|$ given in Lemma
3.4. We can verify that applying $\varphi$ on each coordinate gives us an
equivariant map $\|{\mathsf{G}}^{2}_{\Delta}\|\to\widehat{\mathsf{K}}$. Since
$\mathsf{G}$ is not planar, Lemma 4.9 implies that
$\textnormal{ind}_{\mathbb{Z}_{2}}(\widehat{\mathsf{K}})=2$, therefore
$\mathsf{K}$ is not $2$-Matoušek by Proposition 2.4.

###  4.4 $d$-Matoušek and $d$-collapsibility

As we have seen, every $d$-representable complex is both $d$-collapsible and
$d$-Matoušek (for every $d\in\mathbb{N}$). This raises the question of the
relation between $d$-collapsibility and $d$-Matoušek. We give examples that
prove the two properties are independent (neither implies the other), and even
together don’t coincide with $d$-representable complexes. In other words, we
give some examples of simplicial complexes which are $d$-collapsible but not
$d$-Matoušek, other complexes which are $d$-Matoušek but not $d$-collapsible,
and others which are both $d$-Matoušek and $d$-collapsible but not
$d$-representable.

####  4.4.1 Complexes that are $d$-collapsible but not $d$-Matoušek

There exist complexes which are $d$-collapsible but not $d$-Matoušek even for
$d=1$: simply take a complex which is $1$-collapsible but not
$1$-representable. Such a complex is $1$-Matoušek by Theorem 4.4.

Using asteroidal maps, we may generalize one such complex to arbitrary
dimension:

######  Example 4.10.

For any $d\in\mathbb{N}$, let $\mathsf{L}$ have vertices $V=0,1,\ldots,d+2$,
and maximal faces given by

$\mathsf{m}_{\mathsf{L}}=\left\\{F\subseteq V\ :\ |F|=d+1\ ,\ 0\in
F\right\\}.$

Note that $\mathsf{L}$ is the cone of $\Delta_{d+1}^{(d-1)}$. This allows us
to define a map $g:\|\Delta_{d+1}^{(d)}\|\to\|\mathsf{L}\|$: we take $g$ as
the identity on any simplex of dimension $\leq d-1$, and extend to
$d$-dimensional simplices by contracting the boundary into the vertex $0$.

Now let $\mathsf{K}$ be the barycentric subdivision of $\mathsf{L}$. It is
straightforward to verify that $\mathsf{K}$ is $d$-collapsible, and that by
composing $g$ with the canonical homeomorphism between $\mathsf{L}$ and
$\mathsf{K}$ we obtain an asteoridal map
$f:\|\Delta_{d+1}^{(d)}\|\to\|\mathsf{K}\|$. Thus by Proposition 4.2,
$\mathsf{K}$ is not $d$-Matoušek. For $d=1$, the complex $\mathsf{K}$ is the
(unique) minimal non-$1$-representable tree.

####  4.4.2 Complexes that are $d$-Matoušek but not $d$-collapsible

######  Example 4.11.

In [9], Tancer constructed a good cover in $\mathbb{R}^{2}$, which has a nerve
$\mathsf{K}$ that isn’t $2$-collapsible. We know that $\mathsf{K}$ is
$2$-Matoušek from Corollary 3.10.

####  4.4.3 Complexes that are $d$-Matoušek and $d$-collapsible, but not
$d$-representable

The previous examples show that $d$-Matoušek and $d$-collapsibility are
“independent” properties of $d$-representable complexes. Even combining both
notions, we still remain with some complexes which are not $d$-representable,
even when $d=2$. We give two examples of such complexes, one which is the
nerve of a good cover in $\mathbb{R}^{2}$, and one which is not.

######  Example 4.12.

Let $\mathsf{K}$ be the triangulation of the Mobius strip, with vertices
$V=[6]$, and maximal faces

$\mathsf{m}_{\mathsf{K}}\ =\ \left\\{124\ ,\ 125\ ,\ 156\ ,\ 235\ ,\ 236\ ,\
246\ ,\ 345\ ,\ 456\right\\}.$

This complex was given by Eckhoff in [3] as an example of a (strongly)
$2$-collapsible complex that is not $2$-representable. The complex is the
nerve of a good cover in $\mathbb{R}^{2}$, outlined in Figure 3:

123444456666

Figure 3

The sets $U_{1},\ldots,U_{6}\subseteq\mathbb{R}^{2}$ should be open and
slightly overlap the boundaries indicated in the figure. We deduce that
$\mathsf{K}$ is $2$-Matoušek from Corollary 3.10.

######  Example 4.13.

We will now give an example of a graph $\mathsf{K}$ which is $2$-collapsible
and $2$-Matoušek, but not the nerve of any good cover in $\mathbb{R}^{2}$.
A graph $\mathsf{G}$ is a _string graph_ if it is the nerve of a collection of
curves in $\mathbb{R}^{2}$. In [4], Kratochvíl and Matoušek constructed string
graphs for which the number of intersection points between any such collection
of curves in $\mathbb{R}^{2}$ is exponential in the number of curves. One
instance of this construction will be our example. Consider the graph
$\mathsf{G}$ drawn in $\mathbb{R}^{2}$ in [4] (some vertices have been
renamed), reproduced as Figure 4 below:

$c_{0}$$c_{0}$$c_{1}$$c_{2}$$c_{3}$$c_{4}$$l$$u_{2}$$u_{1}$$v_{2}$$u_{0}$$v_{1}$$r$$d_{0}$$a$$b$$d_{1}$

Figure 4

Note that $\mathsf{G}$ includes the edges drawn using dashed lines
($u_{1}v_{1}$ and $u_{2}v_{2}$) as well as those drawn with solid lines.

We define our simplicial complex $\mathsf{K}$ as the barycentric subdivision
of $\mathsf{G}$, with the addition of pairs of $\mathsf{G}$-edges which
intersect in Figure 4. In other words $V(\mathsf{K})=V(\mathsf{G})\cup
E(\mathsf{G})$, and the maximal faces of $\mathsf{K}$ are all pairs of the
form $\left\\{x,xy\right\\}$ where $xy$ is an edge in $\mathsf{G}$, along with
$4$ additional pairs:

$\ \
\left\\{ab,u_{1}v_{1}\right\\}\quad,\quad\left\\{ab,u_{2}v_{2}\right\\}\quad,\quad\left\\{u_{0}v_{1},u_{2}v_{2}\right\\}\quad,\quad\left\\{v_{1}y,u_{2}v_{2}\right\\}\
.$

Note that $\mathsf{K}$ is itself a graph (maximal faces of $\mathsf{K}$ are
$1$-dimensional), so $\mathsf{K}$ is trivially $2$-collapsible. Our next goal
is to prove that $\mathsf{K}$ is $2$-Matoušek (in fact we show that there
exists a faithful map $\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{2}$), and that
$\mathsf{K}$ is not the nerve of any good cover in $\mathbb{R}^{2}$. Both of
these facts can be deduced by examining the drawing of $\mathsf{G}$ in
$\mathbb{R}^{2}$.

To ease notation, for every $x\in V(\mathsf{G})$, let
$\overline{x}\in\mathbb{R}^{2}$ denote the position of $x$ indicated by the
drawing, and for every edge $xy\in E(\mathsf{G})$ let
$\overline{xy}\subseteq\mathbb{R}^{2}$ denote all points on the indicated path
from $\overline{x}$ to $\overline{y}$.

The drawing shows that $\mathsf{K}$ is “almost” the nerve of a good cover in
$\mathbb{R}^{2}$: for every $x\in V(\mathsf{G})$, let
$U_{x}\subseteq\mathbb{R}^{2}$ be an open ball of radius $\varepsilon>0$
centered around $\overline{x}$. For every $e=xy\in E(\mathsf{G})$, let
$U_{e}\subseteq\mathbb{R}^{2}$ be the $\frac{\varepsilon}{10}$-neighborhood of
the points in $\overline{xy}\setminus\left(U_{x}\cup U_{y}\right)$. We may
choose $\varepsilon$ to be sufficiently small as not to introduce unwanted
intersections (i.e. intersections among collections not in $\mathsf{K}$). For
example, $U_{ab}\cap U_{b}$ and $U_{bu_{0}}\cap U_{b}$ must be nonempty, but
$U_{ab}\cap U_{bu_{0}}=\emptyset$. Clearly the _only_ intersection of sets
among the collection $\left\\{U_{w}\right\\}_{w\in V(\mathsf{K})}$ which is
neither empty nor contractible is $U_{ab}\cap U_{u_{2}v_{2}}$, which is a
disjoint union of $2$ contractible sets. Nevertheless, the proof of Lemma 3.9
still applies (see Remark 3.11), we obtain a faithful map
$\|\mathsf{K}^{\prime}\|\to\mathbb{R}^{2}$, and $\mathsf{K}$ is $2$-Matoušek
by Proposition 3.6.

Finally, assume by contradiction that $\mathsf{K}$ is the nerve of a
collection $\left\\{U_{w}\right\\}_{w\in V(\mathsf{K})}$ which is a good cover
in $\mathbb{R}^{2}$. We draw $\mathsf{G}$ in $\mathbb{R}^{2}$, (that is we
construct a map $\|\mathsf{G}\|\to\mathbb{R}^{2}$), such that every vertex
$x\in V(\mathsf{G})$ is mapped to some $\overline{x}\in U_{x}$, and every edge
$e=xy\in E(\mathsf{G})$ is drawn on as a piecewise linear curve
$\overline{xy}\subseteq U_{x}\cup U_{e}\cup U_{y}$.

Let $\mathsf{H}$ be the graph obtained from $\mathsf{G}$ by removing the edges
$u_{1}v_{1}$ and $u_{2}v_{2}$ (so $\mathsf{H}$ is the graph drawn in Figure 4
without dashed edges). Note that the induced drawing of $\mathsf{H}$ must be
planar, and $\mathsf{H}$ has the same faces in every planar drawing (so the
faces are the same as in Figure 4). Specifically, $\overline{u_{1}v_{1}}$ must
cross $\overline{ab}$, and $U_{ab}\setminus U_{u_{1}v_{1}}$ must have (at
least) two connected components (one bounded in the interior of the curve
$\overline{u_{1}}\overline{v_{1}}\overline{u_{0}}\overline{v_{2}}\overline{u_{1}}$,
the other in the exterior). Finally, observe that $\overline{u_{2}v_{2}}$ must
intersect $\overline{ab}$ at least twice (once in each connected component) as
in Figure 4 - thus $U_{ab}\cap U_{u_{2}v_{2}}$ is not path-connected,
contradicting our assumption.

##  5 Further questions

######  Question 5.1.

Are there $d$-Matoušek complexes that aren’t $d$-Leray?

We know of the complex in Example 4.11 due to Tancer, which is $2$-Matoušek
but not $2$-collapsible. But it is unknown to the author whether there exists
a simplicial complex which is $d$-Matoušek but not $d$-Leray, even for $d=2$.
By Remark 3.8, the question is equivalent to: does there exist a $d$-Matoušek
complex with non-trivial $m$-homology for some $m\geq d$?

Furthermore, if such complexes exist, can we find a property similar to
$d$-Matoušek, which also implies that a complex is $d$-Leray?

######  Question 5.2.

Under what conditions is a $d$-Matoušek complex also $d$-representable?

For example, are there some conditions on $d$ and $\dim\mathsf{K}$ or
$\dim\mathsf{K}^{\prime}$? A theorem of Weber from 1967 (see [1, chapter 7])
states that if $2d\geq 3\left(\dim\mathsf{L}+1\right)$ and there exists an
equivariant map $\|{\mathsf{L}}^{2}_{\Delta}\|\to S^{d-1}$, then there also
exists an embedding $\|\mathsf{L}\|\to\mathbb{R}^{d}$. This implies that for
simplicial complexes $\mathsf{K}$ that satisfy
$\widehat{\mathsf{K}}=\|{\left(\mathsf{K}^{\prime}\right)}^{2}_{\Delta}\|$,
for any $d$ such that $2d\geq 3\left(\dim\mathsf{K}^{\prime}+1\right)$, we
have: $\mathsf{K}$ is $d$-Matoušek iff $\mathsf{K}$ is the nerve of a good
cover in $\mathbb{R}^{d}$. Such complexes exist by Lemma 3.15, however there
are cases where $\widehat{\mathsf{K}}$ is substantially “smaller” than
$\|{\left(\mathsf{K}^{\prime}\right)}^{2}_{\Delta}\|$.

######  Question 5.3.

What new results can we obtain by using more general methods?

While stronger topological methods exist (see [13]), we choose to use the
Borsuk–Ulam method as it doesn’t require as much intensive topology (the
Borsuk–Ulam theorem is the only topological result needed), while still
proving new results. The disadvantage is that our results might not be as
strong as they could be. We therefore welcome improvements to the “correct”
definition or a $d$-Matoušek complex. In vague terms, if a $d$-Leray complex
is one which has no homological reason not to be $d$-representable, a
$d$-Matoušek complex as defined in this paper is one for which a
$d$-representation lifted to the configuration space of pairs doesn’t
contradict the Borsuk–Ulam theorem. The true spirit of the definition for a
$d$-Matoušek complex should be a complex which doesn’t exhibit similarly
defined topological barriers to $d$-representability.

## References

  * [1] M. Adachi, Embeddings and immersions, American Mathematical Society, Providence, 1993.
  * [2] E. G. Bajmóczy, I. Bárány, On a common generalization of Borsuk’s and Radon’s theorem, Acta Mathematica Hungarica, 34 (1979), 347–350.
  * [3] J. Eckhoff, An upper-bound theorem for families of convex sets, Geometriae Dedicata, 19 (1985), 217–227.
  * [4] J. Kratochvíl, J. Matoušek, String graphs requiring exponential representations, Journal of Combinatorial Theory, Series B, 53 (1991), 1–4.
  * [5] C. Lekkerkerker, J. Boland, Representation of a finite graph by a set of intervals on the real line, Fundamenta Mathematicae, 51 (1962), 45–64.
  * [6] J. Matoušek, A Helly-type theorem for unions of convex sets, Discrete & Computational Geometry, 18 (1997), 1–12.
  * [7] J. Matoušek, Using the Borsuk–Ulam theorem: lectures on topological methods in combinatorics and geometry, Springer, Berlin, 2008.
  * [8] K. S. Sarkaria, Kneser colorings of polyhedra, Illinois Journal of Mathematics, 33 (1989), 592–620.
  * [9] M. Tancer, A counterexample to Wegner’s conjecture on good covers, Discrete & Computational Geometry, 47 (2012), 266–274.
  * [10] M. Tancer, d-representability of simplicial complexes of fixed dimension, Journal of Computational Geometry, 2 (2011), 183–188.
  * [11] M. Tancer, Intersection patterns of convex sets via simplicial complexes: a survey, In: Thirty essays on geometric graph theory, Springer, New York, 2013, pp. 521–540.
  * [12] G. Wegner, d-Collapsing and nerves of families of convex sets, Archiv der Mathematik, 26 (1975), 317–321.
  * [13] R.T. Živaljević, User’s guide to equivariant methods in combinatorics. II., Publications de l’Institut Mathématique. Nouvelle Série, 64 (1998), 107–132.


11institutetext:  Departamento de Física, Facultad de Ciencias Básicas,
Universidad de Antofagasta, Casilla 170, Antofagasta, Chile.

Classical black holes Quantum aspects of black holes, evaporation,
thermodynamics Exact solutions

# A new model of regular black hole in $(2+1)$ dimensions

Milko Estrada 11 Francisco Tello-Ortiz 1111

###### Abstract

We provide a new regular black hole solution in $(2+1)$ dimensions with
presence of matter fields in the energy momentum tensor. The inclusion of our
proposed energy density plus a negative cosmological constant allows that the
solution can have both flat as de–Sitter or Anti de–Sitter core. This latter
is a proper characteristic of our solution, because other models of regular
black holes have only a single type of core. Since the first law of
thermodynamics for regular black holes is modified by the presence of the
matter fields, we provide a new version of the first law, where a local
definition of the variation of energy is defined, and, where the entropy and
temperature are consistent with the previously known in literature. At the
hypothetical limit when the horizon radius $r_{+}\to\infty$ the usual first
law $dM=TdS$ is recovered. The effectiveness of the formalism used to compute
the mass of our regular black holes in $(2+1)$ suggests the potential
applicability of this method to calculate the mass of other models of regular
black holes in $d\geq 4$ dimensions.

###### pacs:

04.70.Bw

###### pacs:

04.70.Dy

###### pacs:

04.20.Jb

##  1 Introduction

The recent detection of gravitational waves through the collision of two
rotating black holes [1, 2], together with the also recent assignment of the
Nobel Prize, have positioned the black holes as one of the most interesting
and intriguing objects in gravitation. In this connection, the fact that the
black holes, due to quantum fluctuations emit as black bodies, where its
temperature is related to its surface gravity [3], shows that in these objects
the geometry and thermodynamics are directly connected.

The Schwarzschild black hole has a central singularity where the laws of
physics cease to operate. From the classical point of view, Bardeen in
reference [4] proposed the first model of regular black holes (RBHs). In this
model the singularity is avoided by changing the mass parameter in the
Schwarzschild solution by a radial mass function such that near the origin the
function behaves as $f\approx 1-Cr^{2}$, i.e. the solution has a de sitter
core. After this, several models of regular black holes have been constructed,
where the Einstein field equations are coupled to non–linear electromagnetic
sources. Examples of this can lead to RBHs with de–Sitter core [5] or flat
core [6, 7].

Furthermore, it is possible to construct RBHs solutions including matter
fields in the energy momentum tensor. Examples of this are the models of
references [8, 9]. These models are solutions of General Relativity and have
de–Sitter core. We can see a review of these types of RBHs in reference [10].
Other models of RBHs based in the presence of matter fields in the energy
momentum tensor for higher curvature theories, with a de–Siter core, can be
found in references [11, 12, 13].

It is well known that the first law of thermodynamics is modified for RBHs due
to the presence of matter fields in the energy momentum tensor. In reference
[14] it have been shown a way of writing the first law including a correction
factor, which corresponds to an integration of the radial coordinate up to
infinity. However, the first law of thermodynamics for RBHs is still an open
question of physical interest.

On the other hand, models of gravity in $(2+1)$ dimensions have drawn high
attention in the last years, due to the simplicity of its equations of motion.
These models can be viewed as toy models, with the hope that its results can
help the understanding of $(3+1)$ and higher dimensional models of gravity
[15]. Models of black holes in $(2+1)$ also have been widely studied in the
last years. BTZ model [16, 17] is a vacuum black hole solution of the Einstein
field equations in $(2+1)$ dimensions, which has been studied to find some
conceptual issues in quantum gravity, string theory and AdS/CFT correspondence
[18]. In this solution the curvature invariants are regular everywhere.
Regular black holes in $(2+1)$ dimensions coupled to a non–linear
electrodynamics were studied in reference [19, 20]. Nevertheless, in the
literature also there are solutions in $(2+1)$ dimensions with a central
singularity, as for example the solutions of reference [21] for Massive
Gravity [22], or the solution given in [23]. See references [24, 25, 26] for a
generalization of BTZ model with other fields and references [27, 28] for an
interesting higher dimensional generalization of BTZ and geodesic analysis. We
can see more recent studies in references [29, 30, 31, 32, 33, 34].

Thus, it is undoubtedly interesting the seeking of new black hole solutions in
$(2+1)$ dimensions with matter fields in the energy momentum tensor and the
study of its thermodynamics properties. In this work we will provide a new
model of energy density in $(2+1)$ dimensions, which will lead to a new
regular black hole solution. Furthermore we will propose a structure for the
first law of thermodynamics for RBHs and compute the mass of the solution
using a definition of conserved charged appeared recently in the literature
[35, 36]. Furthermore we will study the stability of our solution.

##  2 $(2+1)$ Einstein Field Equations

The Einstein field equation are given by:

$G^{\mu}_{\,\,\,\nu}+\Lambda\delta^{\mu}_{\,\,\,\nu}=8\pi\bar{G}T^{\mu}_{\,\,\,\nu},$
(1)

where the cosmological constant is equal to $\Lambda=-\dfrac{1}{l^{2}}$. The
Newton constant, in the natural system of units, has units of length
$[\bar{G}]=\ell$. For simplicity, we consider arbitrarily that this constant
has a magnitude equal to 1. The energy momentum tensor describing a perfect
fluid, is given by:

$T^{\mu}_{\,\,\,\nu}=\mbox{diag}(-\rho,p_{r},p_{\theta}).$ (2)

We shall study the following spherically symmetric space time:

$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\theta.$ (3)

This form of the metric has the following consequence on the energy momentum
tensor through the Einstein field equations:

$\rho=-p_{r}.$ (4)

So, for the line element (3), the $(t,t)$ and $(r,r)$ component of the
Einstein field equations are:

$\frac{1}{2r}\frac{df}{dr}+\Lambda=-8\pi\rho$ (5)

On the other hand, from the conservation law, $\nabla_{\mu}T^{\mu\nu}=0$, one
gets

$p_{\theta}=r\frac{d}{dr}p_{r}+p_{r}.$ (6)

We will chose the following ansatz:

$f(r)=1-8\bar{G}m(r)-\Lambda r^{2}.$ (7)

Next, replacing the ansatz (7) into the equation (5), one arrives to:

$\frac{d}{dr}m(r)=2\pi r\rho(r).$ (8)

##  3 The new model

The mass function for the $(2+1)$ case is defined as:

$m(r)=2\pi\int_{0}^{r}x\rho(x)dx.$ (9)

As was mentioned above, in this work we will provide a new RBH solution based
in the inclusion of matter field in the energy momentum tensor. For this, we
will propose a new model of energy density. In order to have a well posed
physical situation, the energy density model that we will choose, follows the
requirements described in reference [12]:

  * •

The energy density must be positive and continuously differentiable to avoid
singularities.

  * •

The energy density must have a finite single maximum at the origin. The
finiteness of $\rho(0)$ forbids the presence of a central singularity.

  * •

In order to guarantee a well defined asymptotic behavior, the energy density
must be a decreasing radial function and must vanish at infinity. Furthermore,
the mass function must reach its finite maximum value at infinity, i.e

$\displaystyle\displaystyle\lim_{r\to\infty}\rho$ $\displaystyle=0$ (10)
$\displaystyle\displaystyle\lim_{r\to\infty}m(r)$
$\displaystyle=\mbox{constant}.$ (11)

Thus, we propose the following model for energy density:

$\rho=\frac{2LM^{2}}{\pi(2LM+r^{2})^{2}},$ (12)

and replacing into the equation (9) we obtain the mass function

$m(r)=\frac{Mr^{2}}{2LM+r^{2}},$ (13)

where $L$ is a positive constant of units $[L]=\ell^{3}$. This constant must
be positive to ensure that $\rho(0)=1/(2\pi L)>0$. The parameter $M$, is the
so called mass parameter, whit units of $[M]=\ell^{-1}$. This constant also
must be positive to ensure the absence of singularities in the energy density.

Considering the previous analysis, the mass function has units
$[m(r)]=\ell^{-1}$ and since $[\bar{G}]=\ell$, the factor $\bar{G}m(r)$ is
dimensionless, where the magnitude of $\bar{G}$ is equal to the unity.

This model could be viewed as an extension in $(2+1)$ dimensions of the higher
dimensional model of reference [12], which for the $(3+1)$ dimensional case,
coincides with the Hayward metric [9].

For the Hayward metric in $(3+1)$ dimensions, when the energy density is of
the order of the Planck units near the origin, the formation of a de–Sitter
core is associated with quantum fluctuations. These models are known as Planck
Stars [37, 38]. In reference [39], using radio astronomy data, it is
conjectured that Planck Stars represent a speculative but realistic
possibility to testing quantum gravity effects. Thus, if our energy density
were of order of the Planck units near the origin, our model could serve as a
toy model to study these ideas in a future work due to the simplicity of the
equations of motion in $(2+1)$ dimensions. As we shall see below, the
inclusion of a negative cosmological constant not only provides the formation
of a dS core, but also provides the formation of a flat/AdS core. This latter
feature could be analyzed for Planck Stars models elsewhere. It is worth
mentioning that the formation of a flat or AdS core is a new feature for RBHs
without non–linear electromagnetic sources.

The solution is:

$f(r)=1-\Lambda
r^{2}-8\bar{G}m(r)=1+\frac{r^{2}}{l^{2}}-\frac{8Mr^{2}}{2LM+r^{2}}.$ (14)

For this solution, the Ricci and Kretschmann invariants are given by:

$R=-\frac{6}{l^{2}}+\frac{48M}{2LM+r^{2}}-\frac{112Mr^{2}}{(2LM+r^{2})^{2}}+\frac{64Mr^{4}}{(2LM+r^{2})^{3}},$
(15)

$\displaystyle K$
$\displaystyle=\left(\frac{2}{l^{2}}-\frac{16M}{2LM+r^{2}}+\frac{80Mr^{2}}{(2LM+r^{2})^{2}}-\frac{64Mr^{4}}{(2LM+r^{2})^{3}}\right)^{2}$
$\displaystyle+2\left(\frac{2}{l^{2}}-\frac{16M}{2LM+r^{2}}+\frac{16Mr^{2}}{(2LM+r^{2})^{2}}\right)^{2}.$
(16)

We understand by regular when the curvature invariants are free of
singularities. In our case both Ricci and Kretschmann don’t diverge at $r=0$
nor other value of $r$ due that $L>0$ and $M>0$.

###  3.1 behavior near the origin

From equation (14) it is evident that this function behaves near the origin
as:

$f\big{|}_{r\approx 0}\approx
1+\left(\frac{1}{l^{2}}-\frac{4}{L}\right)r^{2},$ (17)

thus, we can define the effective radius ($\tilde{l}$):

$\frac{1}{\tilde{l}}=\left(\frac{1}{l^{2}}-\frac{4}{L}\right)$ (18)

  * •

For $\frac{1}{l^{2}}=\frac{4}{L}$ the behavior near the origin corresponds to
a flat space time. Regular solutions with a flat core have been studied for
$(3+1)$ dimensions with a non–linear electromagnetic source in reference [7,
6]. However, for the $(2+1)$ case, with a nonzero energy density, this
behavior is a proper feature of our solution.

  * •

For $\frac{1}{l^{2}}<\frac{4}{L}$ the behavior near the origin corresponds to
a dS space time. Several models of regular black holes share this feature [9,
11, 12, 37, 38, 13]. In this case the equation (18) represents an effective dS
radius.

  * •

For $\frac{1}{l^{2}}>\frac{4}{L}$ the behavior near the origin corresponds to
an AdS space time. So, in our model this possibility could be valid, which
differs from the most of the RBHs models. Due to the simplicity of the $(2+1)$
models, the physical consequences of this particular feature could be testing
elsewhere. In this case the equation (18) represents an effective AdS radius.

Thus, our model of energy density added to a negative cosmological constant
allow that the solution can have both flat as de–Sitter or Anti de–Sitter
core, depending on the values of the $l$ and $L$. This latter is a proper
characteristic of our solution, because other models of regular black holes
have only a single type of core.

##  4 Thermodynamic analysis

###  4.1 Conserved Charges

Recently, in references [35, 36] has appeared a new definition of conserved
charges. In [35] the energy and momentum can be computed by integrating a
covariantly conserved current $J^{\mu}=T^{\mu}_{\nu}\xi^{\nu}$ in a volume
integral. It is worth to mention that the definition of reference [35] is
reduced to the definition of conserved energy of reference [36] for a Killing
vector $\xi^{\mu}=-\delta^{\mu}_{0}$. Following [35], the energy is defined
as:

$E=\int d^{d-1}x\sqrt{-g}J^{0}=\int d^{d-1}x\sqrt{-g}T^{0}_{\nu}\xi^{\nu},$
(19)

where $\xi^{\nu}$ is a Killing vector and $d$ is the number of dimensions.

It is worth mentioning that this definition is diffeo-invariant. Furthermore
the current is covariantly conserved due that $\nabla_{\mu}T^{\mu}_{\nu}=0$
and $\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}=0$.

Following [35], we choose $\xi^{\mu}=-\delta^{\mu}_{0}$. Evaluating our
solution into the equation (19):

$\displaystyle
E=-2\pi\int^{\infty}_{0}rdrT^{0}_{0}=2\pi\int^{\infty}_{0}rdr\rho(r)=\lim_{r\to\infty}m(r)=M$
(20)

Thus, the parameter $M$ represents the total energy of the black hole. On the
other hand, the obtained mass $M$ coincides with the value of the vacuum BTZ
solution computed in reference [40] using other definition of conserved
charges.

So, it is computed the mass of the black hole in $(2+1)$ without adding
boundary terms as in references [40, 41]. The effectiveness of this formalism
for computing the mass of our $(2+1)$ RBHs model could make us think about
using this formalism for computing the mass of another models of regular black
holes in $d\geq 4$ dimensions, where the energy density also fulfills the
conditions mentioned before. Some examples of these models are [9, 37, 38, 8,
12].

###  4.2 The first law of thermodynamic in $(2+1)$ dimensions with matter
fields

It is well known that the first law of thermodynamics is modified for RBHs due
to the presence of matter fields in the energy momentum tensor. Following [42,
43] we define the thermodynamics pressure as $p=-\dfrac{\Lambda}{8\pi}$. In
order to propose a structure for the first law of thermodynamics for RBHs, we
will use the conditions $f(r_{+},M,p)=0$ and $\delta f(r_{+},M,p)=0$ , which
can be viewed as constraints on the evolution along the space parameters [11,
33]. Thus, from the condition

$0=\frac{\partial f}{\partial r_{+}}dr_{+}+\frac{\partial f}{\partial
M}dM+\frac{\partial f}{\partial p}dp$ (21)

it is straightforward to check that for the present solution (14), the first
law takes the form:

$\frac{\partial m}{\partial
M}dM=\left(\frac{1}{4\pi}f^{\prime}\big{|}_{r=r_{+}}\right)d\left(\frac{\pi
r_{+}}{2}\right)+(\pi r_{+}^{2})dp.$ (22)

The above equation can be rewritten as:

$du=TdS+Vdp,$ (23)

where the temperature and entropy are easily computed as:

$\displaystyle T$ $\displaystyle=\frac{1}{4\pi}f^{\prime}\big{|}_{r=r_{+}}$
(24) $\displaystyle S$ $\displaystyle=\frac{\pi r_{+}}{2}$ (25) $\displaystyle
V$ $\displaystyle=\pi r_{+}^{2},$ (26)

where our definition of volume coincide with [42].

One can notice that $p$, the pressure, has units of $[p]=\ell^{-2}$, equation
(14). Likewise, one can check that the factor $[\bar{G}m]=\ell^{0}$ is
dimensionless and $[S]=\ell^{1}$. Following Euler’s theorem [44], one can
construct the Smarr formula:

$TS-2pV=0$ (27)

which coincide with the vacuum $(2+1)$ solution of reference [42] for the non
rotating case. In a future work, could be studied the case where the constant
$L$ is a thermodynamics parameter.

It is direct to check that the temperature for our model (14) is:

$T=\frac{2}{r_{+}}m(r_{+})-\frac{2}{r_{+}}-8\frac{dm}{dr}\Big{|}_{r=r_{+}}.$
(28)

On the other hand, the heat capacity at constant pressure is computed as:

$C_{p}=T\frac{dS}{dT}\bigg{|}_{p=cte}=T\frac{dS}{dr_{+}}\left(\frac{dT}{dr_{+}}\right)^{-1}\bigg{|}_{p=cte}$
(29)

In the vacuum case, the first law is of the form $dM=TdS$, where $M$ is the
energy of the system. However in our case the left side of the first law is
modified in order to obtain the correct value of the entropy due to the
presence of matter fields in the energy momentum tensor. So, in our case, the
term of the left side, $du$, corresponds to a local definition of the
variation of the energy.

The modification of the first law for regular black holes was studied in
reference [14], without our constraint in the space of parameters, by
including a correction factor which corresponds to an integration of the
radial coordinate up to infinity. Unlike this latter reference, in our case
both terms, $du$ and $dS$, are local variables defined at the horizon.

It is easy to check that the factor $dm/dM$ in equation (23) is always
positive, and thus, the sign of the variation of $du$ always coincides with
the sign of the variation of the total energy $dM$. Furthermore, at infinity
it is fulfilled that $\displaystyle\lim_{r\to\infty}dm/dM=1$, therefore the
variation of the local and total energy are similar at the asymptotically
region. Thus, at the hypothetical limit when the horizon radius
$r_{+}\to\infty$, the usual first law $dM=TdS$ is recovered.

In a future work, following [45], we could test a possible relation between
$l$ and $L$ to study the ensemble dependency problem.

###  4.3 Stability of our solutions

In figure 1 are shown the behavior of the mass parameter (top panel),
temperature (middle panel) and heat capacity (bottom panel). It is direct to
check that this behavior is generic for other values of the parameter $l$.

The equation (14) can take the form $f(r)=0$, which taking arbitrarily fixed
$l$ and $L$, can have zero roots, two roots or one root, depending on the
value of the mass parameter $M$.

In the top panel of figure 1, following the analysis of references [46, 11],
we plot the mass parameter $M(R)$, where $r=R$ corresponds to the root of the
equation $f(r)=0$. $R$ can take the values $r_{-}$ and $r_{+}$, which
correspond to the internal horizon and the black hole horizon, respectively.
There is a critical value of the mass parameter $M_{*}=M(r_{*})$, which
corresponds to the minimum value on the curve, and where there is an extremal
black hole. At this point the internal horizon $r_{-}$ and the black hole
horizon $r_{+}$ coincide, i.e. $r_{*}=r_{-}=r_{+}$. For $M<M_{*}$ the solution
does not have horizons and for $M>M_{*}$ the solution has both internal and
black hole horizon.

In the middle panel of figure 1, we note that the temperature is vanishing
just in the extremal black hole. This does not have a linear dependence on
$r_{+}$ as the spinless BTZ solution [47] and its dependence on $r_{+}$ also
differs from the rotating BTZ solution [16].

In the bottom panel of figure 1, is displayed the heat capacity, which is
always positive, therefore the solution is always stable. Its behavior also
differs from the BTZ solution whose heat capacity is constant for the static
case [47].

Figure 1:  Top panel: The Mass Parameter. Middle panel: The Temperature.
Bottom Panel: The Heat Capacity. All these plots were obtained by considering
$l=L=\bar{G}=1$

.

##  5 Discussion and Conclusions

We have obtained a new regular black hole solution in $(2+1)$ dimensions with
presence of matter fields in the energy momentum tensor. For this, we have
proposed a new model of energy density following the requirements described in
reference [12]. This energy density has a finite value at the center, avoiding
the formation of a central singularity.

Our solution differs from BTZ-like geometry of reference [48], because in our
solution is present the mass function, unlike this reference, where is only
present the constant mass parameter. Furthermore, in reference [48] the BTZ
horizon is characterized for small values of the mass and cosmological
constant. In our model the energy density tends to zero at infinity, equation
(10), this differs from the model of reference [49], where the energy density
diverges at infinity. Furthermore, our solution differs from the reference
[50], which is based in the matching between the internal geometry obtained by
means the Gravitational Decoupling algorithm and the external BTZ geometry.

The inclusion of our type of energy density plus a negative cosmological
constant allows that the solution can have both flat as de–Sitter or Anti
de–Sitter core, depending on the values of the $l$ and $L$. This latter is a
proper characteristic of our solution, because other models of regular black
holes have only a single type of core.

Regular solutions with a flat core have been studied in $(3+1)$ dimensions
with a non–linear electromagnetic source in references [6, 7]. However, for
the $(2+1)$ case, with a nonzero energy density, the flat core is a proper
feature of our solution. In $(3+1)$ dimensions, regular black hole solutions
with a dS core have been widely studied in the literature. Our model with a dS
core could be viewed as a $(2+1)$ extension of the Hayward metric. Since the
Hayward model has been used to testing quantum gravity effects [39], our model
could serve to study these effects in elsewhere due to the simplicity of the
equations of motion in $(2+1)$ dimension. Furthermore our solution admits a
core with AdS structure, which is a not studied feature so far.

Using the recently definition of conserved charges of references [35, 36] we
have computed the total energy of our regular black hole solution, which is
equal to the mass parameter $M$. The effectiveness of the definition used for
computing the mass of our $(2+1)$ regular black hole model suggests that this
formalism could be used to compute the mass of another models of regular black
holes in $d\geq 4$ dimensions with presence of matter in the energy momentum
tensor, as for example the models of references [9, 37, 38, 8, 12].

The structure of the first law of thermodynamic for regular black holes is
modified by the presence of matter fields in the energy momentum tensor [14].
So, we have provided a new version of the first law of thermodynamic (23),
where it is defined a local definition of the variation of the energy, namely
$du$, and, where the values of entropy and temperature are consistent with the
previously known [16]. We have showed that the sign of the local variation of
the energy $du$ always coincides with the sign of variation of the total
energy $dM$. Furthermore, at the hypothetical limit when the horizon radius
$r_{+}\to\infty$ the usual first law $dM=TdS$ is recovered.

## Acknowledgements

F. Tello-Ortiz acknowledges financial support by CONICYT PFCHA/DOCTORADO-
NACIONAL/2019-21190856 project ANT–1756 at Universidad de Antofagasta, Chile.
F. Tello-Ortiz thanks the PhD program Doctorado en Física mención en Física
Matemática de la Universidad de Antofagasta for continuous support and
encouragement.

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