# Equivariance in Approximation by Compact Sets

Saugata Basu  and  Alison Rosenblum

###### Abstract.

We adapt a construction of Gabrielov and Vorobjov for use in the symmetric
case. Gabrielov and Vorobjov had developed a means by which one may replace an
arbitrary set $S$ definable in some o-minimal expansion of $\mathbb{R}$ with a
compact set $T$. $T$ is constructed in such a way that for a given $m>0$ we
have epimorphisms from the first $m$ homotopy and homology groups of $T$ to
those of $S$. If $S$ is defined by a boolean combination of statements
$h(x)=0$ and $h(x)>0$ for various $h$ in some finite collection of definable
continuous functions, one may choose $T$ so that these maps are isomorphisms
for $0\leq k\leq m-1$. In this case, $T$ is also defined by functions closely
related to those defining $S$.

In this paper we study sets $S$ symmetric under the action of some finite
reflection group $G$. One may see that in the original construction, if $S$ is
defined by functions symmetric relative to the action of $G$, then $T$ will be
as well. We show that there is an equivariant map $T\rightarrow S$ inducing
the aforementioned epimorphisms and isomorphisms of homotopy and homology
groups. We use this result to strengthen theorems of Basu and Riener
concerning the multiplicities of Specht modules in the isotypic decomposition
of the cohomology spaces of sets defined by polynomials symmetric relative to
$\mathfrak{S}_{n}$.

##  1\. Introduction

Fix an o-minimal structure expanding a real closed field $R$ (on occasion, we
will specify that this real closed field is $\mathbb{R}$; one may if desired
assume as much throughout). We will take all sets, maps, etc. to be definable
in this structure.

Gabrielov and Vorobjov in [9] develop a construction by which one may replace
an arbitrary definable set $S$ by a closed and bounded set $T$, such that we
have epimorphisms (and in a special case isomorphisms) from the homotopy and
homology groups of the approximating set to those of the original set. In this
special case, termed the _constructible case_ , we take $S$ to be defined by a
quantifier-free formula with atoms $h>0$ or $h=0$ for some finite number of
continuous definable functions $h$. In this case, the procedure described in
[9] gives an explicit description of, and control over the number of, the
functions defining $T$. Gabrielov and Vorobjov’s paper describes the
ramifications of this in calculating upper bounds on Betti numbers for sets
definable in certain structures.

In [3], Basu and Riener leverage properties of symmetry (particularly,
symmetry relative to the standard action of $\mathfrak{S}_{n}$ on $R^{n}$) to
study the Betti numbers of semialgebraic sets defined by symmetric polynomials
of bounded degree, developing algorithms with favorable complexity bounds. The
paper makes use of the construction of Gabrielov and Vorobjov presented in
[9], replacing $S$ with a set $T$ defined by closed conditions and then
applying various results requiring closedness in order to determine the
structure of the cohomology spaces of $T$. From Gabrielov and Vorobjob’s
results, we know that the cohomology spaces of $T$ are isomorphic to those of
$S$. If we assume $S$ is defined by symmetric polynomials of degree bounded by
some $d$ at least 2, then the set produced by the Gabrielov-Vorobjov
construction is also defined by symmetric polynomials of degree no more than
$d$. However, it is not immediately clear whether the maps on the level of
homotopy and homology are equivariant. Without equivariance, we cannot
conclude that the cohomology spaces of $S$ and $T$ have the same structures as
$\mathfrak{S}_{n}$-modules, meaning that results for $T$ only translate back
to $S$ in part.

This paper establishes an equivariant version of the construction of Gabrielov
and Vorobjov in [9]. We introduce a few pieces of notation in order to present
our main theorem.

######  Notation 1.1.

Let $S=\\{\mathbf{x}\in R^{n}\mid\mathcal{F}(\mathbf{x})\\}$ be a set defined
by a formula $\mathcal{F}$. Say $\mathcal{P}=\\{h_{1},\ldots,h_{s}\\}$ is a
finite collection of continuous definable functions $R^{n}\rightarrow R$. If
$\mathcal{F}$ is a boolean combination of statements of the form $h=0$ and
$h>0$ for $h\in\mathcal{P}$, we call $\mathcal{F}$ a _$\mathcal{P}$ -formula_
and $S$ a _$\mathcal{P}$ -set_. If $\mathcal{F}$ is a monotone boolean
combination (i.e. without negations) of statements of the form $h\geq 0$ and
$h\leq 0$ for $h\in\mathcal{P}$, then we say that $\mathcal{F}$ is a
_$\mathcal{P}$ -closed formula_ and $S$ is a _$\mathcal{P}$ -closed set_.

######  Notation 1.2 ([9] Definition 1.7).

When we say a statement holds for

$0<\varepsilon_{0}\ll\varepsilon_{1}\ll\cdots\ll\varepsilon_{m}\ll 1$

we mean that for each $0\leq i\leq m$ there is a definable function
$f_{i}:(0,1)^{m-i}\rightarrow(0,1)$ (i.e. $f_{m}$ is a constant in $(0,1)$)
such that the statement holds for all $\varepsilon_{0},\ldots,\varepsilon_{m}$
which satisfy, for $0\leq i\leq m$, the condition that
$0<\varepsilon_{i}<f_{i}(\varepsilon_{i+1},\ldots,\varepsilon_{m})$.

Our equivariant adaptation of the main theorem of Gabrielov and Vorobjov in
[9] is then as follows.

######  Theorem 1.3.

Let $G$ be a finite reflection group acting on $\mathbb{R}^{n}$, and let
$S\subset\mathbb{R}^{n}$ be a definable set symmetric under the action of $G$.
Choose an integer $m>0$.

  1. (a)

(Definable Case) Say that $S$ is represented in a compact symmetric set $A$ by
families $\\{S_{\delta}\\}_{\delta>0}$ and
$\\{S_{\delta,\varepsilon}\\}_{\delta,\varepsilon>0}$ of compact sets (see
Definition 3.1), and that each $S_{\delta}$ and $S_{\delta,\varepsilon}$ is
also symmetric under the action of $G$. Then for parameters
$0<\varepsilon_{0},\delta_{0},\ldots,\varepsilon_{m},\delta_{m}<1$, there
exists a compact symmetric set
$T=T(\varepsilon_{0},\delta_{0},\ldots,\varepsilon_{m},\delta_{m})$ such that,
for $0<\delta_{0}\ll\varepsilon_{0}\ll\ldots\ll\varepsilon_{m}\ll\delta_{m}\ll
1$, we have an equivariant map $\psi:T\rightarrow S$ inducing equivariant
epimorphisms

$\displaystyle\psi_{\\#,k}$
$\displaystyle:\pi_{k}(T,*)\rightarrow\pi_{k}(S,*^{\prime})$
$\displaystyle\psi_{*,k}$ $\displaystyle:H_{k}(T)\rightarrow H_{k}(S)$

for each $1\leq k\leq m$.

  2. (b)

(Constructible Case) Say that $S$ is a $\mathcal{P}$-set for
$\mathcal{P}=\\{h_{1},\ldots,h_{s}\\}$ a collection of continuous definable
functions having the property that $h\circ g\in\mathcal{P}$ for all $g\in G$
and $h\in\mathcal{P}$. For parameters $r>0$ and
$0<\varepsilon_{0},\delta_{0},\ldots,\varepsilon_{m},\delta_{m}<1$, let

$\mathcal{P}^{\prime}=\bigcup_{h\in\mathcal{P}\cup\\{r^{2}-(X_{1}^{2}+\cdots+X_{n}^{2})\\}}\bigcup_{j=0}^{m}\\{h\pm\varepsilon_{j},h\pm\delta_{j}\\}$

Then there exists a $\mathcal{P}^{\prime}$-closed and bounded set $T$ such
that for sufficiently large $r$ and
$0<\delta_{0}\ll\varepsilon_{0}\ll\ldots\ll\varepsilon_{m}\ll\delta_{m}$,
there exists an equivariant map $\psi:T\rightarrow S$ inducing equivariant
homomorphisms

$\displaystyle\psi_{\\#,k}$
$\displaystyle:\pi_{k}(T,*)\rightarrow\pi_{k}(S,*^{\prime})$
$\displaystyle\psi_{*,k}$ $\displaystyle:H_{k}(T)\rightarrow H_{k}(S)$

which are isomorphisms for $1\leq k\leq m-1$ and epimorphisms for $k=m$. If
$m\geq\dim(S)$, then $\psi$ induces a homotopy equivalence $T\simeq S$. Note
that in particular if all functions in $\mathcal{P}$ are symmetric relative to
$G$, then the same is true for $\mathcal{P}^{\prime}$.

Parts (a) and (b) of this theorem are proved in Theorem 6.26 and Corollary
6.33 respectively. Our arguments follow the strategy of [9], with adjustments
to ensure equivariance. The maps composed to obtain $\psi$ in the definable
and constructible cases are displayed in the summary in Subsection 6.5.

To construct the maps of homotopy and homology groups in [9], Gabrielov and
Vorobjov first consider a triangulation adapted to $S$ of some larger compact
set $A$. Thus, a major step in our paper concerns proving the existence of a
triangulation with the needed symmetry and equivariance properties for a given
symmetric, definable set. This is done in Theorem 4.7. We also establish
equivariant versions of a few other theorems as needed.

Section 2 contains background and definitions. Section 3 describes the
construction of the approximating set $T$ in the definable and constructible
cases. The results relating to symmetric triangulation and equivariant
versions of the other background theorems can be found in Sections 4 and 5
respectively. Finally, in Section 6, we assemble our results to prove the
existence of an equivariant map $T\rightarrow S$ inducing the promised
epimorphisms and isomorphisms of homotopy and homology groups. Section 7
discusses the ramifications for the results of Basu and Riener in [3].

The authors wish to express heartfelt appreciation for the advice of Dr.
Gabrielov throughout, and particularly concerning Theorem 5.12.

##  2\. Background

Much of the information here is relatively standard. We include these
definitions for completeness, and to establish a few conventions.

###  2.1. Symmetry

######  Definition 2.1.

Let $G$ be a group acting on a set $X$. A subset $Y\subset X$ is said to be
_symmetric_ with respect to the action of $G$ on $X$ if for each $g\in G$ and
$y\in Y$, we have $g(y)\in Y$.

We extend the definition of a symmetric function, standard for the usual
action of $\mathfrak{S}_{n}$ on $\mathbb{R}^{n}$, to more general group
actions.

######  Definition 2.2.

Let $f:X\rightarrow Y$ for sets $X$ and $Y$, and let $G$ be a group acting on
$X$. We will say that $f$ is _symmetric_ with respect to the action of $G$ if
we have that $f(g(x))=f(x)$ for all $x\in X$ and $g\in G$.

######  Definition 2.3.

Let $f:X\rightarrow Y$ for some sets $X$ and $Y$, and let $G$ be a group which
acts on both $X$ and $Y$. We say $f$ is _equivariant_ with respect to the
action of $G$ if for all $g\in G$, the diagram

${X}$${Y}$${X}$${Y}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{f}$

commutes.

We do need to give consideration to symmetry and equivariance for homotopy
groups. Let $X$ be a topological space, with $G$ a group acting on $X$. Then
for $g\in G$ and basepoint $x_{0}\in X$, the map $g:X\rightarrow X$ induces
maps $g_{\\#,k}:\pi_{k}(X,x_{0})\rightarrow\pi_{k}(X,g(x_{0}))$ of homotopy
groups for each $k\geq 0$. Thus, for a pointed space $(X,*)$, we understand
that the induced action of $g$ on any $\pi_{k}(X,*)$ will also change the
basepoint. For this reason, we will be careful to maintain basepoint notation
in reference to homotopy groups. In particular, we must give attention to our
basepoint when considering equivariance of maps on the level of homotopy
groups.

######  Definition 2.4.

Let $f:\pi_{k}(X,*)\rightarrow\pi_{k}(Y,*^{\prime})$ for topological spaces
$(X,*)$ and $(Y,*^{\prime})$, and let $G$ be a group acting on $X$ and $Y$. We
will say that $f$ is equivariant if the diagram

${\pi_{k}(X,*)}$${\pi_{k}(Y,*^{\prime})}$${\pi_{k}(X,g(*))}$${\pi_{k}(Y,g(*^{\prime}))}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{f}$

commutes.

###  2.2. Reflection Groups

Let $V$ be a finite dimensional real vector space equipped with an inner
product, and let $\mathcal{O}(V)$ denote the group of all orthogonal linear
transformations of $V$. We will for the sake of clarity distinguish a _linear
hyperplane_ (a codimension $1$ vector subspace, which must therefore contain
$\mathbf{0}$) from an _affine hyperplane_ (a codimension $1$ affine subspace,
which need not pass through the origin). If $P$ is a hyperplane (in either
sense), then the complement $V\setminus P$ has two connected components, which
are called (linear or affine, resp.) half-spaces.

######  Definition 2.5 (see [10] Chapter 3).

Let $G$ be a finite subgroup of $\mathcal{O}(V)$. A subset $H\subset V$ is a
_fundamental region_ of $V$ relative to $G$ provided

  1. (1)

$H$ is an open subset of $V$

  2. (2)

$H\cap g(H)=\emptyset$ for all $e\neq g\in G$

  3. (3)

$V=\bigcup_{g\in G}\overline{g(H)}$

######  Theorem 2.6 ([10] Theorem 3.1.2).

Let $G$ be a finite subgroup of $\mathcal{O}(V)$. Then there exists a finite
collection $\\{H_{1},\ldots,H_{k}\\}$ of linear half-spaces such that the
region

$H=\bigcap_{i=1}^{k}H_{i}$

is a fundamental region of $V$ relative to $G$.

We are interested in the case where $G$ is a finite reflection group acting on
$\mathbb{R}^{n}$ (with the standard inner product), i.e. $G$ is generated by
elements $g$ whose action on $\mathbb{R}^{n}$ are given by $g(\mathbf{x})$
reflecting $\mathbf{x}$ across some linear hyperplane $P_{g}$. We will
frequently use the fact that any affine hyperplane $P$ in $R^{n}$ coincides
with some $\\{\mathbf{x}\in R^{n}\mid L(\mathbf{x})=0\\}$, where
$L:R^{n}\rightarrow R$ is of the form
$L(x_{1},\ldots,x_{n})=a_{0}+a_{1}x_{1}+\cdots+a_{n}x_{n}$ for some
$a_{0},\ldots,a_{n}\in R$. $P$ is a linear hyperplane iff $a_{0}=0$.

Say that $G$ is a finite reflection group acting on $\mathbb{R}^{n}$. Then, as
detailed in [10] Chapter 4, we can select a subset $\\{g_{1},\ldots,g_{k}\\}$
of non-identity elements of $G$ such that, if $g_{i}$ acts by reflecting
through the hyperplane $P_{i}$, we can choose functions $L_{i}$ with $P_{i}$
given by $L_{i}(\mathbf{x})=0$ so that

$H=\bigcap_{i=1}^{k}\\{\mathbf{x}\in\mathbb{R}^{n}\mid L_{i}(\mathbf{x})>0\\}$

is a fundamental region of $\mathbb{R}^{n}$ relative to $G$. The sets of the
form $(L_{1}\geq 0)\cap\ldots\cap(L_{i-1}\geq 0)\cap(L_{i}=0)\cap(L_{i+i}\geq
0)\cap\ldots\cap(L_{k}\geq 0)$ are called the walls of this fundamental
region.

In this description, for $g\in G$ we have that there exists a subset
$\lambda_{g}$ of $\\{1,\ldots,k\\}$ such that

$g(\overline{H})=\left(\bigcap_{i\in\lambda_{g}}L_{i}(\mathbf{x})\leq
0\right)\cap\left(\bigcap_{i\in\\{1,\ldots,k\\}\setminus\lambda_{g}}L_{i}(\mathbf{x})\geq
0\right)$

Then $\overline{H}\cap g(\overline{H})$ is given by

$H_{\lambda_{g}}=\left(\bigcap_{i\in\lambda_{g}}L_{i}(\mathbf{x})=0\right)\cap\left(\bigcap_{i\in\\{1,\ldots,k\\}\setminus\lambda_{g}}L_{i}(\mathbf{x})\geq
0\right)$

which is an intersection of walls of $H$. $H_{\lambda_{g}}$ is also the set of
points of $H$ fixed by the action of $g$.

######  Example 2.7.

In our primary example, $G=\mathfrak{S}_{n}$ acts on $\mathbb{R}^{n}$ via the
standard action given by the permutation of coordinates, i.e. for
$\sigma\in\mathfrak{S}_{n}$ and
$\mathbf{x}=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$ we have
$\sigma(\mathbf{x})=(x_{\sigma(1)},\ldots,x_{\sigma(n)})$. Here, we may take
as a fundamental region the interior of the Weyl chamber

$\mathcal{W}^{(n),\mathrm{o}}=\\{(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}\mid
x_{1}<\ldots<x_{n}\\}$

(see [3] Notation 9). The walls of the Weyl chamber correspond to the adjacent
transpositions $s_{i}=(i\;i+1)\in\mathfrak{S}_{n}$ (i.e. the standard Coxeter
generators of $\mathfrak{S}_{n}$). Each $s_{i}$ acts by reflecting through the
hyperplane given by $x_{i}=x_{i+1}$, so we have that the walls of the Weyl
chamber are the sets

$\mathcal{W}^{(n)}_{s_{i}}=\\{(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}\mid
x_{1}\leq\ldots\leq x_{i}=x_{i+1}\leq\ldots\leq x_{n}\\}$

for $1\leq i\leq n-1$ (see [3] Notation 11).

###  2.3. Simplicial Complexes

Our proofs make reference to both abstract and concrete simplicial complexes.
Accordingly, we present both viewpoints here, beginning with the concrete
setting.

######  Definition 2.8 (see [7] Section 4.2).

Let the points $v_{0},\ldots,v_{d}\in R^{n}$ be affine independent. Then the
_open simplex_ (of dimension $d$) with vertices $v_{0},\ldots,v_{d}$ is the
set

$\displaystyle\Delta(v_{0},\ldots,v_{d})=\\{x$ $\displaystyle\in R^{n}\mid
x=t_{1}v_{1}+\cdots+t_{d}v_{d}$ $\displaystyle\text{ for some
}t_{1},\ldots,t_{d}\in(0,1]\text{ with }t_{1}+\cdots+t_{d}=1\\}$

The corresponding _closed simplex_ is

$\displaystyle\overline{\Delta}(v_{0},\ldots,v_{d})=\\{x$ $\displaystyle\in
R^{n}\mid x=t_{1}v_{1}+\cdots+t_{d}v_{d}$ $\displaystyle\text{ for some
}t_{1},\ldots,t_{d}\in[0,1]\text{ with }t_{1}+\cdots+t_{d}=1\\}$

A _face_ of a simplex $\overline{\Delta}(v_{0},\ldots,v_{d})$ is a simplex
$\overline{\Delta}(u_{0},\ldots,u_{d^{\prime}})$ with vertex set
$\\{u_{0},\ldots,u_{d^{\prime}}\\}\subset\\{v_{0},\ldots,v_{d}\\}$.

If we are drawing our vertices from an indexed set $\\{v_{i}\\}_{i\in I}$, we
may simply refer to a simplex $\Delta(v_{i_{0}},\ldots,v_{i_{d}})$ as
$\Delta(i_{0},\ldots,i_{d})$.

######  Definition 2.9 (see [7] Section 4.2).

A (finite) _simplicial complex_ in $R^{n}$ is a finite collection $\Lambda$ of
open simplices in $R^{n}$ with the following properties

  1. (1)

If $\Delta\in\Lambda$, then for each face $\overline{\Delta}^{\prime}$ of
$\overline{\Delta}$, we have that $\Delta^{\prime}\in\Lambda$

  2. (2)

If $\Delta_{1},\Delta_{2}\in\Lambda$, then
$\overline{\Delta}_{1}\cap\overline{\Delta}_{2}=\overline{\Delta}_{3}$ for
some $\Delta_{3}\in\Lambda$.

Given a simplicial complex $\Lambda$ in $R^{n}$, we call
$\bigcup_{\Delta\in\Lambda}\Delta\subset R^{n}$ the _geometric realization_ of
$\Lambda$, and denote it $\lvert\Lambda\rvert$.

If not clear from context, we may include a subscript to indicate where the
realization is taking place. For example, say $\Lambda$ is a simplicial
complex in $R^{n}$ seen as a subset of $R^{n}\times R^{m}$. Then we may wish
to distinguish between $\lvert\Lambda\rvert_{R^{n}}$ and
$\lvert\Lambda\rvert_{R^{n+m}}$, the realizations of $\Lambda$ in $R^{n}$ and
$R^{n+m}$ respectively.

######  Definition 2.10.

Let $\Delta,\Delta^{\prime}\in\Lambda$ for some simplicial complex $\Lambda$.
Then $\Delta^{\prime}$ is a _subsimplex_ of $\Delta$ if
$\Delta^{\prime}\neq\Delta$ and $\Delta^{\prime}\subset\overline{\Delta}$ (in
other words, if $\overline{\Delta}^{\prime}$ is a proper face of
$\overline{\Delta}$).

######  Definition 2.11.

A _$k$ -flag_ of cells in a CW complex (so in particular, of simplices in a
simplicial complex) is a sequence $\sigma_{0},\ldots,\sigma_{k}$ of cells such
that $\sigma_{i}$ is contained in the boundary of $\sigma_{i-1}$ for each
$1\leq i\leq k$.

When speaking of symmetry for any CW complex (and in particular the geometric
realization of a simplicial complex $\Lambda$ in $\mathbb{R}^{n}$), we will in
general consider the action of $G$ on our decomposition to be that induced by
a given action of $G$ on our larger space:

######  Definition 2.12.

Let $G$ be a group acting on $\mathbb{R}^{n}$. A CW complex $X$ is _symmetric_
with respect to the action of $G$ if for each cell $\sigma\in\Lambda$ and each
$g\in G$, $g(\sigma)=\\{g(x)\mid x\in\sigma\\}$ is again a cell of $X$.

In particular, this means that $X$ is symmetric as a subset of
$\mathbb{R}^{n}$.

Now, we shift our attention to abstract simplicial complexes.

######  Definition 2.13 (from [14]).

An _abstract simplicial complex_ is a set $V$ of vertices and a set $\Lambda$
of finite nonempty subsets of $V$ which we consider to be simplices, having
the properties

  1. (1)

$\\{v\\}\in\Lambda$ for each $v\in V$ (each vertex is a simplex)

  2. (2)

any nonempty subset of a simplex is a simplex

One can and often does ignore the distinction between an abstract simplicial
complex and its set of simplices. An abstract simplicial complex also comes
with a geometric realization, which we will explicitly describe for the sake
of later use.

######  Definition 2.14 (see [14]).

Let $\Lambda$ be a nonempty abstract simplicial complex. The _geometric
realization_ of $\Lambda$, denoted by $\lvert\Lambda\rvert$, is the space
whose points are functions $\alpha$ from the set of vertices of $\Lambda$ to
the interval $[0,1]\subset\mathbb{R}$ such that

  1. (1)

The set $\\{v\in\Lambda\mid\alpha(v)\neq 0\\}$ is a simplex of $\Lambda$

  2. (2)

$\sum_{v\in\Lambda}\alpha(v)=1$

appropriately topologized (as described in [14]).

It will be convenient to refer to the point $\alpha$ using the notation
$\sum_{v\in\Lambda}\alpha(v)v$.

######  Definition 2.15 (from [14] Chapter 3 Section 2).

Let $\Lambda$ be any simplicial complex and let $X$ be a topological space
which is a subset of a real vector space, appropriately topologized (see [14]
for details; in particular euclidean space and geometric realizations of
simplicial complexes meet the criteria). A continuous map
$f:\lvert\Lambda\rvert\rightarrow X$ is said to be _linear_ (on $\Lambda$) if
for $\alpha\in\lvert\Lambda\rvert$, we have

$f(\alpha)=\sum_{v\in\Lambda}\alpha(v)f(v)$

Following [6], we will say an abstract simplicial complex is symmetric with
respect to a group $G$ acting on its set of vertices if the map given by $g$
is simplicial (i.e. $g$ carries simplices to simplices) for each $g\in G$. An
action on $\Lambda$ induces a linear action on $\lvert\Lambda\rvert$: namely,
$g(\alpha)$ is the function from $\Lambda$ to $[0,1]$ given by
$g(\alpha)(v)=\alpha(g(v))$.

Any simplicial complex $\Lambda$ comes with a _face poset_
$\mathscr{F}(\Lambda)$, where the simplicies of $\Lambda$ are ordered by
$\sigma_{1}\leq\sigma_{2}$ if $\overline{\sigma_{1}}$ is a face of
$\overline{\sigma_{2}}$. For $X$ a CW complex, we will still use the notation
$\mathscr{F}(X)$ for the cell poset of $X$ (which has as points the cells
$\sigma$ of $X$ and order given by $\sigma_{1}\leq\sigma_{2}$ if
$\sigma_{1}\subset\overline{\sigma_{2}}$).

###  2.4. Miscellaneous Background

The following notions will appear later in the paper.

######  Definition 2.16 (from [8] Chapter 9 Section 2).

Let $A\subset R^{n}$ and $x\in R^{n}\setminus A$. Then the _cone_ with vertex
$x$ and base $A$ is the set

$\\{ta+(1-t)x\mid a\in A,t\in[0,1]\\}$

(the union of all line segments from $x$ to a point in $A$).

######  Definition 2.17 (from [7] Section 3.2).

We say a family $\\{S_{x}\\}_{x\in R^{m}}$ of subsets of some real closed
field $R^{n}$ is a _definable family_ (with respect to some o-minimal
structure on $R$) if there is a definable set $S^{\prime}\subset R^{m+n}$ such
that $S_{x}$ is equal to $\\{y\in R^{n}\mid(x,y)\in S^{\prime}\\}$ for each
$x\in R^{m}$. It follows that each $S_{x}$ is a definable set.

The reader unfamiliar with o-minimal geometry is also encouraged to review the
cell decomposition theorem ([8] Chapter 3 Section 2 or [7] Section 2.2) before
approaching our equivariant triangulation proofs in Section 4.

For a connected topological space $X$, we let $\pi_{k}(X)$ denote the $k$th
homotopy group. Let $H_{k}(X)$ be the $k$th singular homology group with
coefficients in some fixed Abelian group, and denote by
$b_{k}(X)=\operatorname{rank}(H_{k}(X))$ the $k$th Betti number of $X$.
Throughout, we will use $\simeq$ to denote homotopy equivalence and $\cong$ to
denote group isomorphism.

The following two theorems are used heavily in Gabrielov and Vorobjov’s proofs
in [9] and referenced also in our own arguments. These do not require
equivariant versions; if a map $f:X\rightarrow Y$ is equivariant, the induced
homomorphisms of homotopy and homology groups will be as well.

######  Theorem 2.18 (Whitehead Theorem on Weak Homotopy Equivalence, [14]
7.6.24).

A map $f:X\rightarrow Y$ between connected CW complexes is a weak homotopy
equivalence (i.e. the induced homomorphism of homotopy groups
$f_{\\#,k}:\pi_{k}(X)\rightarrow\pi_{k}(Y)$ is an isomorphism for each $k>0$)
iff $f$ is a homotopy equivalence.

######  Theorem 2.19 (Whitehead Theorem on Homotopy and Homology, [14] 7.5.9).

Let $f:X\rightarrow Y$ be a continuous map between path connected topological
spaces. If there is a $k>0$ such that the induced homomorphism of homotopy
groups $f_{\\#,j}:\pi_{j}(X)\rightarrow\pi_{j}(Y)$ is an isomorphism for $j<k$
and an epimorphism for $j=k$, then the induced homomorphism of homology groups
$f_{*,j}:H_{j}(X)\rightarrow H_{j}(Y)$ is an isomorphism for $j<k$ and an
epimorphism for $j=k$.

##  3\. The Gabrielov-Vorobjov Construction

In this section, we describe the construction of the approximating set $T$, as
given by Gabrielov and Vorobjov in [9]. We also discuss the implications when
symmetry is introduced to the construction.

In order to construct $T$, we begin with families of compact sets
$\\{S_{\delta}\\}$ and $\\{S_{\delta,\varepsilon}\\}$ which represent $S$ in
some larger compact set $A$.

######  Definition 3.1 ([9] Definition 1.1).

Let $S\subset\mathbb{R}^{n}$ be definable, and let $A\subset\mathbb{R}^{n}$ be
a compact definable set with $S\subset A$. Let $\\{S_{\delta}\\}_{\delta>0}$
and $\\{S_{\delta,\varepsilon}\\}_{\delta,\varepsilon>0}$ be definable
families of compact subsets of $A$. We say $S$ is _represented_ by
$\\{S_{\delta}\\}_{\delta>0}$ and
$\\{S_{\delta,\varepsilon}\\}_{\delta,\varepsilon>0}$ in $A$ if we have that

  1. (a)

for all $\delta^{\prime},\delta\in(0,1)$, if $\delta^{\prime}>\delta$, then
$S_{\delta^{\prime}}\subset S_{\delta}$

  2. (b)

$S=\bigcup_{\delta>0}S_{\delta}$

and furthermore for each $\delta>0$

  1. (i)

for all $\varepsilon^{\prime},\varepsilon\in(0,1)$, if
$\varepsilon^{\prime}>\varepsilon$, then $S_{\delta,\varepsilon}\subset
S_{\delta,\varepsilon^{\prime}}$

  2. (ii)

$S_{\delta}=\bigcap_{\varepsilon>0}S_{\delta,\varepsilon}$

  3. (iii)

for all $\delta^{\prime}$ sufficiently smaller than $\delta$ and for all
$\varepsilon^{\prime}>0$ there exists a set $U\subset A$ with $U$ open in $A$
and $S_{\delta}\subset U\subset S_{\delta^{\prime},\varepsilon^{\prime}}$

We then take $T$ to be the union of a selection of finitely many of the sets
$S_{\delta,\varepsilon}$.

######  Definition 3.2 ([9] Definition 1.8).

For any nonnegative integer $m$ and parameters
$\varepsilon_{0},\delta_{0},\varepsilon_{1},\delta_{1},\ldots,\varepsilon_{m},\delta_{m}$,
we denote

$T=T(\varepsilon_{0},\delta_{0},\varepsilon_{1},\delta_{1},\ldots,\varepsilon_{m},\delta_{m}):=S_{\delta_{0},\varepsilon_{0}}\cup
S_{\delta_{1},\varepsilon_{1}}\cup\cdots\cup S_{\delta_{m},\varepsilon_{m}}$

Gabrielov and Vorobjov refer to the general case, in which $S$ is an arbitrary
definable set together with any representing families $\\{S_{\delta}\\}$ and
$\\{S_{\delta,\varepsilon}\\}$, as the _definable case_. If $S$ is a
$\mathcal{P}$-set for some collection $\mathcal{P}=\\{h_{1},\ldots,h_{s}\\}$
of definable continuous functions $h_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}$,
they prescribe particular choices of $A$, $\\{S_{\delta}\\}$, and
$\\{S_{\delta,\varepsilon}\\}$ (see Definition 3.6), which grants an extra
property (termed separability; see Definition 6.27 or [9] Section 5.2) that
allows for the stronger version of the results. We call this case the
_constructible case_. The choices in the constructible case also guarantee
that our sets $S_{\delta}$ and $S_{\delta,\varepsilon}$ are described by
definable functions closely related to the original functions $h_{i}$.

In these settings, Gabrielov and Vorobjov prove the following theorem. Note
that they show ([9] Lemma 1.9) that provided $m$ is at least $1$, there exists
a one-to-one correspondence between the connected components of $S$ and $T$.
This allows Gabrielov and Vorobjov to consider each connected component
individually and so ignore basepoint considerations, a luxury we do not have.

######  Theorem 3.3 ([9] Theorem 1.10).

  1. (i)

(Definable Case) For
$0<\varepsilon_{0}\ll\delta_{0}\ll\cdots\ll\varepsilon_{m}\ll\delta_{m}\ll 1$
and every $1\leq k\leq m$, there are epimorphisms

$\displaystyle\psi_{\\#,k}$ $\displaystyle:\pi_{k}(T)\rightarrow\pi_{k}(S)$
$\displaystyle\psi_{*,k}$ $\displaystyle:H_{k}(T)\rightarrow H_{k}(S)$

and in particular,
$\operatorname{rank}(H_{k}(S))\leq\operatorname{rank}(H_{k}(T))$

  2. (ii)

(Constructible Case) In the constructible case, for
$0<\varepsilon_{0}\ll\delta_{0}\ll\cdots\ll\varepsilon_{m}\ll\delta_{m}\ll 1$
and every $1\leq k\leq m-1$, $\psi_{\\#,k}$ and $\psi_{*,k}$ are isomorphisms.
In particular, $\operatorname{rank}(H_{k}(S))=\operatorname{rank}(H_{k}(T))$.
Moreover, if $m\geq\dim(S)$, then $T\simeq S$.

In applications, even in the definable case one may via this construction
obtain improved upper bounds on the first $m$ Betti numbers of a given set
$S$. Our primary setting of interest, the one considered by Basu and Reiner in
[3], utilizes the constructible case.

Our main theorem (Theorem 1.3) is an equivariant version of the above theorem.
It is clear from the definition of $T$ that, so long as each set in the family
$\\{S_{\delta,\varepsilon}\\}$ is symmetric relative to the action of some
group $G$, then $T$ will be as well. In the definable case, we need only
assume that we have chosen our families to consist of symmetric sets. In the
constructible case, we will show that if $S$ is symmetric and the collection
$\\{h_{1},\ldots,h_{s}\\}$ is invariant under the action of $G$, our choices
will produce symmetric sets. In subsequent sections, we show that for $G$ a
finite reflection group acting on $\mathbb{R}^{n}$, we can in fact construct
an equivariant map $\psi:T\rightarrow S$ which induces the desired
isomorphisms and epimorphisms $\psi_{\\#,k}$ and $\psi_{*,k}$ on the level of
homotopy and homology. The remainder of this section is devoted to describing
the sets $S_{\delta}$ and $S_{\delta,\varepsilon}$ in the constructible case.

###  3.1. Constructible Case: Bounding $S$

Let $S\subset R^{n}$ be definable, and assume $S$ is unbounded. Via the
conical structure at infinity of definable sets, there exists an $r\in R$,
$r>0$, such that $S$ is (definably) homotopy equivalent to
$S\cap\overline{B(0,r)}$. Basu, Pollack, and Roy in [1] show this for
semialgebraic sets. However, their proof holds for any o-minimal structure in
which addition and mulitpication are definable. The proof in [1] centers on
the local conical structure specifically of semialgebraic sets, but this
property (a consequence of Hardt triviality) holds for any o-minimal expansion
of some real closed field (see [8] Chapter 9 Theorem 2.3). The remaining steps
of their proof may be performed in any o-minimal structure provided it
contains addition and mulitpication.

######  Proposition 3.4 (Conical Structure at Infinity, see [1] Proposition
5.49).

Let $A\subset R^{n}$ be definable. Then there exists an $r\in R$, $r>0$, such
that $A$ is definably homotopy equivalent to $A\cap\overline{B(0,r)}$.
Specifically, there exists a continuous definable function $h:[0,1]\times
A\rightarrow A$ with $h(0,-)=\operatorname{id}_{A}$, $h(1,-)$ having image
contained in $A\cap\overline{B(0,r)}$, and with $h(t,a)=a$ for each
$t\in[0,1]$ and $a\in A\cap\overline{B(0,r)}$.

In our case, for any set $S\subset\mathbb{R}^{n}$ that is symmetric under the
action of a finite reflection group $G$, certainly $S\cap\overline{B(0,r)}$ is
as well. The inclusion $S\cap\overline{B(0,r)}\hookrightarrow S$ is also
clearly equivariant, and so induces equivariant isomorphisms of homotopy and
homology groups

$\displaystyle\pi_{k}(S\cap\overline{B(0,r)},*)$
$\displaystyle\rightarrow\pi_{k}(S,*)$ $\displaystyle
H_{k}(S\cap\overline{B(0,r)})$ $\displaystyle\rightarrow H_{k}(S)$

for each $k\geq 0$.

We now replace $S$ by $S\cap\overline{B(0,r)}$ to assume henceforth that $S$
is bounded. If $S$ was a $\mathcal{P}$-set for some
$\mathcal{P}=\\{h_{1},\ldots,h_{s}\\}$, we replace $\mathcal{P}$ with
$\mathcal{P}\cup\\{r^{2}-(X_{1}^{2}+\cdots X_{n}^{2})\\}$, and so increase $s$
by one. Since $r^{2}-(X_{1}^{2}+\cdots+X_{n}^{2})$ is symmetric relative to
the action of any finite reflection group $G$, our collection $\mathcal{P}$
remains invariant (and in fact if each $h_{i}\in\mathcal{P}$ was symmetric,
the same remains true after including this new function in our collection).
Note also that in the case that each $h_{i}(x)$ is a polynomial, the only
instance in which we have increased the maximum degree among our collection is
if each of $\\{h_{1},\ldots,h_{s}\\}$ had degree one.

This is slightly different from the procedure employed by Gabrielov and
Vorobjov in [9], where the larger compact set $A$ was taken as the definable
one-point compactification of $\mathbb{R}^{n}$. This choice aids certain
results on Betti numbers in [9], but we find our own method more convenient
when tracking the group’s action. Since Gabrielov and Vorobjov bound their
sets $\\{S_{\delta}\\}$ and $\\{S_{\delta,\varepsilon}\\}$ in the
constructible case by intersecting with closed balls of radius $1/\delta$, our
method does not stray too far from the intuition of the original. Our
assumption will increase the number of equations needed to define $T$, but
only slightly.

###  3.2. Constructible Case: the Sets $\\{S_{\delta}\\}$ and
$\\{S_{\delta,\varepsilon}\\}$

In the constructible case (with the assumption that $S$ is bounded), we form
families $\\{S_{\delta}\\}_{\delta>0}$ and
$\\{S_{\delta,\varepsilon}\\}_{\delta,\varepsilon>0}$ by decomposing $S$ into
sign sets of the functions defining $S$, and then contracting inequalities and
expanding equalities by a factor of $\delta$ or $\varepsilon$ respectively, in
a manner we now proceed to describe.

######  Definition 3.5 ([9] Definition 1.5).

Let $\mathcal{P}=\\{h_{1},\ldots,h_{s}\\}$ be a finite collection of functions
with each $h_{i}:R^{n}\rightarrow R$. Let $\\{I_{0},I_{+},I_{-}\\}$ be a
partition of $\\{1,\ldots,s\\}$. Then the set
$B_{\mathcal{P},(I_{0},I_{+},I_{-})}$ given by

$\\{\mathbf{x}\in R^{n}\mid\bigwedge_{i\in
I_{0}}(h_{i}(\mathbf{x})=0)\wedge\bigwedge_{i\in
I_{+}}(h_{i}(\mathbf{x})>0)\wedge\bigwedge_{i\in
I_{-}}(h_{i}(\mathbf{x})<0)\\}$

will be called the _sign set_ of $\mathcal{P}$ corresponding to the tuple
$(I_{0},I_{+},I_{-})$.

Note that any two distinct sign sets of some
$\mathcal{P}=\\{h_{1},\ldots,h_{s}\\}$ are disjoint, and that for $S$ a
$\mathcal{P}$-set, we may write $S$ as a union of sign sets of the functions
in $\mathcal{P}$. Though for a given collection $\mathcal{P}$ of functions,
some tuples $(I_{0},I_{+},I_{-})$ will produce empty sign sets, we will
exclude these tuples from the sign set decomposition of any $\mathcal{P}$-set.

In the constructible case, we make the following choices for $A$,
$\\{S_{\delta}\\}$ and $\\{S_{\delta,\varepsilon}\\}$.

######  Definition 3.6 (from [9]).

Let $S\subset\mathbb{R}^{n}$ be a bounded $\mathcal{P}$-set for some
collection $\mathcal{P}=\\{h_{1},\ldots,h_{s}\\}$ of continuous definable
functions $h:\mathbb{R}^{n}\rightarrow\mathbb{R}$. Let $\mathcal{B}$ denote
the set of tuples $(I_{0},I_{+},I_{-})$ corresponding to elements in the sign
set decomposition of $S$.

Let $r>0$ be such that $S\subset\overline{B(0,r)}$. Then take
$A=\overline{B(0,r^{\prime})}$ for some $r^{\prime}$ sufficiently larger than
$r$. For each $\delta>0$, we let $S_{\delta}$ be the union of sets defined by

$\bigwedge_{i\in I_{0}}(h_{i}=0)\wedge\bigwedge_{i\in
I_{+}}(h_{i}\geq\delta)\wedge\bigwedge_{i\in I_{-}}(h_{i}\leq-\delta)$

over all tuples $(I_{0},I_{+},I_{-})\in\mathcal{B}$. For
$\delta,\varepsilon>0$, we let $S_{\delta,\varepsilon}$ be the union of the
sets given by

$\bigwedge_{i\in I_{0}}(-\varepsilon\leq
h_{i}\leq\varepsilon)\wedge\bigwedge_{i\in
I_{+}}(h_{i}\geq\delta)\wedge\bigwedge_{i\in I_{-}}(h_{i}\leq-\delta)$

again over all tuples $(I_{0},I_{+},I_{-})\in\mathcal{B}$.

One may check that $S$ is indeed represented by the families
$\\{S_{\delta}\\}_{\delta>0}$ and
$\\{S_{\delta,\varepsilon}\\}_{\delta,\varepsilon>0}$ in $A$. It remains to
consider the conditions needed to ensure that the sets $S_{\delta}$ and
$S_{\delta,\varepsilon}$ are symmetric.

######  Proposition 3.7.

Assume we are in the constructible case. If $S$ is symmetric under the action
of $G$ and furthermore we have that for each $g\in G$, $\\{h_{1}\circ
g,\ldots,h_{s}\circ g\\}=\\{h_{1},\ldots,h_{s}\\}$, then every set in the
families $\\{S_{\delta}\\}_{\delta>0}$ and
$\\{S_{\delta,\varepsilon}\\}_{\delta,\varepsilon>0}$ is symmetric.

###### Proof.

Let $\mathcal{P}=\\{h_{1},\ldots,h_{s}\\}$. Note that for a sign set
$B=B_{\mathcal{P},(I_{0},I_{+},I_{-})}$ and any $g\in G$,
$g(B)=\\{g(\mathbf{x})\mid\mathbf{x}\in B\\}$ is given by

$\\{\mathbf{x}\in\mathbb{R}^{n}\mid\bigwedge_{i\in
I_{0}}(h_{i}(g^{-1}(\mathbf{x}))=0)\\\ \wedge\bigwedge_{i\in
I_{+}}(h_{i}(g^{-1}(\mathbf{x}))>0)\wedge\bigwedge_{i\in
I_{-}}(h_{i}(g^{-1}(\mathbf{x}))<0)\\}$

For each $g\in G$, we define a function on the indices
$\\{1,\ldots,s\\}\rightarrow\\{1,\ldots,s\\}$, given by $g(i)=j$ if
$h_{j}=h_{i}\circ g^{-1}$.

Let $\mathcal{B}$ be the collection of tuples $(I_{0},I_{+},I_{-})$
corresponding to the sign sets contained in $S$. We claim $\mathcal{B}$ has
the property that for each $g\in G$, if $(I_{0},I_{+},I_{-})\in\mathcal{B}$
then $(g(I_{0}),g(I_{+}),g(I_{-}))\in\mathcal{B}$. Indeed, if $B$ is the sign
set corresponding to $(I_{0},I_{+},I_{-})$, we have that $B\subset S$ and
hence $g(B)\subset S$. But we have seen above that $g(B)$ is the sign set
corresponding to $(g(I_{0}),g(I_{+}),g(I_{-}))$. The symmetry of each
$S_{\delta}$ and each $S_{\delta,\varepsilon}$ follows from this property and
the descriptions of each $S_{\delta}$ and $S_{\delta,\varepsilon}$ given in
Definition 3.6. ∎

In particular, if each $h_{i}$ is a symmetric function, then $h_{i}\circ
g=h_{i}$, and so clearly $\\{h_{1}\circ g,\ldots,h_{s}\circ
g\\}=\\{h_{1},\ldots,h_{s}\\}$ for every $g\in G$.

###  3.3. Connected Components

From the definitions (in both the definable and constructible cases), we see a
clear correspondence between the connected components of $T$ and those of $S$.
Let $S$ be represented in $A$ by any compact families $\\{S_{\delta}\\}$ and
$\\{S_{\delta,\varepsilon}\\}$, and let
$T=T(\varepsilon_{0},\delta_{0},\ldots,\varepsilon_{m},\delta_{m})$ be as
described in Definition 3.2. For
$0<\varepsilon_{0}\ll\delta_{0}\ll\cdots\ll\varepsilon_{m}\ll\delta_{m}\ll 1$,
let $\mathbf{S}$ and $\mathbf{T}$ denote the sets of connected components of
$S$ and $T$ respectively. Gabrielov and Vorobjov describe a map
$C:\mathbf{T}\rightarrow\mathbf{S}$. If $S^{\prime}$ is a connected component
of $S$, then those elements of $\mathbf{T}$ which map to $S^{\prime}$ under
$C$ are exactly those components which would make up the approximating set
defined relative to $S^{\prime}$ alone. Thus, by the symmetry of $T$ and $S$,
$C$ is equivariant. Gabrielov and Vorobjov demonstrate in [9] Lemma 1.9 that,
provided $m>0$, we may choose the conditions upon the parameters in such a way
that $C$ is bijective. Hence, though we can’t simply reduce to the case of a
connected set without risking a loss of symmetry, we may when needed use $C$
to pair the connected components of $S$ and $T$ and then apply the needed
theorems to each pair individually.

##  4\. Symmetric Triangulation

The construction of the maps in [9] relies upon triangulating the given set
$S$ inside the larger compact set $A$. Definable sets are triangulable (see
for example [7] Theorem 4.4, quoted below). However, in order to ensure
equivariance, we will need a triangulation that respects the action of our
group. That is the aim of this section.

###  4.1. Triangulation of Definable Sets

######  Definition 4.1 ([7] Section 4.3).

Let $A$ be a closed bounded definable subset of $R^{n}$. Then a
_triangulation_ $(\Lambda,\Phi)$ of $A$ is a finite simplicial complex
$\Lambda$ in $R$ together with a definable homeomorphism
$\Phi:\lvert\Lambda\rvert\rightarrow A$.

If we wish to decompose a set which is not necessarily closed into images of
simplices, we must do so within some larger compact set. For $A\subset R^{n}$
compact and $S\subset A$, we will say a triangulation $(\Lambda,\Phi)$ of $A$
is _adapted_ to $S$ if $S$ is a union of images by $\Phi$ of simplices of
$\Lambda$. Triangulations of compact definable sets always exist, and may be
adapted to any finite collection of subsets.

######  Theorem 4.2 (Triangulation of Definable Sets ([7] Theorem 4.4)).

Let $A$ be a closed and bounded definable subset of $R^{n}$, and let
$S_{1},\ldots,S_{l}$ be definable subsets of $A$. Then there exists a
triangulation $(\Lambda,\Phi)$ of $A$ adapted to each of $S_{1},\ldots,S_{l}$.
Furthermore, we may choose all vertices of $\Lambda$ to be in
$\mathbb{Q}^{n}$.

We need to triangulate the symmetric set $S$ in a manner that retains symmetry
relative to our group of interest. Specifically, we would like a triangulation
with the following properties:

######  Definition 4.3.

Let $G$ be a finite reflection group acting on $R^{n}$ and let $A\subset
R^{n}$ be a closed bounded definable set symmetric relative to $G$. A
triangulation $(\Lambda,\Phi)$ of $A$ is said to be an _equivariant
triangulation_ if $\Lambda$ is symmetric as a simplicial complex and the map
$\Phi:\lvert\Lambda\rvert\rightarrow A$ is equivariant.

Our general strategy for obtaining an equivariant triangulation of $A$ adapted
to $S$ involves triangulating a fundamental region of $\mathbb{R}^{n}$, and
then applying the action of $G$. This strategy has appeared in other contexts;
see in particular [12] which proves such a result for smooth manifolds. We
would like to establish a proof in the definable case. Furthermore, we would
like our proof to follow the spirit of [7] Theorem 4.4, in which we create a
concrete simplicial complex in $\mathbb{R}^{n}$ symmetric relative to our
existing action of $G$.

In order to make this approach feasible in the concrete setting, we must
triangulate the portion of $A$ lying in our fundamental region in such a way
that the resulting simplicial complex lies within the same fundamental region
of $R^{n}$, with points lying in its walls remaining as such. This motivates
the following definition.

######  Definition 4.4.

Let $B\subset R^{n}$. If a triangulation $(\Lambda,\Phi)$ of $A\subset R^{n}$
is adapted to $A\cap B$ and furthermore we have $x\in A\cap B$ if and only if
$\Phi^{-1}(x)\in\lvert\Lambda\rvert\cap B$, then we say the triangulation
_respects_ the set $B$.

To show that we may find a triangulation respecting our fundamental region and
its walls, we will in fact prove something slightly stronger: for an
arrangement of affine hyperplanes given by some collection
$\\{L_{1}(\mathbf{x})=0,\ldots,L_{k}(\mathbf{x})=0\\}$ of linear functions, we
can find a triangulation of $A$ that respects any of the various half-spaces
determined by our hyperplanes (for convenience, we will phrase this as
respecting sign sets of $\\{L_{1},\ldots,L_{k}\\}$). We lose the property that
all vertices lie in $\mathbb{Q}^{n}$, but the vertices of our triangulation
will be rational expressions in the coefficients of $L_{1},\ldots,L_{k}$.
Since our application does not in fact require any such condition on the
vertices, this is not an issue here.

The reader may wish to study the proof of the following lemma in conjunction
with Example 4.6, in order to see concrete usage of the notation set forth in
the proof.

######  Lemma 4.5.

Let $A\subset R^{n}$ be closed, bounded, and definable, and let
$S_{1},\ldots,S_{l}$ be definable subsets of $A$. Let
$\\{L_{1},\ldots,L_{k}\\}$ be a collection of functions with each
$L_{i}:R^{n}\rightarrow R$ given by
$L_{i}(\mathbf{x})=a_{i,0}+a_{i,1}x_{1}+\cdots+a_{i,n}x_{n}$ for some
$a_{i,j}\in R$. Then there exists a triangulation $(\Lambda,\Phi)$ of $A$ a
which is adapted to each $S_{1},\ldots,S_{l}$ and which respects all sign sets
of $\\{L_{1},\ldots,L_{k}\\}$. Furthermore, we can choose this triangulation
so that the coordinates of all vertices are in $\mathbb{Q}(\\{a_{i,j}\mid
1\leq i\leq k,0\leq j\leq n\\})^{n}$.

###### Proof.

We can obtain a triangulation with the desired properties by way of a few
slight modifications to the proof of Theorem 4.2 presented in [7].
Specifically, the proof there chooses a number of points $b_{\nu}$ which serve
as vertices in the simplicial complex $\Lambda$. We must add the requirement
that our points $b_{\nu}$ belong to the correct sign sets, and confirm that
our additional claims hold with this adjustment.

The proof in [7] proceeds by induction on dimension. In the case $n=1$, we
have that $L_{i}=a_{i,0}+a_{i,1}x_{1}$ for each $i$. We may assume without
loss of generality that $a_{i,1}\neq 0$ for every $i$. Set
$c_{i}=-\frac{a_{i,0}}{a_{i,1}}$. Reorder and remove duplicates to assume we
have points $c_{1}<\cdots<c_{k}\in R$. Then the sign sets we must respect are
the points and intervals $(-\infty,c_{1})$, $\\{c_{i}\\}$ for each $i$,
$(c_{i},c_{i+1})$ for $1\leq i\leq k-1$, and $(c_{k},\infty)$. As [7]
describes, we choose points $\xi_{1}<\cdots<\xi_{p}\in R$ so that each of $A$,
$S_{1}\ldots,S_{l}$, and each nonempty intersection of $A$ and a sign set of
$\\{L_{1},\ldots,L_{k}\\}$ can be written as a union of points and intervals
$\\{\xi_{\mu}\\}$ and $(\xi_{\mu},\xi_{\mu+1})$.

We define a map $\tau:\\{\xi_{1},\ldots,\xi_{p}\\}\rightarrow R$ as follows:

  * •

If $k=0$, then let $\tau(\xi_{i})=i-1$ for each $i$.

Otherwise,

  * •

If $\xi_{\mu}=c_{i}$ for some $i$, let $\tau(\xi_{\mu})=c_{i}$

  * •

Let $\xi_{0,1}<\cdots<\xi_{0,p_{0}}$ with
$\\{\xi_{0,1},\ldots,\xi_{0,p_{0}}\\}=\\{\xi_{1},\ldots,\xi_{p}\\}\cap(-\infty,c_{1})$.
Then set $\tau(\xi_{0,\mu})=c_{1}-p_{0}+\mu-1$

  * •

Let $\xi_{i,1}<\cdots<\xi_{i,p_{i}}$ with
$\\{\xi_{i,1},\ldots,\xi_{i,p_{i}}\\}=\\{\xi_{1},\ldots,\xi_{p}\\}\cap(c_{i},c_{i+1})$
(where $1\leq i\leq k-1$). Then set
$\tau(\xi_{i,\mu})=c_{i}+\mu\frac{c_{i+1}-c_{i}}{p_{i}+1}$

  * •

Let $\xi_{k,1}<\cdots<\xi_{k,p_{k}}$ with
$\\{\xi_{k,1},\ldots,\xi_{k,p_{k}}\\}=\\{\xi_{1},\ldots,\xi_{p}\\}\cap(c_{k},\infty)$.
Then set $\tau(\xi_{k,\mu})=c_{k}+\mu$

In either case, we have that $\tau$ preserves order and respects containment
in sign sets of $\\{L_{1},\ldots,L_{k}\\}$. Each $\tau(\xi_{\mu})$ is also a
rational expression in the coefficients of the $L_{i}$’s. We extend $\tau$
piecewise linearly to obtain a definable order preserving homeomorphism
$\tau^{\prime}:R\rightarrow R$, and take our simplicial complex $\Lambda$ to
be $\tau^{\prime}(A)$ and
$\Phi=(\tau^{\prime})^{-1}_{\restriction\lvert\Lambda\rvert}:\lvert\Lambda\rvert\rightarrow
A$.

Assume now that $n>1$, and that our claim holds in dimension $n-1$. We can
safely follow the procedure in [7] to assume each $S_{i}$ is closed. We denote
the boundary of $A$ by $F_{0}$, and the boundary of $S_{i}$ by $F_{i}$ for
each $1\leq i\leq l$. Choose a cell decomposition of $R^{n}$ adapted to each
of $F_{0},\ldots,F_{l}$ and also to the sign sets of
$\\{L_{1},\ldots,L_{k}\\}$. If we let $p:R^{n}\rightarrow R^{n-1}$ be the
projection onto the first $n-1$ coordinates, we have that our cell
decomposition of $R^{n}$ gives us a decomposition of $p(A)$ into definably
connected definable subsets $X_{\alpha}$, and for each $X_{\alpha}$ a finite
number of continuous functions
$\zeta_{\alpha,1}<\ldots<\zeta_{\alpha,m_{\alpha}}:X_{\alpha}\rightarrow R$.
This means each $F_{i}$ can be written as a union of graphs
$\zeta_{\alpha,\mu}$. Note that because our cell decomposition of $R^{n}$ is
adapted to the sign sets of $\\{L_{1},\ldots,L_{k}\\}$, each cell in $A$ is
contained in precisely one such sign set. Furthermore for each sign set $B$ of
$\\{L_{1},\ldots,L_{k}\\}$, we have that $A\cap B$ is a union of cells in the
decomposition.

Consider the sets given by $\bigcap_{i\in\lambda}\\{L_{i}(\mathbf{x})=0\\}$
for various selections of $\lambda\subset\\{1,\ldots,k\\}$. If for a given
$\lambda$,
$P^{\prime}_{\lambda}=p\left(\bigcap_{i\in\lambda}\\{L_{i}(\mathbf{x})=0\\}\right)$
has dimension $n-2$, then $P^{\prime}_{\lambda}$ is an affine hyperplane of
$R^{n-1}$, and therefore is the zero set for some linear function
$L_{\lambda}^{\prime}(\mathbf{x})=a_{\lambda,0}^{\prime}+a_{\lambda,1}^{\prime}x_{1}+\cdots+a_{\lambda,n-1}^{\prime}x_{n-1}:R^{n-1}\rightarrow
R$. Furthermore, we can choose $L_{\lambda}^{\prime}$ such that each
$a_{\lambda,j}^{\prime}\in R$ is contained in $\mathbb{Q}(\\{a_{i,j}\mid
i\in\lambda,0\leq j\leq n\\})$. Removing any duplicates, we obtain a
collection $\\{L_{1}^{\prime},\ldots,L_{k^{\prime}}^{\prime}\\}$ of functions
$L_{i}^{\prime}(\mathbf{x})=a_{i,0}^{\prime}+a_{i,1}^{\prime}x_{1}+\cdots+a_{i,n-1}^{\prime}x_{n-1}:R^{n-1}\rightarrow
R$ having the property that all coefficients $a_{i,j}^{\prime}$ are rational
expressions in the coefficients $\\{a_{i,j}\\}$ of the functions $L_{i}$.
Furthermore, if $B$ is a sign set of $\\{L_{1},\ldots,L_{k}\\}$, then $p(B)$
is a union of sign sets of
$\\{L_{1}^{\prime},\ldots,L_{k^{\prime}}^{\prime}\\}$. Since our cell
decomposition was adapted to all sign sets $B$ of $\\{L_{1},\ldots,L_{k}\\}$,
we have that each $X_{\alpha}$ is contained in exactly one sign set of
$\\{L_{1}^{\prime},\dots,L_{k^{\prime}}^{\prime}\\}$.

Let $(\Lambda_{n-1},\Psi_{n-1})$ be a triangulation of $p(A)\subset R^{n-1}$
which is adapted to each $X_{\alpha}$ and respects each sign set of
$\\{L_{1}^{\prime},\ldots,L_{k^{\prime}}^{\prime}\\}$. Consider (as [7] does)
the set

$A^{\prime}=\\{(x^{\prime},x_{n})\in\lvert\Lambda_{n-1}\rvert\times
R\mid(\Psi_{n-1}(x^{\prime}),x_{n})\in A\\}$

Note that the map $\Psi=(\Psi_{n-1},\operatorname{id}):A^{\prime}\rightarrow
A$ is a definable homeomorphism. Let $\\{\delta_{\beta}\\}$ denote the
(finite) collection of all simplices of $\Lambda_{n-1}$. Since $\Psi_{n-1}$ is
such that for each cell $X_{\alpha}\subset p(A)$,
$\Psi_{n-1}^{-1}(X_{\alpha})$ is a union of simplices of $\Lambda_{n-1}$, we
have that for each $\delta_{\beta}$ there is precisely one $\alpha$ so that
$\Psi_{n-1}(\delta_{\beta})\subset X_{\alpha}$. Consider the graphs
$\zeta_{\alpha,1}<\cdots<\zeta_{\alpha,m_{\alpha}}$ defined on each
$X_{\alpha}$ in our cell decomposition of $A$. Then for a given $\beta$ and
$1\leq\mu\leq m_{\beta}:=m_{\alpha}$,
$\xi_{\beta,\mu}=\zeta_{\alpha,\mu}\circ\Psi_{n-1}\restriction_{\delta_{\beta}}:\delta_{\beta}\rightarrow
R$ is such that $\Psi\circ\xi_{\beta,\mu}=\zeta_{\alpha,\mu}\circ\Psi_{n-1}$
on $\delta_{\beta}$. We also have that on each $\delta_{\beta}$,
$\xi_{\beta,1}<\cdots<\xi_{\beta,m_{\beta}}$. Let $\\{C_{\nu}\\}$ be the
(finite) collection of all graphs $\xi_{\beta,\mu}:\delta_{\beta}\rightarrow
R$ and bands $(\xi_{\beta,\mu},\xi_{\beta,\mu+1})\subset\delta_{\beta}\times
R$ which are contained in $A^{\prime}$. Note that for each $\nu$, there exists
one sign set $B$ of $\\{L_{1},\ldots,L_{m}\\}$ such that $\Psi(C_{\nu})\subset
B$.

For each $\delta_{\beta}$, let
$b(\delta_{\beta})=((b(\delta_{\beta}))_{1},\ldots,(b(\delta_{\beta}))_{n-1})$
denote the barycenter of the simplex $\delta_{\beta}$. For each cell $C_{\nu}$
in $A^{\prime}$, we will choose a point $b_{\nu}$ to serve as a vertex in our
new simplicial complex $\Lambda$. Consider a simplex $\delta_{\beta}$ in
$\Lambda_{n-1}$. Let

$\displaystyle I$ $\displaystyle=\\{1\leq i\leq k\mid
L_{i}(\Psi_{n-1}(b(\delta_{\beta})),x_{n})=0\text{ has a unique solution}\\}$
$\displaystyle=\\{1\leq i\leq k\mid L_{i}(b(\delta_{\beta}),x_{n})=0\text{ has
a unique solution}\\}$

For $i\in I$, let $c_{\beta,i}\in R$ be such that
$L_{i}(\Psi_{n-1}(b(\delta_{\beta})),c_{\beta,i})=0$, and let
$c_{\beta,i}^{\prime}\in R$ be such that
$L_{i}(b(\delta_{\beta}),c_{\beta,i}^{\prime})=0$. Because our triangulation
of $A^{\prime}$ respects
$\\{L_{1}^{\prime},\ldots,L_{k^{\prime}}^{\prime}\\}$, we know that the map of
subsets $\\{c_{\beta,i}\mid i\in I\\}\rightarrow\\{c^{\prime}_{\beta,i}\mid
i\in I\\}$ of $R$ given by $c_{\beta,i}\mapsto c^{\prime}_{\beta,i}$ is a
well-defined, order preserving bijection. So, let
$\operatorname{card}(\\{c_{\beta,i}\mid i\in I\\})=k_{\beta}$, and reorder and
remove skipped indices and duplicates, to assume
$c_{\beta,1}<\cdots<c_{\beta,k_{\beta}}$ (and in this same indexing
$c_{\beta,1}^{\prime}<\cdots<c_{\beta,k_{\beta}}^{\prime}$).

For each cell $C_{\nu}$ defined on $\delta_{\beta}$, let

$b_{\nu}=((b(\delta_{\beta}))_{1},\ldots,(b(\delta_{\beta}))_{n-1},(b_{\nu})_{n})$

where $(b_{\nu})_{n}$ is assigned as follows. First, assume $C_{\nu}$ is the
graph of the function $\xi_{\beta,\mu}$. Then we assign $(b_{\nu})_{n}$
according to a schema similar to the one appearing in the $n=1$ case:

  * •

If $k_{\beta}=0$, let $(b_{\nu})_{n}=\mu-1$

Otherwise,

  * •

If $\xi_{\beta,\mu}(b(\delta_{\beta}))=c_{\beta,i}$ for some $i$, let
$(b_{\nu})_{n}=c_{\beta,i}^{\prime}$

  * •

Let $\xi_{\beta,0,1}<\cdots<\xi_{\beta,0,p_{0}}$ be those graphs with
$\xi_{\beta,0,\mu}(b(\delta_{\beta}))<c_{\beta,1}$. Then for $C_{\nu}$
corresponding to $\xi_{\beta,0,\mu}$, set
$(b_{\nu})_{n}=c_{\beta,1}^{\prime}-p_{0}+\mu-1$

  * •

For each $1\leq i<k_{\beta}$, let $\xi_{\beta,i,1}<\cdots<\xi_{\beta,i,p_{i}}$
be those graphs with
$c_{\beta,i}<\xi_{\beta,0,\mu}(b(\delta_{\beta}))<c_{\beta,i+1}$. Then for
$C_{\nu}$ corresponding to $\xi_{\beta,i,\mu}$, set
$(b_{\nu})_{n}=c_{\beta,i}^{\prime}+\mu\frac{c_{\beta,i+1}^{\prime}-c_{\beta,i}^{\prime}}{p_{i}+1}$

  * •

Let $\xi_{\beta,k_{\beta},1}<\cdots<\xi_{\beta,k_{\beta},p_{k_{\beta}}}$ be
those graphs such that
$c_{\beta,k_{\beta}}<\xi_{\beta,0,\mu}(b(\delta_{\beta}))$. Then for $C_{\nu}$
corresponding to $\xi_{\beta,k_{\beta},\mu}$, set
$(b_{\nu})_{n}=c_{\beta,k_{\beta}}^{\prime}+\mu$

Finally, if $C_{\nu}$ is a band $(\xi_{\beta,\mu},\xi_{\beta,\mu+1})$, set
$(b_{\nu})_{n}=\frac{(b_{\nu_{0}})_{n}+(b_{\nu_{1}})_{n}}{2}$, where
$C_{\nu_{0}}$ and $C_{\nu_{1}}$ correspond to the graphs $\xi_{\beta,\mu}$ and
$\xi_{\beta,\mu+1}$ respectively (note that since $A^{\prime}$ is bounded, all
of our bands appearing among the sets $C_{\nu}$ are bounded). Then for any
cell $C_{\nu}$, we have that the point $b_{\nu}$ is in the same sign set of
$\\{L_{1},\ldots,L_{k}\\}$ as $\Psi(C_{\nu})$ is. Note also that each
$c_{\beta,i}^{\prime}$ is a rational expression in the coordinates of
$b(\delta_{\beta})$ and the coefficients of $L_{i}$. By our inductive
hypothesis, all vertices in $\Lambda_{n-1}$ have coordinates which are
rational expressions in the coefficients of
$L_{1}^{\prime},\ldots,L_{k^{\prime}}^{\prime}$, and hence of
$L_{1},\ldots,L_{k}$. Hence the same applies to our barycenters
$b(\delta_{\beta})$ for $\delta_{\beta}\in\Lambda_{n-1}$, and so by the
definition of $(b_{\nu})_{n}$ above, each $b_{\nu}$ has coefficients which are
in $\mathbb{Q}(\\{a_{i,j}\\})^{n}$.

For each cell $C_{\nu}$, [7] next builds a polyhedron $\overline{D}_{\nu}$
together with its subdivision into simplices. The procedure occurs inductively
on the dimension of $\overline{C}_{\nu}$: if $\overline{C}_{\nu}$ is a point,
take $\overline{D}_{\nu}$ to be $\\{b_{\nu}\\}$. Otherwise, take
$\overline{D}_{\nu}$ as the cone from $b_{\nu}$ to the union of all
$\overline{D}_{\nu^{\prime}}$ with
$\overline{C}_{\nu^{\prime}}\subset\partial(\overline{C}_{\nu})$. The
decomposition of $\overline{D}_{\nu}$ into simplices comes from taking cones
with vertex $b_{\nu}$ and base a simplex contained in
$\overline{D}_{\nu^{\prime}}$ for some
$D_{\nu^{\prime}}\subset\partial(\overline{D}_{\nu})$.

Taking all these simplices together, we obtain our desired simplicial complex
$\Lambda$ with $\lvert\Lambda\rvert=\bigcup\overline{D}_{\nu}$. Of primary
note is the fact that the vertices of all simplices come from among the points
$b_{\nu}$. Hence because $L_{1}=0,\ldots,L_{k}=0$ define affine subspaces of
$R^{n}$, we have that for a given sign set $B$ of $\\{L_{1},\ldots,L_{k}\\}$,
a (relatively open) simplex $\Delta(b_{\nu_{1}},\ldots,b_{\nu_{m}})$ in
$\Lambda$ is either contained in $B$ (if all $b_{\nu_{i}}$ are in $B$) or
disjoint from $B$ (else).

For each $\overline{D}_{\nu}$, we define a preparatory homeomorphism
$\theta_{\nu}:\overline{D}_{\nu}\rightarrow\overline{C}_{\nu}$ as follows: if
$\overline{C}_{\nu}$ is a graph $\overline{\xi}_{\beta,\mu}$ (see [7] for a
justification of why we may continuously extend $\xi_{\beta,\mu}$ to the
closed simplex $\overline{\delta}_{\beta}$), let
$\theta_{\nu}(x^{\prime},x_{n})=(x^{\prime},\overline{\xi}_{\beta,\mu}(x^{\prime}))$.
If $C_{\nu}$ is a closed band
$[\overline{\xi}_{\beta,\mu},\overline{\xi}_{\beta,\mu+1}]$, we map each
segment $(\\{x^{\prime}\\}\times R)\cap\overline{D}_{\nu}$ affinely to the
corresponding segment
$\\{x^{\prime}\\}\times[\overline{\xi}_{\beta,\mu}(x^{\prime}),\overline{\xi}_{\beta,\mu+1}(x^{\prime})]$.
Composing with $\Psi$ (and by convexity of our sign sets), we have that for a
simplex $\Delta$ of $\Lambda$ with $\Delta\subset\overline{D}_{\nu}$,
$\Psi\circ\theta_{\nu}(\Delta)\subset B\cap A$ iff $\Delta\subset B$.

Unfortunately, we cannot simply piece together our maps
$\Psi\circ\theta_{\nu}$ to obtain the desired homeomorphism
$\Phi:\lvert\Lambda\rvert\rightarrow A$. Instead, we construct a new
$\Phi^{\prime}:\lvert\Lambda\rvert\rightarrow A^{\prime}$, inducting on the
dimension of $\overline{D}_{\nu}$. If $\overline{D}_{\nu}$ is a point, take
$\Phi^{\prime}_{\nu}:\overline{D}_{\nu}\rightarrow\overline{C}_{\nu}$ to be
$\theta_{\nu}$. Otherwise, we can construct a homeomorphism
$\rho_{\nu}:\partial(\overline{D}_{\nu})\rightarrow\partial(\overline{D}_{\nu})$
by specifying that
${\rho_{\nu}}\restriction_{\overline{D}_{\nu^{\prime}}}=\theta_{\nu}^{-1}\circ\Phi^{\prime}_{\nu^{\prime}}$
for each $\overline{D}_{\nu^{\prime}}\subset\partial(\overline{D}_{\nu})$. We
use the conic structure of $\overline{D}_{\nu}$ to extend $\rho_{\nu}$ to a
homeomorphism $\eta_{\nu}:\overline{D}_{\nu}\rightarrow\overline{D}_{\nu}$,
and set $\Phi^{\prime}_{\nu}=\theta_{\nu}\circ\eta_{\nu}$. Now,
$\Phi^{\prime}$ given by
$\Phi^{\prime}_{\restriction\overline{D}_{\nu}}=\Phi^{\prime}_{\nu}$ is well
defined even on the boundaries of the sets $\overline{D}_{\nu}$. Finally, set
$\Phi:\lvert\Lambda\rvert\rightarrow A$ to be $\Psi\circ\Phi^{\prime}$.

That the triangulation $(\Lambda,\Phi)$ is adapted to $S_{1},\ldots,S_{l}$ and
also $A\cap B$ for each sign set $B$ of $\\{L_{1},\ldots,L_{m}\\}$ follows as
from the proof in [7]. We must still check that the triangulation respects all
of our sign sets. Let $x\in\lvert\Lambda\rvert$, and let $\Delta\in\Lambda$ be
the unique simplex with $x\in\Delta$. Take $\overline{D}_{\nu}$ to be of
minimal dimension with $\Delta\subset\overline{D}_{\nu}$. If
$\overline{D}_{\nu}$ is a point, then
$\overline{D}_{\nu}=\Delta=\\{x\\}=\\{b_{\nu}\\}$, and we have already
established that for a given sign set $B$, $x=b_{\nu}\in B$ iff
$\Phi(x)=\Psi\circ\theta_{\nu}(x)\in B$. Say that
$\operatorname{dim}(\overline{D}_{\nu})>0$. Then
$\Delta=\Delta(b_{\nu_{0}},\ldots,b_{\nu_{q-1}},b_{\nu_{q}}=b_{\nu})$, and
there is some $\nu^{\prime}$ such that
$\overline{D}_{\nu^{\prime}}\subset\partial(\overline{D}_{\nu})$ and
$\overline{\Delta^{\prime}}=\overline{\Delta}(b_{\nu_{0}},\ldots,b_{\nu_{q-1}})\subset\overline{D}_{\nu^{\prime}}$.
Then assuming we have established that $\Phi$ respects sign sets of
$\\{L_{1},\ldots,L_{m}\\}$ on $\overline{D}_{\nu^{\prime}}$, we know that
${\rho_{\nu}}\restriction_{\overline{D}_{\nu^{\prime}}}=\theta_{\nu}^{-1}\circ\Phi^{\prime}_{\nu^{\prime}}$
carries $\Delta^{\prime}$ to a subset of some $B\cap\lvert\Lambda\rvert$ iff
$\Delta^{\prime}$ is already a subset of $B\cap\lvert\Lambda\rvert$. Writing
any $y\in\overline{D}_{\nu}$ (uniquely) as $ty^{\prime}+(1-t)b_{\nu}$ for some
$t\in[0,1]$ and the proper choice of
$y^{\prime}\in\partial(\overline{D}_{\nu})$, we have that
$\eta_{\nu}(y)=t\rho(y^{\prime})+(1-t)b_{\nu}$. Then for our given
$x\in\Delta$, $x\in B$ iff $b_{\nu_{1}},\ldots,b_{\nu_{q}}\in B$ iff
$x^{\prime}\in\Delta^{\prime}\subset B$ and $b_{\nu}\in B$ iff
$\rho_{\nu}(x^{\prime})\in B$ and $b_{\nu}\in B$ iff $\eta_{\nu}(x)\in B$.
Then since $\Phi^{\prime}_{\nu}=\theta_{\nu}\circ\eta_{\nu}$, we have that
$x\in B$ iff $\Phi(x)=\Psi\circ\Phi^{\prime}(x)\in B$. ∎

######  Example 4.6.

Let $A=\overline{B(0,1)}\subset\mathbb{R}^{2}$. We will illustrate how Lemma
4.5 may be applied to give a triangulation of $A$ respecting sign sets of
$\\{L_{1}=y,L_{2}=x-y\\}$. Throughout, we will associate objects to their
notation in the proof of Lemma 4.5, so that the example may aid in the parsing
of the proof.

Figure 1. Cell decomposition adapted to $A$ and to sets $\\{y=x\\}$ and
$\\{y=0\\}$

We begin with a cell decomposition of $\mathbb{R}^{2}$ adapted to $A$ and to
all sign sets of $\\{L_{1},L_{2}\\}$. Following the most obvious choice of
decomposition, we obtain a subdivision of $\mathbb{R}^{1}$ into points
$\\{-1\\}$, $\\{-\sqrt{2}/2\\}$, $\\{0\\}$, $\\{\sqrt{2}/2\\}$, and $\\{1\\}$
and the intervals between them. The full cell decomposition is shown in Figure
1. Projecting to $\mathbb{R}^{1}$ for our induction, we must obtain a
triangulation $(\Lambda_{n-1},\Psi_{n-1})$ of $[-1,1]$ adapted to the sets
$\\{-1\\}$, $(-1,-\sqrt{2}/2)$, $\\{-\sqrt{2}/2\\}$, $(-\sqrt{2}/2,0)$,
$\\{0\\}$, $(0,\sqrt{2}/2)$, $\\{\sqrt{2}/2\\}$, $(\sqrt{2}/2,1)$, and
$\\{1\\}$ and respecting the sets $(-\infty,0)$, $\\{0\\}$, and $(0,\infty)$.

We apply the $n=1$ case of the algorithm outlined in the proof of Lemma 4.5.
In that notation, we have $k=1$, $c_{1}=0$, and
$\xi_{1}=-1,\xi_{2}=-\sqrt{2}/2,\xi_{3}=0,\xi_{4}=\sqrt{2}/2,\xi_{5}=1$. Then
the map $\tau$ is given by $\xi_{1}\mapsto-2,\xi_{2}\mapsto-1,\xi_{3}\mapsto
0,\xi_{4}\mapsto 1,\xi_{5}\mapsto 2$. Hence our triangulation
$(\Lambda_{n-1},\Psi_{n-1})$ of $p(A)$ is such that
$\lvert\Lambda_{n-1}\rvert=[-2,2]$ and
$\Psi_{n-1}:\lvert\Lambda_{n-1}\rvert\rightarrow p(A)$ is induced by piecewise
linearly extending the pairings given by $\tau$.

To construct our triangulation of $A$ itself, we first identify our vertices
$b_{\nu}$. For the sake of example, we will concentrate on the simplex
$\delta_{\beta}=(0,1)$ of $\Lambda_{n-1}$. We have that
$\Psi_{n-1}(\delta_{\beta})\subset X_{\alpha}=(0,\sqrt{2}/2)$ in our cell
decomposition of $p(A)$ (in fact, in this case
$\Psi(\delta_{\beta})=X_{\alpha}$), and that the barycenter
$b(\delta_{\beta})=\frac{1}{2}\in\mathbb{R}^{1}$. In the cell decomposition of
$A$, we have four graphs defined on $(0,\sqrt{2}/2)$: let $\zeta_{1}$ be the
lower semicircle, $\zeta_{2}$ be the line $y=0$, $\zeta_{3}$ be the line
$y=x$, and $\zeta_{4}$ be the upper semicircle. There are a total of seven
cells (graphs and bands) defined on this interval and contained in $A$. In
$A$, the line $\\{x=\Psi_{n-1}(b(\delta_{\beta}))\\}$ (that is,
$\\{x=\sqrt{2}/4\\}$) meets $\\{L_{1}=0\\}$ and $\\{L_{2}=0\\}$ at $y$-values
of $c_{\beta,1}=0$ and $c_{\beta,2}=\sqrt{2}/4$ respectively. Translating to
our triangulation built upon $\Lambda_{n-1}$, we have that
$\\{x=b(\delta_{\beta})\\}$ meets $\\{L_{1}=0\\}$ and $\\{L_{2}=0\\}$ at
$c^{\prime}_{\beta,1}=0$ and $c^{\prime}_{\beta,2}=\frac{1}{2}$. In order to
preserve this correspondence, we assign our vertices
$b_{\nu}\in\mathbb{R}^{2}$ as follows.

  * •

$C_{\nu}$ corresponds to the graph $\zeta_{1}$: $b_{\nu}=(\frac{1}{2},-1)$

  * •

$C_{\nu}$ corresponds to the graph $\zeta_{2}$: $b_{\nu}=(\frac{1}{2},0)$

  * •

$C_{\nu}$ corresponds to the graph $\zeta_{3}$:
$b_{\nu}=(\frac{1}{2},\frac{1}{2})$

  * •

$C_{\nu}$ corresponds to the graph $\zeta_{4}$:
$b_{\nu}=(\frac{1}{2},\frac{3}{2})$

  * •

$C_{\nu}$ corresponds to the band $(\zeta_{1},\zeta_{2})$:
$b_{\nu}=(\frac{1}{2},-\frac{1}{2})$

  * •

$C_{\nu}$ corresponds to the band $(\zeta_{2},\zeta_{3})$:
$b_{\nu}=(\frac{1}{2},\frac{1}{4})$

  * •

$C_{\nu}$ corresponds to the band $(\zeta_{3},\zeta_{4})$:
$b_{\nu}=(\frac{1}{2},1)$

We use these vertices and those obtained by applying the same process to the
remaining cells to build polyhedra $\overline{D}_{\nu}$, each of which comes
with a subdivision into simplices. Taken together, these simplices give us our
desired complex $\Lambda$ respecting sign sets of $\\{L_{1},L_{2}\\}$, as
shown in Figure 2. There are a total of 10 2-dimensional polyhedra, with 64
2-dimensional simplices.

Figure 2. Triangulation of $A$ respecting sets $\\{y=x\\}$ and $\\{y=0\\}$

Now we prove our symmetric triangulation theorem.

######  Theorem 4.7.

Let $A$ be a closed and bounded subset of $\mathbb{R}^{n}$ symmetric under the
action of a finite reflection group $G$, and let $S_{1},\ldots,S_{l}$ be
symmetric sets which are subsets of $A$. Then there exists an equivariant
triangulation $(\Lambda,\Phi)$ of $A$ adapted to $S_{1},\ldots,S_{l}$.

Assume we have fixed a collection $\\{L_{1},\ldots,L_{k}\\}$ of functions
$L_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ given by
$L_{i}(\mathbf{x})=a_{i,1}x_{1}+\cdots a_{i,n}x_{n}$, so that

$H=\bigcap_{i=1}^{k}\\{L_{i}(\mathbf{x})\geq 0\\}$

is a fundamental region of $\mathbb{R}^{n}$ with respect to $G$. Then we may
choose our triangulation so that vertices of $\Lambda$ are in
$\mathbb{Q}(\\{a_{i,j}\mid 1\leq i\leq k,1\leq j\leq n\\})^{n}$.

###### Proof.

Let $G$ be a finite reflection group acting on $\mathbb{R}^{n}$, and if not
already selected, let $\\{L_{1},\ldots,L_{k}\\}$ be linear functions defining
a fundamental region $H$ of $\mathbb{R}^{n}$ with respect to $G$. We apply
Lemma 4.5 to the closed, bounded, definable set $A\cap\overline{H}$, definable
subsets $S_{1}\cap\overline{H},\ldots,S_{l}\cap\overline{H}$, and our
collection of functions $\\{L_{1},\ldots,L_{k}\\}$. The lemma gives us a
finite simplicial complex $\Lambda_{\operatorname{id}}$ with vertices in
$\mathbb{Q}(\\{a_{i,j}\\})$ and a definable homeomorphism
$\Phi_{\operatorname{id}}:\lvert\Lambda\rvert\rightarrow A\cap\overline{H}$.
Note that since $H$ and all its walls are sign sets of
$\\{L_{1},\ldots,L_{k}\\}$, the triangulation
$(\Lambda_{\operatorname{id}},\Phi_{\operatorname{id}})$ respects $H$ and all
its walls. In particular, $\lvert\Lambda\rvert\subset\overline{H}$.

We now define our proposed triangulation of $S$. Let

$\Lambda=\bigcup_{g\in
G}\\{g(\Delta)\mid\Delta\in\Lambda_{\operatorname{id}}\\}$

and let $\Phi:\lvert\Lambda\rvert\rightarrow A$ be given by
$\Phi(\mathbf{x})=g(\Phi_{\operatorname{id}}(\mathbf{x}^{\prime}))$ where
$\mathbf{x}^{\prime}\in\lvert\Lambda_{\operatorname{id}}\rvert$ and $g\in G$
are such that $g(\mathbf{x}^{\prime})=\mathbf{x}$. We claim that (everything
is well-defined and) $\Lambda$ and $\Phi$ give an equivariant triangulation of
$A$ adapted to $S_{1},\ldots,S_{l}$.

$\Lambda$ is a symmetric simplicial complex: We have that if
$\Delta\in\Lambda_{\operatorname{id}}$ and $g\in G$, $g(\Delta)$ remains a
simplex in $\mathbb{R}^{n}$ by linearity of $g$, and so $\Lambda$ is a
collection of simplices. The symmetry property for $\Lambda$ then holds by
construction, so it remains to show that for
$\Delta_{i},\Delta_{j}\in\Lambda$,
$\overline{\Delta}_{i}\cap\overline{\Delta}_{j}$ is the closure of some
simplex in $\Lambda$. Without loss of generality, assume
$\Delta_{1}\in\Lambda_{\operatorname{id}}$ and
$\Delta_{2}=g(\Delta_{2}^{\prime})$ for
$\Delta_{2}^{\prime}\in\Lambda_{\operatorname{id}}$ and $g\in G$. This means
$\overline{\Delta}_{1}\cap\overline{g(\Delta_{2}^{\prime})}\subset\overline{H}\cap
g(\overline{H})$. As described in Subsection 2.1, $\overline{H}\cap
g(\overline{H})=H_{\lambda_{g}}$ is a set of the form

$H_{\lambda_{g}}=\left(\bigcap_{i\in\lambda_{g}}\\{L_{i}(\mathbf{x})=0\\}\right)\cap\left(\bigcap_{i\in\\{1,\ldots,k\\}\setminus\lambda_{g}}\\{L_{i}(\mathbf{x})\geq
0\\}\right)$

which in particular is a sign set of $\\{L_{1},\ldots,L_{k}\\}$. Since
$(\Lambda_{\operatorname{id}},\Phi_{\operatorname{id}})$ is hence adapted to
$A\cap H_{\lambda_{g}}$, we have that $\overline{\Delta}_{1}\cap
H_{\lambda_{g}}$ is also the closure of a simplex of
$\Lambda_{\operatorname{id}}$. Since points of $H_{\lambda_{g}}$ are fixed
under the action of $g$, $\overline{\Delta}_{2}\cap
H_{\lambda_{g}}=\overline{\Delta}_{2}^{\prime}\cap H_{\lambda_{g}}$ is the
closure of a simplex of $\Lambda_{\operatorname{id}}$ as well. Then
$\overline{\Delta}_{1}\cap\overline{\Delta}_{2}=(\overline{\Delta}_{1}\cap
H_{\lambda_{g}})\cap(\overline{\Delta}_{2}\cap H_{\lambda_{g}})$ is an
intersection of closures of simplices of $\Lambda_{\operatorname{id}}$, and
hence a common face of $\overline{\Delta}_{1}$ and $\overline{\Delta}_{2}$.

Vertices of $\Lambda$ are in $\mathbb{Q}(\\{a_{i,j}\\})^{n}$: We will show
that, for $g\in G$ and any point $\mathbf{x}$ in
$\mathbb{Q}(\\{a_{i,j}\\})^{n}$,
$g(\mathbf{x})\in\mathbb{Q}(\\{a_{i,j}\\})^{n}$. Since any element $g$ of $G$
can be written as a product of those elements $g_{1},\ldots,g_{k}$, where the
action of $g_{i}$ is reflection through the linear hyperplane $L_{i}=0$, we
may assume $g=g_{i}$ for some $1\leq i\leq k$. The reflection of $\mathbf{x}$
through $L_{i}=0$ is given by

$g_{i}(\mathbf{x})=\mathbf{x}-2\frac{(\mathbf{x},\mathbf{r}_{i})\mathbf{r}_{i}}{(\mathbf{r}_{i},\mathbf{r}_{i})}$

where we take $\mathbf{r}_{i}$ to be the vector
$\left<a_{i,1},\ldots,a_{i,n}\right>$ (which is perpendicular to the
hyperplane $L_{i}=0$). Then since we have assumed that we are using the
standard inner product on $\mathbb{R}^{n}$, the coordinates of
$g_{i}(\mathbf{x})$ are all also in $\mathbb{Q}(\\{a_{i,j}\\})$, as desired.

$\Phi$ is well-defined: Let $\mathbf{x}\in\lvert\Lambda\rvert$ and say that
$\mathbf{x}=g_{1}(\mathbf{x}_{1})=g_{2}(\mathbf{x}_{2})$ for
$\mathbf{x}_{1},\mathbf{x}_{2}\in\lvert\Lambda_{\operatorname{id}}\rvert$ and
$g_{1},g_{2}\in G$. We must show that
$g_{1}(\Phi_{\operatorname{id}}(\mathbf{x}_{1}))=g_{2}(\Phi_{\operatorname{id}}(\mathbf{x}_{2}))$.
Because $\mathbf{x}_{1}=g_{1}^{-1}(g_{2}(\mathbf{x}_{2}))$ with
$\mathbf{x}_{1},\mathbf{x}_{2}\in\lvert\Lambda_{\operatorname{id}}\rvert$, we
have that $\mathbf{x}_{1}\in\lvert\Lambda_{\operatorname{id}}\rvert\cap
g_{1}^{-1}\circ g_{2}(\lvert\Lambda_{\operatorname{id}}\rvert)\subset
H_{\lambda_{g_{1}^{-1}g_{2}}}$. Since $g_{1}^{-1}\circ g_{2}$ fixes the points
of $H_{\lambda_{g_{1}^{-1}g_{2}}}$, we obtain that
$\mathbf{x}_{1}=\mathbf{x}_{2}$. Since $\Phi_{\operatorname{id}}$ carries
$H_{\lambda_{g_{1}^{-1}g_{2}}}$ to itself, we have that
$\Phi_{\operatorname{id}}(\mathbf{x}_{1})\in H_{\lambda_{g_{1}^{-1}g_{2}}}$,
as is $g_{1}^{-1}\circ g_{2}(\Phi_{\operatorname{id}}(\mathbf{x}_{1}))$, and
so $\Phi_{\operatorname{id}}(\mathbf{x}_{1})=g_{1}^{-1}\circ
g_{2}(\Phi_{\operatorname{id}}(\mathbf{x}_{1}))$, i.e. we have obtained that
$g_{1}(\Phi_{\operatorname{id}}(\mathbf{x}_{1}))=g_{2}(\Phi_{\operatorname{id}}(\mathbf{x}_{2}))$.
That the image of $\Phi$ is $A$ follows from the symmetry of
$A=\\{g(\mathbf{x})\mid\mathbf{x}\in A\cap\overline{H}\text{ and }g\in G\\}$.

$\Phi$ is a homeomorphism: We have already established that the surjectivity
of $\Phi$ follows from the surjectivity of $\Phi_{\operatorname{id}}$ onto
$A\cap\overline{H}$. To show injectivity, say
$\Phi(\mathbf{x}_{1})=\Phi(\mathbf{x}_{2})$ for some
$\mathbf{x}_{1},\mathbf{x}_{2}\in\lvert\Lambda\rvert$, i.e.
$g_{1}(\Phi_{\operatorname{id}}(\mathbf{x}_{1}^{\prime}))=g_{2}(\Phi_{\operatorname{id}}(\mathbf{x}_{2}^{\prime}))$
for some
$\mathbf{x}_{1}^{\prime},\mathbf{x}_{2}^{\prime}\in\lvert\Lambda_{\operatorname{id}}\rvert$
and $g_{1},g_{2}\in G$. Then since
$\Phi_{\operatorname{id}}(\mathbf{x}_{1}^{\prime})=g_{1}^{-1}\circ
g_{2}(\Phi_{\operatorname{id}}(\mathbf{x}_{2}^{\prime}))$, both are in
$H_{\lambda_{g_{1}^{-1}g_{2}}}$, and so
$\Phi_{\operatorname{id}}(\mathbf{x}_{1}^{\prime})=\Phi_{\operatorname{id}}(\mathbf{x}_{2}^{\prime})$.
By the injectivity of $\Phi_{\operatorname{id}}$, this means
$\mathbf{x}_{1}^{\prime}=\mathbf{x}_{2}^{\prime}$. Finally, since
$\mathbf{x}_{1}^{\prime}=\mathbf{x}_{2}^{\prime}$ is in
$H_{\lambda_{g_{1}^{-1}g_{2}}}$, $\mathbf{x}_{1}^{\prime}=g_{1}^{-1}\circ
g_{2}(\mathbf{x}_{2}^{\prime})$, so we have
$\mathbf{x}_{1}=g_{1}(\mathbf{x}_{1}^{\prime})=g_{2}(\mathbf{x}_{2}^{\prime})=\mathbf{x}_{2}$.
Continuity of $\Phi$ follows from continuity on
$g(\Lambda_{\operatorname{id}})$ for each $g\in G$ and agreement on the
boundaries.

$\Phi$ is equivariant under the action of $G$ by construction. The preimage
under $\Phi$ of each set $S_{i}$ among $S_{1},\ldots,S_{l}$ is the union of
the simplices $g(\Delta)$ for $g\in G$ and
$\Delta\in\Lambda_{\operatorname{id}}$ such that
$\Delta\subset\Phi_{\operatorname{id}}^{-1}(S_{i}\cap\overline{H})$. Hence,
$(\Lambda,\Phi)$ gives our desired triangulation. ∎

###  4.2. Triangulation of Definable Functions

In the proofs in [9], to ensure that the triangulation we use is properly
compatible with the family of sets $\\{S_{\delta}\\}_{\delta>0}$, the authors
invoke the triangulation of definable functions. The original theorem from [7]
is below. We will proceed to prove a version for functions symmetric relative
to the action of some finite reflection group $G$.

######  Theorem 4.8 (Triangulation of Definable Functions, [7] Theorem 4.5).

Let $A$ be a closed and bounded definable subset of $R^{n}$ and
$f:A\rightarrow R$ a continuous definable function. Then there exists a finite
simplicial complex $\Lambda$ in $R^{n+1}$ and a definable homeomorphism
$\rho:\lvert\Lambda\rvert\rightarrow A$ such that $f\circ\rho$ is an affine
function on each simplex of $\Lambda$. Moreover, given $S_{1},\ldots,S_{l}$
definable subsets of $A$, we may choose the triangulation
$\rho:\lvert\Lambda\rvert\rightarrow A$ to be adapted to the $S_{i}$.

Following [7], we will prove a more general result concerning the
triangulation of symmetric definable subsets of
$\mathbb{R}\times\mathbb{R}^{n}$, which we will apply to what is essentially
the graph of our function $f$. Our procedure is similar to the one used for
symmetric definable sets. Let $\pi:R\times R^{n}\rightarrow R$ denote
projection on the first coordinate.

######  Lemma 4.9 (ref [7] Proposition 4.6 and Proposition 4.8).

Let $A$ be a closed, bounded, definable subset of $R\times R^{n}$, and let
$S_{1},\ldots,S_{l}$ be definable subsets of $A$. Let
$\\{L_{1},\ldots,L_{k}\\}$ be a collection of functions with $L_{i}:R\times
R^{n}\rightarrow R$ given by $L_{i}:(y,x_{1},\ldots,x_{n})\mapsto
a_{i,0}+a_{i,1}x_{1}+\cdots+a_{i,n}x_{n}$. Then there exists a triangulation
$(\Lambda,\Phi)$ of $A$ adapted to $S_{1},\ldots,S_{l}$, respecting all sign
sets of $\\{L_{1},\ldots,L_{k}\\}$, and having vertices of $\Lambda$ in
$\mathbb{Q}\times\mathbb{Q}(\\{a_{i,j}\mid 1\leq i\leq k,0\leq j\leq
n\\})^{n}$, as well as a definable homeomorphism $\tau:R\rightarrow R$, such
that $\tau\circ\pi\circ\Phi=\pi_{\restriction\lvert\Lambda\rvert}$.

###### Proof.

We induct on $n$. In the $n=0$ case, since we are assuming all functions in
our collection $\\{L_{1},\ldots,L_{k}\\}$ to be independent of the first
coordinate, we have nothing more to prove than [7] does, and so may as there
choose a finite partition of $R$ adapted to $A$ and $S_{1},\ldots,S_{l}$.
Letting $x_{1}<\cdots<x_{p}$ be the points in $R$ defining the partition, we
take $\Lambda$ to be the points $i\in\\{1,\ldots,p\\}$ and intervals $[i,i+1]$
such that ${x_{i}}\in A$ or $[x_{i},x_{i+1}]\subset A$. We let $\tau(x_{i})=i$
and extend piecewise affinely to a map $\tau:R\rightarrow R$.

Assume $n>0$ and that the statement holds for $n-1$. We may again assume all
$S_{i}$ are closed (as described in the proof of [7] Theorem 4.4). We set
$F^{\prime}_{0}$ to be the boundary of $A$, $F^{\prime}_{i}$ to be the
boundary of $S_{i}$ for each $i$, and $F^{\prime}=F^{\prime}_{0}\cup
F^{\prime}_{1}\cup\ldots\cup F^{\prime}_{l}$. Since $F^{\prime}$ is definable
with dimension at most $n$, we have finitely many $c\in R$ for which the set
$\\{\mathbf{x}\in R^{n}\mid(c,\mathbf{x})\in F^{\prime}\\}$ has dimension $n$.
Let $C$ be the set of all such $c$. For reasons of dimension, the proof in [7]
considers sets $F_{i}$ given by taking the union of
$\overline{F^{\prime}_{i}\setminus(C\times R^{n})}$ with the boundary of
$F^{\prime}_{i}\cap(C\times R^{n})$ in $C\times R^{n}$. We choose a cell
decomposition of $R\times R^{n}$ adapted to $F_{0},\ldots,F_{l}$, the sets
$\\{c\\}\times R^{n}$ for each $c\in C$, and all sign sets of
$\\{L_{1},\ldots,L_{k}\\}$.

Let $p:R\times R^{n}\rightarrow R\times R^{n-1}$ be the projection on the
first $n$ coordinates. Our cell decomposition partitions $p(A)$ into definably
connected subsets $X_{\alpha}$. By our inductive hypothesis, we have a
triangulation $(\Lambda_{n-1},\Psi_{n-1})$ of $R\times R^{n-1}$ which is
adapted to each $X_{\alpha}$, respects sign sets of
$\\{L_{1}^{\prime},\ldots,L_{m^{\prime}}^{\prime}\\}$ (where this collection
is defined in a manner analogous to the proof of Lemma 4.5), and with vertices
of $\Lambda_{n-1}$ in $\mathbb{Q}\times\mathbb{Q}(\\{a_{i,j}^{\prime}\mid
1\leq i\leq m^{\prime},0\leq j\leq
n-1\\})^{n-1}=\mathbb{Q}\times\mathbb{Q}(\\{a_{i,j}\mid 1\leq i\leq m,0\leq
j\leq n\\})^{n-1}$. We also have a map $\tau^{\prime}:R\rightarrow R$, having
the property that
$\tau^{\prime}\circ\pi_{n,1}\circ\Psi_{n-1}={\pi_{n,1}}\restriction_{\lvert\Lambda_{n-1}\rvert}$
(where $\pi_{n,1}$ is the projection $R\times R^{n-1}\rightarrow R$ on the
first coordinate). Now, if we follow the remaining steps in the proof of Lemma
4.5, we obtain a triangulation $(\Lambda,\Phi)$ of $A$ which is adapted to
$S_{1},\ldots,S_{l}$, respects sign sets of $\\{L_{1},\ldots,L_{k}\\}$, and
has vertex coordinates in $\mathbb{Q}\times\mathbb{Q}(\\{a_{i,j}\\})^{n}$.
$\Lambda$ and $\Phi$ are such that $p\circ\Phi=\Psi_{n-1}\circ
p_{\restriction\lvert\Lambda\rvert}$. Taking $\tau=\tau^{\prime}$, this means
that our property
$\tau\circ\pi\circ\Phi=\pi_{\restriction\lvert\Lambda\rvert}$ holds. ∎

######  Lemma 4.10 (ref [7] Proposition 4.6 and Proposition 4.8).

Let $A$ be a closed, bounded, definable subset of
$\mathbb{R}\times\mathbb{R}^{n}$ symmetric under the action of some finite
reflection group $G$ on $\mathbb{R}^{n}$ extended to
$\mathbb{R}\times\mathbb{R}^{n}$, and let $S_{1},\ldots,S_{l}$ be definable
symmetric subsets of $A$. Then there exists an equivariant triangulation
$(\Lambda,\Phi)$ of $A$ adapted to $S_{1},\ldots,S_{l}$ and a definable
homeomorphism $\tau:\mathbb{R}\rightarrow\mathbb{R}$ having the property that
$\tau\circ\pi\circ\Phi=\pi_{\restriction\lvert\Lambda\rvert}$.

Assume we have fixed a collection $\\{L_{1},\ldots,L_{k}\\}$ of functions
$L_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ given by
$L_{i}(\mathbf{x})=a_{i,1}x_{1}+\cdots a_{i,n}x_{n}$, so that

$H=\bigcap_{i=1}^{k}\\{L_{i}(\mathbf{x})>0\\}$

is a fundamental region of $\mathbb{R}^{n}$ with respect to $G$. Then we may
choose our triangulation so that vertices of $\Lambda$ are in
$\mathbb{Q}\times\mathbb{Q}(\\{a_{i,j}\mid 1\leq i\leq k,1\leq j\leq
n\\})^{n}$.

###### Proof.

This is analogous to the proof of Theorem 4.7. Let $H$ be a fundamental region
of $\mathbb{R}^{n}$ with respect to $G$. If not already specified, we take
$\\{L_{1},\ldots,L_{k}\\}$ with
$L_{i}(\mathbf{x})=a_{i,1}x_{1}+\cdots+a_{i,n}x_{n}$ to be a collection
functions defining $H$. For each $i$, let
$\tilde{L}_{i}:\mathbb{R}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$ with
$\tilde{L}_{i}(y,\mathbf{x})=L_{i}(\mathbf{x})$. We apply Lemma 4.11 to
$A\cap(\mathbb{R}\times\overline{H})$, subsets
$S_{1}\cap(\mathbb{R}\times\overline{H}),\ldots,S_{l}\cap(\mathbb{R}\times\overline{H})$,
and linear functions $\\{\tilde{L}_{1},\ldots,\tilde{L}_{k}\\}$. We obtain a
triangulation $(\Lambda_{\operatorname{id}},\Phi_{\operatorname{id}})$ with
$\lvert\Lambda_{\operatorname{id}}\rvert\subset\mathbb{R}\times\overline{H}$
and coordinates of vertices of $\Lambda_{\operatorname{id}}$ in
$\mathbb{Q}\times\mathbb{Q}(\\{a_{i,j}\\})^{n}$, and
$\Phi_{\operatorname{id}}:\lvert\Lambda_{\operatorname{id}}\rvert\rightarrow
A\cap(\mathbb{R}\times\overline{H})$, and also obtain a map
$\tau:\mathbb{R}\rightarrow\mathbb{R}$, having the property that
$\tau\circ\pi\circ\Phi_{\operatorname{id}}=\pi_{\restriction\lvert\Lambda_{\operatorname{id}}\rvert}$.
Again, we take

$\Lambda=\bigcup_{g\in
G}\\{g(\Delta)\mid\Delta\in\Lambda_{\operatorname{id}}\\}$

and $\Phi:\lvert\Lambda\rvert\rightarrow A$ given by
$\Phi(\mathbf{x})=g(\Phi_{\operatorname{id}}(\mathbf{x}^{\prime}))$, where
$g\in G$ and $\mathbf{x}^{\prime}\in\lvert\Lambda_{\operatorname{id}}\rvert$
are such that $\mathbf{x}=g(\mathbf{x}^{\prime})$. As in the proof of Theorem
4.7, this provides a symmetric triangulation of $A$ adapted to
$S_{1},\ldots,S_{l}$, with all vertices of $\Lambda$ in
$\mathbb{Q}\times\mathbb{Q}(\\{a_{i,j}\\})^{n}$.

It remains to show that
$\tau\circ\pi\circ\Phi=\pi_{\restriction\lvert\Lambda\rvert}$. Take
$\mathbf{x}\in\lvert\Lambda\rvert$. Then we have that
$\mathbf{x}=g(\mathbf{x}^{\prime})$ for some $g\in G$ and
$\mathbf{x}^{\prime}\in\lvert\Lambda_{\operatorname{id}}\rvert$. Note that by
the definition of our action of $G$ extended to
$\mathbb{R}\times\mathbb{R}^{n}$, we have that $\pi$ is symmetric relative to
this action. Then

$\displaystyle\tau\circ\pi\circ\Phi(\mathbf{x})$
$\displaystyle=\tau\circ\pi(g(\Phi_{\operatorname{id}}(\mathbf{x}^{\prime})))$
$\displaystyle=\tau\circ\pi(\Phi_{\operatorname{id}}(\mathbf{x}^{\prime}))$
$\displaystyle=\pi(\mathbf{x}^{\prime})$
$\displaystyle=\pi(g(\mathbf{x}^{\prime}))=\pi(\mathbf{x})$

as desired. ∎

Our theorem now follows, applying Lemma 4.10 to
$A^{\prime}=\\{(f(\mathbf{x}),\mathbf{x})\mid\mathbf{x}\in A\\}$.

######  Theorem 4.11.

Let $A$ be a closed, bounded, definable subset of $\mathbb{R}^{n}$ symmetric
under the action of a finite reflection group $G$, and let
$f:A\rightarrow\mathbb{R}$ be a continuous definable function symmetric
relative to the action of $G$. Then there exists a finite symmetric simplicial
complex $\Lambda$ in $\mathbb{R}^{n+1}$ and a definable equivariant
homeomorphism $\rho:\lvert\Lambda\rvert\rightarrow A$ such that $f\circ\rho$
is an affine function on each simplex of $\Lambda$. Moreover, given
$S_{1},\ldots,S_{l}$ definable symmetric subsets of $A$, we may choose the
triangulation $\rho:\lvert\Lambda\rvert\rightarrow A$ to be adapted to the
sets $S_{i}$.

Assume we have fixed a collection $\\{L_{1},\ldots,L_{k}\\}$ of functions
$L_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ given by
$L_{i}(\mathbf{x})=a_{i,1}x_{1}+\cdots a_{i,n}x_{n}$, so that

$H=\bigcap_{i=1}^{k}\\{L_{i}(\mathbf{x})>0\\}$

is a fundamental region of $\mathbb{R}^{n}$ with respect to $G$. Then we may
choose our simplicial complex $\Lambda$ so that all vertices of $\Lambda$ are
in $\mathbb{Q}\times\mathbb{Q}(\\{a_{i,j}\mid 1\leq i\leq k,1\leq j\leq
n\\})^{n}$.

###### Proof.

Consider the set
$A^{\prime}=\\{(f(\mathbf{x}),\mathbf{x})\in\mathbb{R}\times\mathbb{R}^{n}\mid\mathbf{x}\in
A\\}$. Since $f$ is symmetric relative to the action of $G$ on
$\mathbb{R}^{n}$, $A^{\prime}$ is a symmetric set relative to the action
induced by $G$ on $\mathbb{R}\times\mathbb{R}^{n}$, and projection on the last
$n$ coordinates gives an equivariant definable homeomorphism
$p:A^{\prime}\rightarrow A$. By Lemma 4.10, we have a symmetric triangulation
$(\Lambda,\Phi)$ of $A^{\prime}$ which is adapted to the sets
$S_{1}^{\prime},\ldots,S_{l}^{\prime}$ (with
$S_{i}^{\prime}=\\{(f(\mathbf{x}),\mathbf{x})\mid\mathbf{x}\in S_{l}\\}$) and
has vertices in $\mathbb{Q}\times\mathbb{Q}(\\{a_{i,j}\\})^{n}$, and a
definable function $\tau:\mathbb{R}\rightarrow\mathbb{R}$ such that
$\tau\circ\pi\circ\Phi=\pi_{\restriction\lvert\Lambda\rvert}$ (where
$\pi:\mathbb{R}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$ is projection on the
first coordinate). Applying $f$ to $\mathbf{x}\in A$ is equivalent to applying
$\pi$ to $(f(\mathbf{x}),\mathbf{x})\in A^{\prime}$, and so taking
$\rho:\lvert\Lambda\rvert\rightarrow A$ to be $p\circ\Phi$, we have that
$f\circ\rho=\tau^{-1}\circ\pi_{\restriction\lvert\Lambda\rvert}$, which is an
affine map on each simplex of $\Lambda$ by construction. ∎

###  4.3. Equivariance and Hardt Triviality

We will also want an equivariant version of Hardt Triviality for o-minimal
sets. Because the proof uses the same sort of argument as equivariant
triangulation, we include it within this section.

######  Definition 4.12 ([8] Chapter 9 Definitions 1.1 and 1.8).

Let $X\subset R^{n}$ and $A\subset R^{m}$ be definable sets and
$f:X\rightarrow A$ a continuous definable function. For $A^{\prime}\subset A$,
we say that $f$ is _definably trivial_ over $A^{\prime}$ if for any $a\in
A^{\prime}$ there is a definable homeomorphism
$h:f^{-1}(A^{\prime})\rightarrow f^{-1}(a)\times A^{\prime}$ such that the
diagram

${f^{-1}(A^{\prime})}$${f^{-1}(a)\times
A^{\prime}}$${A^{\prime}}$$\scriptstyle{f}$$\scriptstyle{h}$$\scriptstyle{\pi}$

commutes (where $\pi$ is the projection on the second coordinate).

For $X_{1},\ldots,X_{l}$ definable subsets of $X$, we say that the definable
trivialization $h$ _respects_ $X_{1},\ldots,X_{l}$ if $h$ maps each $X_{j}\cap
f^{-1}(A^{\prime})$ homeomorphically to $(X_{j}\cap f^{-1}(a))\times
A^{\prime}$.

######  Theorem 4.13 ([8] Chapter 9 Theorem 1.7).

Let $X\subset R^{n}$ and $A\subset R^{m}$ be definable sets, $f:X\rightarrow
A$ a continuous definable function, and $X_{1},\ldots,X_{l}$ definable subsets
of $X$. Then we may partition $A$ into a finite number of definable subsets
$A_{i}$ such that $f$ is definably trivial over each $A_{i}$ in a manner that
respects each of $X_{1},\ldots,X_{l}$.

We would like to show that, if $f$ is a symmetric function, then the
homeomorphisms guaranteed by the trivialization are equivariant.

######  Theorem 4.14.

Let $G$ be a finite reflection group acting on $\mathbb{R}^{n}$, and let
$X\subset\mathbb{R}^{n}$ and $X_{1},\ldots,X_{l}\subset X$ be symmetric
relative to $G$. Say that $A\subset\mathbb{R}^{m}$ is definable and
$f:X\rightarrow A$ is a continuous, definable function symmetric relative to
$G$. Then we may partition $A$ into a finite number of definable subsets
$A_{i}$ such that for each $i$ there is an equivariant homeomorphism
$h_{i}:f^{-1}(A_{i})\rightarrow f^{-1}(a_{i})\times A_{i}$ giving a definable
trivialization of $f$ over $A_{i}$ (where $a_{i}$ is any element of $A_{i}$,
and the action on $f^{-1}(a_{i})\times A$ is given by $g(x,y)=(g(x),y)$).
Furthermore, each trivialization $h_{i}$ respects the subsets
$X_{1},\ldots,X_{l}$.

###### Proof.

Let $H$ be a fundamental region of $\mathbb{R}$ with respect to $G$. We apply
Theorem 4.13 to $f_{\restriction\overline{H}}:\overline{H}\cap X\rightarrow
A$, asking that our definable trivializations respect the sets
$\overline{H}\cap X_{1},\ldots,\overline{H}\cap X_{l}$ as well as each
$H_{\lambda}\cap X$ for $H_{\lambda}$ an intersection of walls of $H$. We
obtain a finite partition $\\{A_{i}\\}$ of $A$, as well as homeomorphisms
$h_{\operatorname{id},i}:f^{-1}_{\restriction\overline{H}}(A_{i})\rightarrow
f^{-1}_{\restriction\overline{H}}(a_{i})\times A_{i}$ which respect both the
portions of each $X_{j}$ within $\overline{H}$ and the intersections of $X$
with the walls of $H$.

Since $f$ is a symmetric function, we know that for $a_{i}\in A_{i}$,

$f^{-1}(a_{i})=\bigcup_{g\in G}g(f^{-1}_{\restriction\overline{H}}(a_{i}))$

and that the same holds true for $f^{-1}(A_{i})$. We define
$h_{i}:f^{-1}(A_{i})\rightarrow f^{-1}(a_{i})\times A_{i}$ by
$h_{i}(\mathbf{x})=g(h_{\operatorname{id},i}(\mathbf{x}^{\prime}))$, where
$\mathbf{x}^{\prime}\in f^{-1}(A_{i})\cap\overline{H}$ and $g\in G$ are such
that $g(\mathbf{x}^{\prime})=\mathbf{x}$. We must show that $h_{i}$ is a
definable trivialization of $f$.

We may use arguments similar to those of Theorem 4.7 to establish that $h_{i}$
is a homeomorphism from $f^{-1}(A_{i})$ to $f^{-1}(a_{i})\times A_{i}$. The
map $h_{i}$ is equivariant by construction, and $h_{i}$ sends each $X_{j}\cap
f^{-1}(A_{i})$ homeomorphically to $(X_{j}\cap f^{-1}(a_{i}))\times A_{i}$ by
construction and the symmetry of each $X_{j}$. Say that $\mathbf{x}\in
f^{-1}(A_{i})$ with $f(\mathbf{x})=\mathbf{y}$. Then if
$\mathbf{x}=g(\mathbf{x}^{\prime})$ for some $g\in G$ and
$\mathbf{x}^{\prime}\in f^{-1}(A_{i})\cap\overline{H}$, by symmetry we know
that $f(\mathbf{x}^{\prime})=\mathbf{y}$ as well, and so
$h(\mathbf{x}^{\prime})=(\mathbf{z},\mathbf{y})$ for some $\mathbf{z}\in
f^{-1}(a_{i})$. Then
$h(\mathbf{x})=g(\mathbf{z},\mathbf{y})=(g(\mathbf{z}),\mathbf{y})$, so
$h_{i}$ is indeed a definable trivialization of $f$ over $A_{i}$. ∎

##  5\. Equivariant Versions of some Topological Results

We address some topological aspects of the construction in this section.
Specifically, for $X$ a regular CW complex whose cells are convex polyhedra
(so specifically, for $X$ a simplicial complex), we explicitly describe
certain maps between $X$ and the order complex of the face poset of $X$, in
aid of demonstrating equivariance. We also establish an equivariant version of
a Nerve Theorem due to Björner.

######  Definition 5.1 ([5] Appendix A2).

Let $P$ be a partially ordered set. The _order complex_ of $P$, $\Delta(P)$,
is the abstract simplicial complex with vertex set the elements of $P$ and
simplices given by finite chains $x_{0}<\cdots<x_{d}$ of elements of $P$.

The order complex of the face poset of a simplicial complex coincides with the
first barycentric subdivision of that simplicial complex. For a regular CW
complex, then, taking the order complex of the face poset serves as a
generalized barycentric subdivision:

######  Theorem 5.2 ([13] Theorem 1.7).

Let $X$ be a regular CW complex. Then $\Delta(\mathscr{F}(X))$ is homeomorphic
to $X$.

The proof involves choosing a point in the interior of each cell $\sigma$ to
act as the barycenter of the cell, and sending the vertex of
$\Delta(\mathscr{F}(X))$ corresponding to $\sigma$ to that barycenter.
Elsewhere, the homeomorphism arises from the fact that the characteristic maps
used to assemble $X$ as a regular CW complex endow each cell with the
structure of a cone with the barycenter as vertex and boundary as base.
Unfortunately, this construction as it stands is not specific enough to ensure
equivariance without some sort of equivariance condition on the characteristic
maps. However, if we know that each cell of $X$ is, for example, a convex
polyhedron, we may make our choices in a manner sufficiently canonical to
ensure equivariance. Note that by polyhedron in this context we mean a bounded
subset of $R^{n}$ obtained by intersecting a finite number of affine half
planes.

######  Definition 5.3.

Let $X\subset R^{n}$ be a regular CW complex in which each cell is a convex
polyhedron. Then we will refer to the homeomorphism $\Psi$ described below as
the _centroidal_ homeomorphism $\lvert\Delta(\mathscr{F}(X))\rvert\rightarrow
X$.

We define $\Psi$ inductively. Let $X^{k}$ denote the $k$-skeleton of $X$. In
the $k=0$ case, let
$\Psi^{0}:\lvert\Delta(\mathscr{F}(X^{0}))\rvert\rightarrow X^{0}$ send the
vertex $v_{\sigma}$ of $\Delta(\mathscr{F}(X^{0}))$ corresponding to the
$0$-dimensional cell $\sigma=\\{x\\}$ to the point $x\in X$.

Now say $k\geq 1$. Assume we have defined
$\Psi^{k-1}:\lvert\Delta(\mathscr{F}(X^{k-1}))\rvert\rightarrow X^{k-1}$ and
established that $\Psi^{k-1}$ is a homeomorphism. We may identify
$\Delta(\mathscr{F}(X^{k-1}))$ with the subset of $\Delta(\mathscr{F}(X^{k}))$
consisting of simplices whose vertices correspond to cells of $X$ of dimension
less than $k$; on this subset, let $\Psi^{k}$ agree with $\Psi^{k-1}$. Now,
assume $\sigma$ is a cell of dimension $k$, and let $v_{\sigma}$ be the vertex
of $\Delta(\mathscr{F}(X^{k}))$ corresponding to $\sigma$. Define
$\Psi^{k}(v_{\sigma})=b_{\sigma}$, where $b_{\sigma}$ is the centroid of the
cell $\sigma$ (which is in the interior of $\sigma$ by convexity). Say
$x\in\lvert\Delta(\mathscr{F}(X^{k}))\rvert$. Then we may uniquely write
$x=tv_{\sigma}+(1-t)x^{\prime}$, where $\sigma$ is a cell of $X$ of dimension
$k$, $x^{\prime}$ is in the realization of some simplex of
$\Delta(\mathscr{F}(X^{k-1}))$, and $t\in[0,1]$. Set
$\Psi^{k}(x)=tb_{\sigma}+(1-t)\Psi^{k-1}(x^{\prime})$. Since each cell
$\sigma$ of $X$ can be seen as a cone with base $\partial\sigma$ and vertex
$b_{\sigma}$, this gives a well-defined homeomorphism to $X^{k}$. Since
$X=\bigcup_{k}X^{k}$, we inductively obtain our desired homeomorphism
$\Psi:\lvert\Delta(\mathscr{F}(X))\rvert\rightarrow X$.

When we refer to the barycentric subdivision of a polyhedral CW complex
$X\subset R^{n}$, we will mean the simplicial complex in $R^{n}$ whose
geometric realization is equal as a set to $X$ and whose cell structure is
inherited from this map.

######  Remark 5.4.

Let $X\subset R^{n}$ be a regular CW complex whose cells are all convex
polyhedra. If $G$ is a group acting linearly on $X$ such that $X$ is symmetric
as a CW complex under the action of $G$, then the action of $G$ on $X$ induces
an action of $G$ on $\Delta(\mathscr{F}(X))$ under which
$\Delta(\mathscr{F}(X))$ is symmetric, and the homeomorphism
$\Psi:\lvert\Delta(\mathscr{F}(X))\rvert\rightarrow X$ of Definition 5.3 is
equivariant.

We will also want to consider subsets of simplicial or polyhedral CW complexes
which are not full subcomplexes. Let $X$ be polyhedral CW complex and let $Y$
be a union of cells of $X$. Gabrielov and Vorobjov in [9] Remark 2.12 describe
how we may consider only those cells of the barycentric subdivision of $X$
which are contained with their closures in $Y$. This gives us a subcomplex of
the barycentric subdivision of $X$ which is homotopy equivalent to $Y$, and
which may replace $Y$ in our applications. Though this procedure is relatively
standard, we will describe the contraction explicitly so that we may ensure
equivariance.

######  Definition 5.5.

Let $X\subset R^{n}$ be a regular CW complex whose cells are all convex
polyhedra, and let $Y$ be a union of cells of $X$. Let $\widehat{X}$ be the
barycentric subdivision of $X$, and let
$\widehat{Y}=\\{\Delta\in\widehat{X}\mid\Delta\subset Y\\}$. Define the
_barycentric retraction_ of $Y$ to be

$\operatorname{br}(Y)=\\{\Delta\in\widehat{X}\mid\overline{\Delta}\subset
Y\\}$

######  Proposition 5.6.

Say $Y$ is a union of cells of some regular CW complex $X\subset R^{n}$ whose
cells are all convex polyhedra. Then $\operatorname{br}(Y)$ is homotopy
equivalent to $Y$

###### Proof.

We construct a homotopy $h_{\operatorname{br}}:[0,1]\times Y\rightarrow Y$,
which we will term the _barycentric retracting map_.

Observe that $\operatorname{br}(Y)\subset Y$ is a subcomplex of $\widehat{X}$,
whose vertex set is precisely those $0$-simplices of $\widehat{X}$
corresponding to cells of $Y$. In fact, the simplices of
$\operatorname{br}(Y)$ correspond precisely to flags of cells in $Y$. This
means that $Y=\emptyset$ iff $\operatorname{br}(Y)=\emptyset$.

Let $\Delta=\Delta(v_{0},\ldots,v_{d})$ be a simplex of $\widehat{Y}$, and
define the set
$J_{\Delta}=\\{v_{i_{0}},\ldots,v_{i_{d^{\prime}}}\\}=\\{v_{i}\in\\{v_{0},\ldots,v_{d}\\}\mid
v_{i}\in\operatorname{br}(Y)\\}$. Then if we let
$\Delta^{\prime}=\Delta(v_{i_{0}},\ldots,v_{i_{d^{\prime}}})$, it is clear
from the correspondence of simplices of $\operatorname{br}(Y)$ to flags of
cells in $Y$ that $J_{\Delta}$ is nonempty and that
$\overline{\Delta}\cap\lvert\operatorname{br}(Y)\rvert=\overline{\Delta}^{\prime}$.
Say that $x\in Y$ with $x\in\overline{\Delta}$. Then we may write $x$ uniquely
as $x=\sum_{i=0}^{d}t_{v_{i}}v_{i}$, where $0\leq t_{v_{i}}\leq 1$ for each
$i$ and $\sum t_{v_{i}}=1$. Then, for $t\in[0,1]$, let

$h_{\operatorname{br}}(t,x)=t\sum_{v_{i}\not\in
J_{\Delta}}t_{v_{i}}v_{i}+\frac{1-t\sum_{v_{i}\not\in
J_{\Delta}}t_{v_{i}}}{\sum_{v_{i}\in J_{\Delta}}t_{v_{i}}}\sum_{v_{i}\in
J_{\Delta}}t_{v_{i}}v_{i}$

From this, we obtain our desired map $h_{\operatorname{br}}:[0,1]\times
Y\rightarrow\lvert\widehat{Y}\rvert$, noting that the definition of
$h_{\operatorname{br}}$ agrees on the boundaries of simplices. Observe that
$h_{\operatorname{br}}(0,-):Y\rightarrow\lvert\operatorname{br}(Y)\rvert$
serves as homotopy inverse to the inclusion
$\lvert\operatorname{br}(Y)\rvert\hookrightarrow Y$. ∎

######  Proposition 5.7.

Let $X\subset R^{n}$ be a regular CW complex whose cells are all convex
polyhedra, and let $Y$ be a union of cells of $X$. If $X$ is symmetric as a CW
complex under the action of some group $G$ acting linearly on $X$ and $Y$ is
also symmetric relative to $G$, then $h_{\operatorname{br}}:[0,1]\times
Y\rightarrow Y$ is equivariant (where the action of $G$ on $[0,1]\times Y$ is
given by $g(t,x)=(t,g(x))$ for all $g\in G$).

###### Proof.

From the symmetry of $X$ and $Y$, we have that $\widehat{X}$, $\widehat{Y}$,
and $\operatorname{br}(Y)$ are all symmetric. Hence, for
$\Delta\in\widehat{Y}$, $g(J_{\Delta})=J_{g(\Delta)}$. Equivariance of
$h_{\operatorname{br}}$ then follows from the linearity of the action of $G$.
∎

Even when $Y$ is not a full subcomplex of $X$, we may make sense of the
simplicial complex $\Delta(\mathscr{F}(Y))$. As described in Definition 5.5,
we may identify $\Delta(\mathscr{F}(Y))$ with $\operatorname{br}(Y)$.
Furthermore, if $X$ and $Y$ are both symmetric, then the homeomorphism
$\lvert\Delta(\mathscr{F}(Y))\rvert\rightarrow\lvert\operatorname{br}(Y)\rvert$
is equivariant.

We conclude by addressing equivariance in Börner’s Nerve Theorem, which will
be used in Section 6.2.

######  Definition 5.8 ([9] Definition 2.9).

Let $\\{X_{i}\\}_{i\in I}$ be a family of sets. The _nerve_ of
$\\{X_{i}\\}_{i\in I}$ is the abstract simplicial complex $\mathcal{N}$ having
vertex set $I$ and simplices given by the finite subsets $\sigma$ of $I$ such
that

$\bigcap_{i\in\sigma}X_{i}\neq\emptyset$

A variety of theorems exist which relate a space covered by a family of sets
to the nerve of that cover. Björner’s Nerve Lemma has the advantage of only
requiring the triviality of sufficiently many homotopy groups of intersections
of sets in the cover (with a slightly weaker conclusion in exchange).

######  Theorem 5.9 (Nerve Theorem, [4] Theorem 6 and Remark 7).

Let $X$ be a regular connected CW complex and $\\{X_{i}\\}_{i\in I}$ a family
of subcomplexes with $X=\bigcup_{i\in I}X_{i}$. Let $\mathcal{N}$ be the nerve
of $\\{X_{i}\\}$

  1. (i)

Say that every finite nonempty intersection $X_{i_{1}}\cap\ldots\cap
X_{i_{t}}$ is $(k-t+1)$-connected. Then there is a map
$f:X\rightarrow\lvert\mathcal{N}\rvert$ such that the induced homomorphism
$f_{\\#,j}:\pi_{j}(X)\rightarrow\pi_{j}(\lvert\mathcal{N}\rvert)$ is an
isomorphism for all $j\leq k$ and an epimorphism for $j=k+1$.

  2. (ii)

If every finite nonempty intersection $X_{i_{1}}\cap\ldots\cap X_{i_{t}}$ is
contractible, then $f$ gives a homotopy equivalence
$X\simeq\lvert\mathcal{N}\rvert$.

A version of Björner’s nerve lemma sufficiently equivariant for our purposes
reads thus:

######  Theorem 5.10.

Let $X$ be a regular connected CW complex which is a subset of a real vector
space, and whose cells are all convex polyhedra. Let $G$ act linearly on $X$
in such a way that $X$ is symmetric as a CW complex under the action of $G$.
Let $\\{X_{i}\\}_{i\in I}$ be a family of subcomplexes with $X=\bigcup_{i\in
I}X_{i}$, and say that for each $g\in G$ and $i\in I$,
$g(X_{i})\in\\{X_{i}\\}_{i\in I}$. Then if $\mathcal{N}$ is the nerve of
$\\{X_{i}\\}_{i\in I}$, we have the following:

  1. (i)

Say every finite nonempty intersection $X_{i_{1}}\cap\ldots\cap X_{i_{t}}$ is
$(k-t+1)$-connected. Then there is an equivariant map
$f:X\rightarrow\lvert\mathcal{N}\rvert$ such that the induced (equivariant)
homomorphism
$f_{\\#,j}:\pi_{j}(X,*)\rightarrow\pi_{j}(\lvert\mathcal{N}\rvert,f(*))$ is an
isomorphism for all $j\leq k$ and an epimorphism for $j=k+1$.

  2. (ii)

If every finite nonempty intersection $X_{i_{1}}\cap\ldots\cap X_{i_{t}}$ is
contractible, then $f$ gives a homotopy equivalence
$X\simeq\lvert\mathcal{N}\rvert$.

###### Proof.

(i): We claim that we may construct an equivariant map in the manner described
in the proof in [4]. By Proposition 5.4, we have an equivariant homeomorphism
$X\rightarrow\lvert\Delta(\mathscr{F}(X))\rvert$. Let
$\varphi:\mathscr{F}(X)\rightarrow\mathscr{F}(\mathcal{N})$ be given by
$\varphi(\sigma)=\\{i\in I\mid\sigma\in X_{i}\\}$ for each cell $\sigma$ of
$X$. This is an order-reversing map of posets. It is also equivariant, since
for a given cell $\sigma$, $\\{g(i)\mid i\in I\text{ with }\sigma\in
X_{i}\\}=\\{i\in I\mid g(\sigma)\in X_{i}\\}$. Then $\varphi$ induces an
equivariant continuous function
$\lvert\Delta(\mathscr{F}(X))\rvert\rightarrow\lvert\Delta(\mathscr{F}(\mathcal{N}))\rvert$.
Since Remark 5.4 gives an equivariant homeomorphism
$\lvert\Delta(\mathscr{F}(\mathcal{N}))\rvert\rightarrow\lvert\mathcal{N}\rvert$,
composing gives us an equivariant continuous function

$f:X\rightarrow\lvert\Delta(\mathscr{F}(X))\rvert\rightarrow\lvert\Delta(\mathscr{F}(\mathcal{N}))\rvert\rightarrow\lvert\mathcal{N}\rvert$

The equivariance of $f$ gives the equivariance of each induced
$f_{\\#,j}:\pi_{j}(X)\rightarrow\pi_{j}(\lvert\mathcal{N}\rvert)$.

Since our equivariant map $f$ is the map from the original proof of the nerve
theorem, part (ii) follows automatically. ∎

Note that though, if following the proof for (ii), one would obtain a map
$g:\lvert\mathcal{N}\rvert\rightarrow X$ which serves as a homotopy inverse to
$f$, we have not guaranteed the equivariance of this or any map in the
opposite direction. Thus there is more work to be done before we may term this
a true equivariant version of Björner’s nerve theorem.

The barycentric retracting map described in Definition 5.5 allows us to apply
this version of the nerve theorem to coverings by sets open in our larger
space. This version of the nerve theorem is used in [9].

######  Theorem 5.11.

Let $X$ be a regular connected CW complex. Let $\\{Y_{i}\\}_{i\in I}$ be a
family of subsets of $X$. Assume each $Y_{i}$ may be written as a union of
cells of $X$, and that each $Y_{i}$ is open in $X$. Let $Y=\bigcup_{i\in
I}Y_{i}$, and let $\mathcal{N}_{Y}$ be the nerve of this family.

  1. (1)

Say that every finite nonempty intersection $Y_{i_{1}}\cap\ldots\cap
Y_{i_{t}}$ is $(k-t+1)$-connected. Then there is a map
$f:Y\rightarrow\lvert\mathcal{N}_{Y}\rvert$ such that the induced homomorphism
$f_{\\#,j}:\pi_{j}(Y)\rightarrow\pi_{j}(\lvert\mathcal{N}_{Y}\rvert)$ is an
isomorphism for all $j\leq k$ and an epimorphism for $j=k+1$.

  2. (2)

If every finite nonempty intersection $Y_{i_{1}}\cap\ldots\cap Y_{i_{t}}$ is
contractible, then $f$ gives a homotopy equivalence
$X\simeq\lvert\mathcal{N}_{Y}\rvert$.

###### Proof.

From $h_{\operatorname{br}}$ in 5.5, we obtain a map
$f_{1}:Y\rightarrow\lvert\operatorname{br}(Y)\rvert$ which induces a homotopy
equivalence. Observe that $\lvert\operatorname{br}(Y)\rvert$ is itself a
regular CW complex, and that each $\lvert\operatorname{br}(Y_{i})\rvert$ is a
subcomplex of $\lvert\operatorname{br}(Y)\rvert$. It is clear that
$\lvert\operatorname{br}(Y)\rvert\supset\bigcup_{i\in
I}\lvert\operatorname{br}(Y_{i})\rvert$. To show the opposite inclusion, say
that $\Delta=\Delta(v_{0},\ldots,v_{d})\in\operatorname{br}(Y)$. This means
that $\overline{\Delta}\in\widehat{Y}$, i.e. that the vertices of $\Delta$,
suitably ordered, correspond to a flag $\sigma_{0},\ldots,\sigma_{d}$ of cells
of $X$, all of which are contained in $Y$. Assume $\sigma_{d}$ has lowest
dimension amongst the cells in this chain, and say that $i\in I$ is such that
$\sigma_{d}\subset Y_{i}$. Since $Y_{i}$ is open in $X$, this means that each
of $\sigma_{0},\ldots,\sigma_{d}$ are contained in $Y_{i}$, and therefore
$\Delta\in\operatorname{br}(Y_{i})$. Hence as desired,
$\lvert\operatorname{br}(Y)\rvert=\bigcup_{i\in
I}\lvert\operatorname{br}(Y_{i})\rvert$.

For $Y_{i_{1}},\ldots,Y_{i_{t}}\in\\{Y_{i}\\}_{i\in I}$, we can see that

$\displaystyle\bigcap_{j=1}^{t}\operatorname{br}(Y_{i_{j}})$
$\displaystyle=\bigcap_{j=1}^{t}\\{\sigma\in\widehat{X}\mid\overline{\sigma}\subset
Y_{i_{j}}\\}$
$\displaystyle=\\{\sigma\in\widehat{X}\mid\overline{\sigma}\subset\bigcap_{j=1}^{t}Y_{i_{j}}\\}$
$\displaystyle=\operatorname{br}(\cap_{j=1}^{t}Y_{i_{j}})$

If $\mathcal{N}_{\operatorname{br}(Y)}$ denotes the nerve of the covering of
$\lvert\operatorname{br}(Y)\rvert$ by the family
$\\{\lvert\operatorname{br}(Y_{i})\rvert\\}_{i\in I}$, this means that
$\mathcal{N}_{Y}=\mathcal{N}_{\operatorname{br}(Y)}$. Furthermore, since via
$h_{\operatorname{br}}$ we have that $\bigcap_{j=1}^{t}Y_{i_{j}}$ is homotopy
equivalent to
$\lvert\operatorname{br}(\bigcap_{j=1}^{t}Y_{i_{j}})\rvert=\bigcap_{j=1}^{t}\lvert\operatorname{br}(Y_{i_{j}})\rvert$,
we may conclude that for any $k$. a finite intersection intersection
$Y_{i_{1}}\cap\cdots\cap Y_{i_{t}}$ is $k$-connected iff
$\lvert\operatorname{br}(Y_{i_{1}})\rvert\cap\cdots\cap\lvert\operatorname{br}(Y_{i_{t}})\rvert$
is.

Let
$f_{2}:\operatorname{br}(Y)\rightarrow\lvert\mathcal{N}_{\operatorname{br}(Y)}\rvert$
be as given by the standard Nerve Theorem (Theroem 5.9). Then composing

$Y\overset{f_{1}}{\rightarrow}\lvert\operatorname{br}(Y)\rvert\overset{f_{2}}{\rightarrow}\lvert\mathcal{N}_{\operatorname{br}(Y)}\rvert=\lvert\mathcal{N}_{Y}\rvert$

we obtain our map $f:Y\rightarrow\lvert\mathcal{N}_{Y}\rvert$ with the desired
properties. ∎

######  Corollary 5.12.

Let $X$, $Y$, and $\\{Y_{i}\\}_{i\in I}$ be as in Theorem 5.11. Assume further
that $X$ is a subset of a real vector space and all cells of $X$ are convex
polyhedra. Let the group $G$ act linearly on our space in such a way that $X$
is symmetric as a CW complex and $\\{Y_{i}\\}$ has the property that for each
$g\in G$ and $i\in I$, $g(Y_{i})\in\\{Y_{i}\\}_{i\in I}$. Let
$\mathcal{N}_{Y}$ denote the nerve of the covering of $Y$ by
$\\{Y_{i}\\}_{i\in I}$. Then the map
$f:Y\rightarrow\lvert\mathcal{N}_{Y}\rvert$ of Theorem 5.11 is equivariant.

###### Proof.

This follows from Proposition 5.7 and Theorem 5.10. ∎

We will want one more equivariant Nerve Theorem. The statement we need was
proved by Hess and Hirsch in [11]. This version is stated for $X$ a simplicial
complex covered by a family for which nonempty intersections are contractible,
but from it we may obtain equivariant maps in both directions.

For $G$ a group and $\sigma$ a simplex in some simplicial complex $X$ which is
symmetric relative to $G$, let $G_{\sigma}$ be the subgroup of $G$ given by
$G_{\sigma}=\\{g\in G\mid g(\sigma)=\sigma\\}$.

######  Theorem 5.13 ([11] Lemma 2.5).

Let $X$ be a simplicial complex symmetric relative to the action of $G$, and
let $\\{X_{i}\\}_{i\in I}$ be a family of subcomplexes with $\lvert
X\rvert=\bigcup_{i\in I}\lvert X_{i}\rvert$ and with
$g(X_{i})\in\\{X_{i}\\}_{i\in I}$ for each $g\in G$ and $i\in I$. Say that
every nonempty finite intersection $\bigcap_{i\in\sigma}\lvert X_{i}\rvert$,
for $\sigma\subset I$, is $G_{\sigma}$-contractible. Then if $\mathcal{N}$ is
the nerve of $\\{X_{i}\\}_{i\in I}$, we have that $\lvert X\rvert$ and
$\lvert\mathcal{N}\rvert$ are $G$-homotopy equivalent.

##  6\. Equivariance in the Gabrielov-Vorobjov Construction

In what follows, we assume that $G$ is a finite reflection group acting on
$\mathbb{R}^{n}$. Let $S$ be the definable set we intend to approximate.
Assume we have also chosen a compact definable set $A\subset\mathbb{R}^{n}$
with $S\subset A$ and families $\\{S_{\delta}\\}_{\delta>0}$ and
$\\{S_{\delta,\varepsilon}\\}_{\delta,\varepsilon>0}$ representing $S$ in $A$
as described in Section 3. The conclusions of Subsections 6.1 and 6.2 (which
concern relations between $S$ and an intermediate set $V$) apply both to the
definable and constructible cases. The distinctions between these two cases
are addressed in Subsections 6.3 and 6.4, which describe relations between $V$
and our approximating set $T$.

###  6.1. Symmetric construction for $V$

Gabrielov and Vorobjov’ argument in [9] first uses a triangulation of $A$
adapted to $S$ to construct, for a given integer $m>0$ and sequence
$0<\varepsilon_{0},\delta_{0},\ldots,\varepsilon_{m},\delta_{m}<1$, an
intermediate set
$V=V(\varepsilon_{0},\delta_{0},\ldots,\varepsilon_{m},\delta_{m})$ and
homomorphisms $\tau_{\\#,k}:\pi_{k}(V)\rightarrow\pi_{k}(S)$ and
$\tau_{*,k}:H_{k}(V)\rightarrow H_{k}(S)$. The set $V$ echoes the behavior of
$T$ relative to the parameters $\varepsilon_{i}$ and $\delta_{i}$. However, it
also has a cover that allows one to liken it to the triangulation of $S$ well
enough to establish the aforementioned maps. We show that, if we start with a
symmetric triangulation, the set $V$ is also symmetric and that there exists
an equivariant map $\tau:V\rightarrow S$ which induces $\tau_{\\#}$ and
$\tau_{*}$.

Let $(\Lambda,\Phi)$ be a symmetric triangulation of $A$ adapted to $S$. We
replace $S$ by $\Phi^{-1}(S)$ to assume that $S$ is a union of simplices of
$\Lambda$. We will primarily work within the first barycentric subdivision of
$\Lambda$, denoted $\widehat{\Lambda}$. If $\Delta(b_{0},\ldots,b_{p})$
corresponds to a simplex in $\widehat{\Lambda}$, we have that for each $b_{i}$
there is a unique simplex $\Delta_{b_{i}}$ in $\Lambda$ such that $b_{i}$ is
the barycenter of $\Delta_{b_{i}}$. We will assume when we write
$\Delta(b_{0},\ldots,b_{p})$ that $b_{0},\ldots,b_{p}$ are ordered so that
$\dim(\Delta_{b_{0}})>\cdots>\dim(\Delta_{b_{p}})$. By $\widehat{S}$ we mean
the set of all simplices of $\widehat{\Lambda}$ which belong to $S$.

Gabrielov and Vorobjov in [9] construct $V$ as a union of sets
$K_{B}(\delta_{i},\varepsilon_{i})$ defined for pairs of simplices
$K\in\widehat{\Lambda}$ and $B\in\widehat{S}$. The set
$K_{B}(\delta_{i},\varepsilon_{i})$ depends on something that Gabrielov and
Vorobjov call the core of the simplex $B=B(b_{0},\ldots,b_{p})$. This in some
sense is meant to capture which faces of $B$ continue to intersect the
preimages of the sets $S_{\delta}$ as $\delta$ shrinks to $0$. To properly
define this notion, though, we must reiterate some technical terminology from
[9].

######  Definition 6.1 ([9] Definition 3.2).

We say $S$ is _marked_ if for each pair of simplices
$(\Delta^{\prime},\Delta)$ in $S$ with $\Delta^{\prime}$ a subsimplex of
$\Delta$, we have designated $\Delta^{\prime}$ as either a _hard_ or _soft_
subsimplex of $\Delta$. For a pair $(\Delta^{\prime},\Delta)$ with
$\Delta^{\prime}$ not in $S$, we always designate $\Delta^{\prime}$ as a soft
subsimplex of $\Delta$.

######  Definition 6.2.

If $A$ and $S$ are symmetric and $(\Lambda,\Phi)$ is an equivariant
triangulation of $A$ adapted to $S$, we say $S$ is _symmetrically marked_ if
$S$ is marked in such a way that for each $g\in G$,
$(g(\Delta^{\prime}),g(\Delta))$ has the same hard/soft designation as
$(\Delta^{\prime},\Delta)$. Since $\Delta^{\prime}$ is in $S$ iff
$g(\Delta^{\prime})$ is also in $S$ by our choice of triangulation, this
stipulation does not interfere with the property that simplices
$\Delta^{\prime}$ not contained in $S$ are always designated as soft
subsimplices.

We will specify separate hard-soft relations for our triangulation depending
on whether we are in the separable case or not (see Subsections 6.3 and 6.4).
For the moment, it is enough to assume that $S$ is symmetrically marked.

######  Definition 6.3 ([9] Definition 3.3).

For a simplex $B=B(b_{0},\ldots,b_{p})$ of $\widehat{\Lambda}$ contained in
$\widehat{S}$, the _core_ of $B$, denoted $C(B)$, is the maximal subset
$\\{b_{0},\ldots,b_{p^{\prime}}\\}$ of $\\{b_{0},\ldots,b_{p}\\}$ so that for
$0\leq\nu\leq p^{\prime}$, we have $\Delta_{b_{\nu}}$ is a hard subsimplex of
$\Delta_{b_{\mu}}$ for every $0\leq\mu<\nu$.

Note that we always have $b_{0}\in C(B)$. We will establish the convention
that if $B$ is not in $\widehat{S}$, $C(B)=\emptyset$.

######  Definition 6.4 ([9] Definition 3.6).

Let $B=B(b_{0},\ldots,b_{p})$ be a simplex in $\widehat{S}$ and
$K=K(c_{0},\ldots,c_{q})$ a simplex in $\widehat{\Lambda}$ with
$B\subset\overline{K}$. Let $I=\\{b_{0},\ldots,b_{p}\\}$ and
$J=\\{c_{0},\ldots,c_{q}\\}$. Then for $0<\delta<1$ and $0<\varepsilon<1$,
define

$K_{B}(\delta,\varepsilon):=\left\\{\sum_{c_{\nu}\in J}t_{c_{\nu}}c_{\nu}\in
K(c_{1},\ldots,c_{q})\mid\sum_{b_{\nu}\in C(B)}t_{b_{\nu}}>\delta,\right.\\\
\left.\sum_{b_{\nu}\in I}t_{b_{\nu}}>1-\varepsilon,\text{ and }\forall
b_{\nu}\in I\;\forall c_{\mu}\in(J\setminus
I),t_{b_{\nu}}>t_{c_{\mu}}\right\\}$

Given $B$ a simplex in $\widehat{S}$, let $S_{B}$ denote the set of all
simplices $B^{\prime}$ of $\widehat{\Lambda}$ with
$B^{\prime}\subset\overline{B}\cap\widehat{S}$.

######  Definition 6.5 ([9] Definition 3.7).

Fix $m>0$ and a sequence
$0<\varepsilon_{0},\delta_{0},\varepsilon_{1},\delta_{1},\ldots,\varepsilon_{m},\delta_{m}<1$.
Then for $B$ a simplex in $\widehat{S}$, let

$V_{B}=\bigcup_{B^{\prime}\in S_{B}}\;\bigcup_{\overline{K}\supset
B^{\prime}}\;\bigcup_{i=0}^{m}K_{B^{\prime}}(\delta_{i},\varepsilon_{i})$

(that is, for each $B^{\prime}\in S_{B}$ the union is taken over all simplices
$K$ in $\widehat{\Lambda}$ with $B^{\prime}\subset\overline{K}$). Let

$V=\bigcup_{B\in\widehat{S}}V_{B}$

######  Proposition 6.6.

Say that $S$ and $A$ are symmetric under the action of $G$, $(\Lambda,\Phi)$
is an equivariant triangulation of $A$ adapted to $S$, and $S$ is
symmetrically marked. Then the family $\\{V_{B}\\}_{B\in\widehat{S}}$ is
symmetric under the induced action of $G$, and hence the set $V$ is symmetric.

###### Proof.

This follows more or less immediately from construction. By symmetry of
$\widehat{\Lambda},\widehat{S}$, and our marking and by linearity of the
action of $G$, we can see that
$g(K_{B^{\prime}}(\delta_{i},\varepsilon_{i}))=g(K)_{g(B^{\prime})}(\delta_{i},\varepsilon_{i})$,
and hence $g(V_{B})=V_{g(B)}$. ∎

Some properties of the sets $K_{B}(\delta,\varepsilon)$, $V_{B}$ and $V$ are
worth noting here.

######  Lemma 6.7.

Let $B=B(b_{0},\ldots,b_{p})$ be a simplex in $\widehat{S}$ and
$K=K(c_{0},\ldots,c_{q})$ a simplex in $\widehat{\Lambda}$ with
$B\subset\overline{K}$. Then for
$0<\delta,\varepsilon,\delta^{\prime},\varepsilon^{\prime}<1$ we have

$\displaystyle K_{B}(\delta,\varepsilon)\cup
K_{B}(\delta^{\prime},\varepsilon^{\prime})$
$\displaystyle=K_{B}(\min\\{\delta,\delta^{\prime}\\},\max\\{\varepsilon,\varepsilon^{\prime}\\})$
$\displaystyle K_{B}(\delta,\varepsilon)\cap
K_{B}(\delta^{\prime},\varepsilon^{\prime})$
$\displaystyle=K_{B}(\max\\{\delta,\delta^{\prime}\\},\min\\{\varepsilon,\varepsilon^{\prime}\\})$

###### Proof.

This follows immediately from the definition. The second line appears as [9]
Lemma 4.1. ∎

######  Proposition 6.8 (see the proof of [9] Theorem 4.8).

For any pair of simplices $B_{1}$ and $B_{2}$ in $\widehat{S}$, one of the
following holds

  1. (i)

$V_{B_{1}}\cap V_{B_{2}}=\emptyset$
($\Leftrightarrow\overline{B}_{1}\cap\overline{B}_{2}\cap\lvert\widehat{S}\rvert=\emptyset$)

  2. (ii)

$V_{B_{1}}\cap V_{B_{2}}=V_{B_{0}}$ where $V_{B_{0}}$ is the unique simplex in
$\widehat{S}$ with
$\overline{B}_{1}\cap\overline{B}_{2}\cap\lvert\widehat{S}\rvert=\overline{B}_{0}\cap\lvert\widehat{S}\rvert$

###### Proof.

The bulk of this statement is taken from [9]. Because we assert something
slightly more detailed, we include a proof.

From the definition, we have that for $B,B^{\prime}\in\widehat{S}$ and
$K,K^{\prime}\in\widehat{\Lambda}$ with $B\subset\overline{K}$ and
$B^{\prime}\subset\overline{K}^{\prime}$, and any $0<\varepsilon,\delta<1$,
then $K_{B}(\delta,\varepsilon)\cap
K^{\prime}_{B^{\prime}}(\delta,\varepsilon)\neq\emptyset$ implies that
$K=K^{\prime}$ and either $B\subset\overline{B}^{\prime}$ or
$B^{\prime}\subset\overline{B}$ (this is [9] Lemma 4.1). This means that

$V_{B_{1}}\cap V_{B_{2}}=\bigcup_{B^{\prime}\in S_{B_{1}}\cap
S_{B_{2}}}\;\bigcup_{\overline{K}\supset
B^{\prime}}\;\bigcup_{i=0}^{m}K_{B^{\prime}}(\delta_{i},\varepsilon_{i})$

Since $S_{B_{1}}\cap S_{B_{2}}=\emptyset$ $\Leftrightarrow$
$\overline{B}_{1}\cap\overline{B}_{2}\cap\widehat{S}=\emptyset$ and otherwise
$S_{B_{1}}\cap S_{B_{2}}=S_{B_{0}}$ where $B_{0}$ is the simplex such that
$\overline{B}_{0}\cap\widehat{S}=\overline{B}_{1}\cap\overline{B}_{2}\cap\widehat{S}$,
the statements follow. ∎

######  Lemma 6.9 ([9] Lemma 4.5).

For $B$ in $\widehat{S}$, $m\geq 1$, and
$0<\varepsilon_{0}\ll\delta_{0}\ll\cdots\ll\varepsilon_{m}\ll\delta_{m}\ll 1$,
the set $V_{B}$ is open in $\lvert\Lambda\rvert$ and $(m-1)$-connected.

Note that Gabrielov and Vorobjov give $V_{B}$ as

$V_{B}=\bigcup_{B^{\prime}\in S_{B}}\;\bigcup_{\overline{K}\supset
B}\;\bigcup_{i=0}^{m}K_{B^{\prime}}(\delta_{i},\varepsilon_{i})$

where the union is only taken over those $K\in\widehat{\Lambda}$ with
$B\subset\overline{K}$. However, we need to define $V_{B}$ as it appears here
in Definition 6.5 in order for Proposition 6.8 to be stated and used as it is
in [9]. Lemma 6.9 holds with the updated definition; in the proof of [9] Lemma
4.5, one needs only to define the sets $U_{B^{\prime},i}$ for $B^{\prime}\in
S_{B}$ as

$U_{B^{\prime},i}=\bigcup_{\overline{K}\supset
B^{\prime}}K_{B^{\prime}}(\delta_{i},\varepsilon_{i})$

rather than

$U_{B^{\prime},i}=\bigcup_{\overline{K}\supset
B}K_{B^{\prime}}(\delta_{i}.\varepsilon_{i})$

(i.e., one must take the union over all simplices $K$ of $\widehat{\Lambda}$
with $B^{\prime}\subset\overline{K}$, and not just those with
$B\subset\overline{K}$). Then the updated sets $U_{B^{\prime},i}$ for
$B^{\prime}\in S_{B}$ and $1\leq i\leq m$ cover the updated $V_{B}$, but the
intersection condition and hence the nerve of this family remains unchanged,
and so the argument in [9] continues to hold.

###  6.2. Equivariance of the map $\tau$

The existence of homomorphisms $\tau_{\\#,k}:\pi_{k}(V)\rightarrow\pi_{k}(S)$
and $\tau_{*,k}:H_{k}(V)\rightarrow H_{k}(S)$ for $0\leq k\leq m$ is given in
[9] Theorem 4.8. In order to show that we may construct these functions
equivariantly, we must dissect the proofs there and describe more explicitly
some of the details involved in applying the nerve theorem to $V$. We will in
fact obtain an equivariant map $\tau:V\rightarrow S$ on the level of sets,
inducing equivariant maps of (pointed) homotopy and homology groups which are
isomorphisms or epimorphisms for the promised indices.

Our first goal is to endow $\lvert\Lambda\rvert$ with a regular CW complex
structure fine enough to allow us to write $V$ as a union of cells.

######  Notation 6.10.

Let $K$ be a simplex of some simplicial complex $\Lambda$, and say $K$ has
vertex set $\\{v_{0},\ldots,v_{d}\\}$.

  * •

Let $\mathcal{C}_{K,\text{vert}}=\\{L\in\widehat{\Lambda}\mid L\subset K\\}$.

  * •

For $I\subset\\{v_{0},\ldots,v_{d}\\}$ and a given $0<\varepsilon<1$, let

    * $\sim$

$C_{K,I,>,\varepsilon}=\\{\sum_{i=1}^{d}t_{i}v_{i}\in K\mid\sum_{i\in
I}t_{i}>\varepsilon\\}$

    * $\sim$

$C_{K,I,<,\varepsilon}=\\{\sum_{i=1}^{d}t_{i}v_{i}\in K\mid\sum_{i\in
I}t_{i}<\varepsilon\\}$

    * $\sim$

$C_{K,I,=,\varepsilon}=\\{\sum_{i=1}^{d}t_{i}v_{i}\in K\mid\sum_{i\in
I}t_{i}=\varepsilon\\}$

We denote
$\mathcal{C}_{K,I,\varepsilon}=\\{C_{K,I,>,\varepsilon},C_{K,I,<,\varepsilon},C_{K,I,=,\varepsilon}\\}$.

######  Lemma 6.11.

Fix a simplex $K$ of some simplicial complex $\Lambda$, let $I$ be a subset of
the vertex set of $K$, and let $0<\varepsilon<1$. For a given subsimplex
$K^{\prime}$ of $K$, let $I_{K^{\prime}}$ be the intersection of $I$ with the
vertex set of $K^{\prime}$. Then

$\bigcup_{K^{\prime}\subset\overline{K}}\mathcal{C}_{K^{\prime},I_{K^{\prime}},\varepsilon}$

gives a regular CW decomposition of $\overline{K}$ into convex polyhedra,
where the union is taken over all simplices $K^{\prime}\subset\overline{K}$.

###### Proof.

Each cell of our collection is a convex polyhedron more or less by definition,
and hence a regular cell in a fairly immediate manner. We have that for each
simplex $K^{\prime}\subset\overline{K}$,
$\mathcal{C}_{K^{\prime},I_{K^{\prime}},\varepsilon}$ gives a partition of
$K^{\prime}$, and so since the simplices $K^{\prime}$ partition
$\overline{K}$, our collection indeed forms a partition of $\overline{K}$.

It remains to show that for a cell $C$ in our collection, $\partial C$ is
contained in a union of cells of lower dimension. Say
$C=C_{K^{\prime},I_{K^{\prime}},=,\varepsilon}$. Then the boundary of $C$ is
the union of those cells
$C_{K^{\prime\prime},I_{K^{\prime\prime}},=,\varepsilon}$,
$K^{\prime\prime}\subsetneq\overline{K^{\prime}}$, which are nonempty (so,
those for which the vertex set of $K^{\prime\prime}$ is neither contained in
nor disjoint from $I$). If $C=C_{K^{\prime},I_{K^{\prime}},>,\varepsilon}$,
then $\partial C$ consists both of
$C_{K^{\prime},I_{K^{\prime}},=,\varepsilon}$ together with the cells in its
boundary and those nonempty cells
$C_{K^{\prime\prime},I_{K^{\prime\prime}},>,\varepsilon}$ corresponding to
simplices $K^{\prime\prime}\subsetneq K^{\prime}$ (i.e. where the vertex set
of $K^{\prime\prime}$ is not disjoint from I). The case of
$C=C_{K^{\prime},I_{K^{\prime}},<,\varepsilon}$ is identical save that in the
condition for nonemptiness we must replace $I$ with its complement. Hence this
collection indeed gives a regular CW decomposition of $\overline{K}$. ∎

We return to our primary setting, in which $\Lambda$ is a symmetric
triangulation of $A$ adapted to $S$. For $K=K(c_{1},\ldots,c_{q})$ a simplex
of $\widehat{\Lambda}$ and $0<\delta,\varepsilon<1$, we let

$\mathcal{C}_{K,\delta,\varepsilon}=\left\\{C=L\cap\bigcap_{I\subset\\{c_{0},\ldots,c_{q}\\}}C_{K,I,\delta}\cap\bigcap_{I\subset\\{c_{0},\ldots,c_{q}\\}}C_{K,I,\varepsilon}\mid\right.\\\
L\in\mathcal{C}_{K,\text{vert}},C_{K,I,\delta}\in\mathcal{C}_{K,I,\delta}\text{
and }C_{K,I,\varepsilon}\in\mathcal{C}_{K,I,\varepsilon}\\\ \left.\text{ for
each }I,\text{ and }C\neq\emptyset\right\\}$

and let
$\mathcal{C}_{\delta,\varepsilon}=\bigcup_{K\in\widehat{\Lambda}}\mathcal{C}_{K,\varepsilon,\delta}$.

Figure 3. Decomposition $\mathcal{C}_{K,\delta,\varepsilon}$ for
$K(v_{0},v_{1},v_{2})$, assuming $\delta<\varepsilon$.
$K_{B}(\delta,\varepsilon)$ is shaded for $B=B(v_{0},v_{1})$ and
$C(B)=\\{v_{0}\\}$

######  Lemma 6.12.

For any $0<\delta,\varepsilon<1$, $\mathcal{C}_{\delta,\varepsilon}$ gives a
regular CW decomposition of $\lvert\widehat{\Lambda}\rvert$ which is symmetric
relative to $G$ and in which each cell is a convex polyhedron.

###### Proof.

We have that for each $K\in\widehat{\Lambda}$, the collection of cells of the
second barycentric subdivision of $\Lambda$ contained in $\overline{K}$ gives
a regular CW decomposition of $K$ into convex polyhedra. Lemma 6.11 shows that
for each subset $I$ of the vertex set of $K$,
$\bigcup_{K^{\prime}\subset\overline{K}}\mathcal{C}_{K^{\prime},I_{K^{\prime}},\varepsilon}$
and
$\bigcup_{K^{\prime}\subset\overline{K}}\mathcal{C}_{K^{\prime},I_{K^{\prime}},\delta}$
each gives a regular CW decomposition of $\overline{K}$. Since each set in
$\mathcal{C}_{\delta,\varepsilon}$ is an intersection of one set from each of
these decompositions across the various simplices $K$ of $\widehat{\Lambda}$,
the cells of $\mathcal{C}_{\delta,\varepsilon}$ remain convex polyhedra. Also
for each $K$, $\mathcal{C}_{K.\delta,\varepsilon}$ remains a partition of $K$,
and so $\mathcal{C}_{\delta,\varepsilon}$ gives a partition of
$\lvert\widehat{\Lambda}\rvert$.

Let $C\subset K$ for some $C\in\mathcal{C}_{\delta,\varepsilon}$. We simplify
the notation above to write $C=C_{1}\cap\cdots\cap C_{l}$ for $C_{i}$ cells of
our various decompositions. Then we have

$\displaystyle\partial C$
$\displaystyle\subset\bigcup_{i=1}^{l}\overline{C}_{1}\cap\ldots\cap\overline{C}_{i-1}\cap\partial
C_{i}\cap\overline{C}_{i+1}\cap\ldots\cap\overline{C}_{l}$
$\displaystyle=\bigcup_{J\subsetneq\\{1,\ldots,l\\}}\left(\bigcap_{i\in
J}C_{i}\cap\bigcap_{i\in\\{1,\ldots,l\\}\setminus J}\partial C_{i}\right)$

after decomposing each $\overline{C}_{i}=\partial C_{i}\cup C_{i}$ and
distributing. We know $\delta C_{i}$ is a union of cells of dimension lower
than that of $C_{i}$ from the decomposition of $\overline{K}$ corresponding to
the decomposition of $K$ from which $C_{i}$ comes. Replacing each instance of
$\partial C_{i}$ with this decomposition for each $1\leq i\leq l$ in the
expression above and again distributing, rewriting, and discarding empty
intersections, we obtain that $\partial C$ is contained in a union of cells of
$\mathcal{C}_{\delta,\varepsilon}$ all of which have dimension lower than that
of $C$. Hence $\mathcal{C}_{\delta,\varepsilon}$ gives the desired CW
decomposition of $\lvert\Lambda\rvert$.

To show symmetry, let $C\in\mathcal{C}_{\delta,\varepsilon}$ with $C\subset K$
for a simplex $K$ in $\widehat{\Lambda}$. Then $C$ is the set of points of the
form $\sum t_{c_{i}}c_{i}$, with the $c_{i}$ being the vertices of $K$, such
that the coefficients $t_{c_{i}}$ satisfy certain conditions. The linearity of
the action of $G$ implies that $g(C)$ then consists of points $\sum
t_{c_{i}}g(c_{i})$ such that the coefficients $t_{c_{i}}$ satisfy those same
conditions. From our definition of $\mathcal{C}_{\delta,\varepsilon}$, this
means $C$ is a cell contained in
$\mathcal{C}_{g(K),\delta,\varepsilon}\subset\mathcal{C}_{\delta,\varepsilon}$,
as desired. ∎

Say $V=V(\varepsilon_{0},\delta_{0},\ldots,\varepsilon_{m},\delta_{m})$, and
let $\delta=\min\\{\delta_{0},\ldots,\delta_{m}\\}$ and
$\varepsilon=\max\\{\varepsilon_{0},\ldots,\varepsilon_{m}\\}$.

######  Lemma 6.13.

For each $B$ a simplex of $\widehat{S}$, $V_{B}$ can be written as a union of
cells of $\mathcal{C}_{\delta,\varepsilon}$, and hence so can $V$.
Furthermore, the set $\\{C\in\mathcal{C}_{\delta,\varepsilon}\mid C\subset
V\\}$ is symmetric under the action of $G$.

###### Proof.

First, observe that if a simplex $K$ has vertex set
$\\{c_{0},\ldots,c_{q}\\}$, then the simplices of the barycentric subdivision
of $\overline{K}$ belonging to $K$ correspond to the subsets of $\overline{K}$
given by

$0<t_{c_{i_{1}}}=\cdots=t_{c_{i_{\lambda_{1}}}}<t_{c_{i_{\lambda_{1}+1}}}=\cdots=t_{c_{i_{\lambda_{2}}}}<\\\
\ldots<t_{c_{i_{\lambda_{l}}}}=\cdots=t_{c_{i_{q}}}<1$

for various $1\leq\lambda_{1}\leq\cdots\leq\lambda_{l}\leq q$ and permutations
$i_{1},\ldots,i_{q}$ of $1,\ldots,q$. Using this observation to translate the
condition “$t_{b_{\nu}}>t_{c_{\nu}}$ for all $b_{\nu}\in I$ and
$c_{\mu}\in(J\setminus I)$” from the definition of
$K_{B}(\delta,\varepsilon)$, we see we have defined
$\mathcal{C}_{\delta,\varepsilon}$ in such a way that for each pair $K$ and
$B$ with $K$ a simplex of $\widehat{\Lambda}$, $B$ a simplex of $\widehat{S}$,
and $B\subset\overline{K}$, each cell of $\mathcal{C}_{\delta,\varepsilon}$ is
either contained in or disjoint from the set $K_{B}(\delta,\varepsilon)$
(compare Notation 6.10 and Definition 6.4). However, by Lemma 6.7, we have
that
$K_{B}(\delta,\varepsilon)=K_{B}(\delta_{0},\varepsilon_{0})\cup\cdots\cup
K_{B}(\delta_{m},\varepsilon_{m})$. Now, for $B$ any simplex contained in
$\widehat{S}$, we have that $V_{B}$ is the union of sets
$K_{B^{\prime}}(\delta,\varepsilon)$ for $K$ in $\widehat{\Lambda}$ with
$B\subset\overline{K}$ and $B^{\prime}\in S_{B}$, and hence is a union of
cells of $\mathcal{C}_{\delta,\varepsilon}$.

Let $C\in\mathcal{C}_{\delta,\varepsilon}$ with $C\subset K$ for $K$ a simplex
in $\widehat{\Lambda}$. If $C\subset K_{B}(\delta,\varepsilon)$ for some
appropriate $K$ and $B$, then $g(C)\subset g(K)_{g(B)}(\delta,\varepsilon)$.
Hence the set of cells of $\mathcal{C}_{\delta,\varepsilon}$ contained in $V$
is symmetric under the action of $G$. ∎

Now, we may begin assembling our equivariant map $\tau:V\rightarrow S$.

######  Lemma 6.14.

Let $\mathcal{N}_{V}$ denote the nerve of the covering of $V$ by the family
$\\{V_{B}\\}_{B\in\widehat{S}}$ (see Definition 6.27). Then for $m\geq 1$ and
$0<\varepsilon_{0}\ll\delta_{0}\ll\cdots\ll\varepsilon_{m}\ll\delta_{m}\ll 1$,
there exists an equivariant map $\psi_{V}:V\rightarrow\mathcal{N}_{V}$ such
that the induced homomorphism
$(\psi_{V})_{\\#,k}:\pi_{k}(V,*)\rightarrow\pi_{k}(\lvert\mathcal{N}_{V}\rvert,\psi_{V}(*))$
is an isomorphism for $k\leq m-1$ and an epimorphism for $k=m$.

###### Proof.

By Lemma 6.12, $\lvert\widehat{\Lambda}\rvert$ together with the collection
$\mathcal{C}$ is a symmetric regular CW complex whose cells are all convex
polyhedra. Each set in $\\{V_{B}\\}_{B\in\widehat{S}}$ is open in