# Topological Posets and Tropical Phased Matroids

Ulysses Alvarez and Ross Geoghegan  Ulysses Alvarez, Department of
Mathematical Sciences, Binghamton University (SUNY), Binghamton, NY
13902-6000, USA Ross Geoghegan, Department of Mathematical Sciences,
Binghamton University (SUNY), Binghamton, NY 13902-6000, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: June 5, 2021)

###### Abstract.

For a discrete poset $\mathcal{X}$, McCord proved that the natural map
$|{\mathcal{X}}|\to{\mathcal{X}}$, from the order complex to the poset with
the Up topology, is a weak homotopy equivalence. Much later, Z̆ivaljević
defined the notion of order complex for a topological poset. For a large class
of such topological posets we prove the analog of McCord’s theorem, namely
that the natural map from the order complex to the topological poset with the
Up topology is a weak homotopy equivalence. A familiar topological example is
the Grassmann poset of proper non-zero linear subspaces of $\mathbb{R}^{n+1}$
partially ordered by inclusion. But our motivation in topological
combinatorics is to apply the theorem to posets associated with tropical
phased matroids over the tropical phase hyperfield.

##  1\. Introduction

###  1.1. Topological Posets

A topological poset $({\mathcal{X}},\leq,{\mathcal{T}})$ is a poset
$({\mathcal{X}},\leq)$ equipped with a Hausdorff topology $\mathcal{T}$ on the
set ${\mathcal{X}}$ such that the order relation
$P\coloneqq\\{(x,y)\in{\mathcal{X}}\times{\mathcal{X}}\mid x\leq y\\}$ is a
closed subspace of ${\mathcal{X}}\times{\mathcal{X}}$. This topological poset
is said to be mirrored, and is called an $M$-poset, if it comes equipped with
a poset map $\mu\colon{\mathcal{X}}\to{\mathcal{R}}$, where ${\mathcal{R}}$ is
a finite poset, satisfying

  * •

$x<y$ implies $\mu(x)<\mu(y)$,

  * •

for all $r\in{\mathcal{R}}$, $X_{r}:=\mu^{-1}(r)$ is a non-empty closed subset
of ${\mathcal{X}}$.

The map $\mu$ is a mirror of $\mathcal{X}$. It follows that ${\mathcal{X}}$ is
the topological sum $\coprod_{r\in{\mathcal{R}}}X_{r}$. An example is the real
Grassmann Poset ${\mathcal{G}}_{n}$; here, ${\mathcal{X}}$ is the set of all
proper linear subspaces of ${\mathbb{R}}^{n+1}$ of positive dimension,
partially ordered by inclusion, ${\mathcal{R}}$ is the set $[n]$ of positive
integers $\leq n$ with the natural ordering, and $\mu$ takes the
$k$-dimensional subspaces to the integer $k$.

######  Notation.

For a subset $A$ of ${\mathcal{X}}$ define

${\operatorname{\uparrow}A}\coloneqq\\{x\in{\mathcal{X}}\mid a\leq x\text{ for
some }a\in A\\}.$

If ${\mathcal{A}}$ is a family of subsets,
$\operatorname{\uparrow}{\mathcal{A}}\coloneqq\\{\operatorname{\uparrow}A\mid
A\in\mathcal{A}\\}$. For $a\in A,$ we use the abbreviation
$\operatorname{\uparrow}a\coloneqq\operatorname{\uparrow}\\{a\\}$. For
$A\subseteq X_{r}$ and $s>r$, we use the abbreviation
$A^{(s)}\coloneqq(\operatorname{\uparrow}A)\bigcap X_{s}$.

Each $X_{r}$ inherits a topology from ${\mathcal{X}}$. We choose a basis
${\mathcal{U}}_{r}$ for the topology of $X_{r}$.

######  Openness Property.

We will assume from now on that our $M$-posets have the property that when $U$
is open in $X_{r}$ then $\operatorname{\uparrow}U$ is open in $\mathcal{X}$;
equivalently, for each $s>r$, the set $U^{(s)}$ is open in $X_{s}$.

The Up topology on ${\mathcal{X}}$ has basis
$\bigcup_{r}\\{\operatorname{\uparrow}{\mathcal{U}}_{r}\mid
r\in{\mathcal{R}}\\}$. The Openness Property implies that this is a basis for
a topology. It is important to distinguish between the Original topology on
${\mathcal{X}}$ (which is Hausdorff) and the Up topology on ${\mathcal{X}}$
(which is $T_{0}$ but, in general, not $T_{1}$.) The $M$-posets discussed in
this paper are considered to have both topologies, the “Up” being derived from
the “Original”. The Openness Assumption ensures that the Up topology agrees
with the ordinary topology on each $X_{r}$; i.e. the inclusion of $X_{r}$ into
$\mathcal{X}$ with the Up topology is a topological embedding.

######  Example.

Let ${\mathcal{X}}$ be the real Grassmann Poset ${\mathcal{G}}_{3}$ Here
$X_{1}$ is the collection of lines in ${\mathbb{R}}^{3}$ containing $0$ and
$X_{2}$ is the collection of planes in ${\mathbb{R}}^{3}$ containing $0$,
partially ordered by inclusion. Separately, the $X_{r}$ have their Original
(usual) topologies. In the Up topology a basic open neighborhood of a line
consists of an open set of nearby lines together with all planes containing
those lines. A basic open neighborhood of a plane in the Up topology is just a
neighborhood of that plane in the ordinary sense.

###  1.2. The Order Complex and the Comparison Map.

In [Ž98], a simplicial space $\Delta({\mathcal{X}})$, called the order
complex111Strictly speaking, this is not a complex, but rather the geometric
realization of a simplicial topological space., is associated with each
topological poset ${\mathcal{X}}$. For $M$-posets where the spaces $X_{r}$ are
locally compact polyhedra (as will always be the case in this paper) there is
a rather simple definition as follows: Each element $z$ of the topological
join ${\Asterisk_{r\in{\mathcal{R}}}}X_{r}$ has the form $z={\sum}t_{i}x_{i}$
where $x_{i}\in X_{i},\\\ t_{i}\geq 0,\text{ and }\sum t_{i}=1$. Define the
support of $z$ to be $\text{supp}(z)\coloneqq\\{i\in{\mathcal{R}}\mid
t_{i}>0\\}$. Then

$\Delta({\mathcal{X}})\coloneqq\\{z\in{\Asterisk_{r\in{\mathcal{R}}}}X_{r}\mid\text{supp}(z)\text{
is a chain in }{\mathcal{R}},\text{ and if }i<j\text{ in supp}(z)\text{ then
}x_{i}<x_{j}\text{ in }{\mathcal{X}}\\}.$

We will sometimes identify $X_{r}$ with the subspace of
$\Delta({\mathcal{X}})$ whose support is the singleton $\\{r\\}$.

######  Example.

Vassiliev [Vas92], [Vas91] proved that for $\mathbb{K}$ the reals, the complex
numbers, or the quaternions, and ${\mathcal{G}}_{n}(\mathbb{K})$ the Grassmann
Poset of proper non-zero linear subspaces of $K^{n+1}$, the corresponding
order complex, $\Delta({\mathcal{G}}_{n})$, is homeomorphic to the sphere
$S^{m}$ where $m={n+1\choose 2}d+n-1$, $d$ being the dimension of $K$ over
$\mathbb{R}$.

The Comparison Map $f\colon\Delta({\mathcal{X}})\to{\mathcal{X}}$ is defined
by $f(z)=\text{max}\\{x_{i}\mid i\in\text{ supp}(z)\\}$. It is not hard to
prove that $f$ is continuous222The word “map” is used in this paper to mean
that the given function is continuous. with respect to the Up topology on
${\mathcal{X}}$. Our aim is to prove that under reasonable topological and
geometric assumptions, and considering $\mathcal{X}$ with the Up topology, the
Comparison Map $f$ is a weak homotopy equivalence. This will be done by
satisfying the hypotheses of the following theorem of McCord [McC67], reproved
independently333May’s paper appears to date from the late 1970’s though the
publication year is 2006. by May [May06].

######  Theorem 1.1.

Let $g\colon C\to D$ be a map between topological spaces, and let
${\mathcal{B}}$ be a basis for the open subsets of $D$. If for all
$B\in{\mathcal{B}}$ the restriction $g|\colon g^{-1}(B)\to B$ is a weak
homotopy equivalence, then $g$ is a weak homotopy equivalence.

###  1.3. Polyhedral $M$-posets

An $M$-poset ${\mathcal{X}}$ is polyhedral if each $X_{i}$ is a locally
compact polyhedron444In Section 3.1 we go into detail about our use of
polyhedral language., and $\mathcal{X}$ has the Polyhedral Diagonal Property,
namely: the order relation $P$ is a (closed) subpolyhedron of
${\mathcal{X}}\times{\mathcal{X}}$; here $\mathcal{X}$ has the ordinary
topology.

######  Remark 1.2.

Checking this property is usually done as follows. There is an $M$-poset
$\mathcal{X}$ which obviously has the property, and one is interested in a
subset $\mathcal{Y}$ such that each $Y_{i}$ is a closed polyhedral subset of
$X_{i}$. Then the order relation for $\mathcal{Y}$ is the polyhedron
$P\bigcap{\mathcal{Y}}\times{\mathcal{Y}}$, so $\mathcal{Y}$ is a polyhedral
$M$-poset.

###  1.4. Convenient Choice of Bases

When the space $X_{r}$ is polyhedral each point has a basic system of
neighborhoods which are piecewise linear (PL) cones on their frontiers. We may
assume that the basis ${\mathcal{U}}_{r}$ consists of such neighborhoods of
points. In particular, each such set is contractible and its closure is a
compact contractible subpolyhedron of $X_{r}$. Our basis
${\mathcal{U}}=\bigcup_{r}\\{\operatorname{\uparrow}{\mathcal{U}}_{r}\\}$ for
the topology of $\mathcal{X}$ will always be understood to consist of such
sets.

###  1.5. Geometric Posets and Main Theorem

A compact subset $C$ of a polyhedron $D$ has trivial shape if $C$ can be
contracted to a point in any of its neighborhoods555This is an intrinsic
property of $C$, independent of $D$. Note that when $C$ is contractible then
$C$ has trivial shape, but trivial shape is more general: for example, the
Topologist’s Sine Curve in ${\mathbb{R}}^{2}$ has trivial shape but has two
path components. in $D$.

######  Notation.

When $(M,d)$ is a metric space, the space of non-empty compact subsets of $M$
with the Hausdorff metric is denoted by $cM$.

A polyhedral $M$-poset $({\mathcal{X}},\leq,{\mathcal{T}})$ is geometric if it
has the following Additional Properties:

  1. A1

: When $r<s$ and $x\in X_{r}$, the set $x^{(s)}$ is compact and non-empty.

  2. A2

: When $r<s$, the map $\pi_{r,s}\colon X_{r}\to cX_{s}$ defined by $x\mapsto
x^{(s)}$ is continuous.

  3. A3

: When $U\in{\mathcal{U}}_{r}$, $s>r$, and $y\in\overline{U}^{(s)}$, then
$\\{x\in{\overline{U}}\mid x<y\\}$ has trivial shape666This implies that every
$y$ has a predecessor $x$..

Our goal in this paper is to prove the following theorem:

######  Theorem 1.3.

(Comparison Theorem) When the $M$-poset ${\mathcal{X}}$ is geometric and
carries the Up topology, the map
$f\colon\Delta({\mathcal{X}})\to{\mathcal{X}}$ is a weak homotopy equivalence.

Our approach will be to prove that for $U\in{\mathcal{U}}_{r}$ both
$\operatorname{\uparrow}U$ and $f^{-1}(\operatorname{\uparrow}U)$ are weakly
contractible777i.e. all homotopy groups are trivial., and then to apply
Theorem 1.1.

######  Remark.

The discrete case of this theorem is due to McCord [McC66]. See Section 8 for
a discussion.

######  Example.

The Grassmann Posets ${\mathcal{G}}_{n}(\mathbb{K})$ are geometric, so Theorem
1.3 and Vassiliev’s result gives us the singular homology of
${\mathcal{G}}_{n}(\mathbb{K})$ with the Up topology.

##  2\. Connection with Matroids

While this paper is about topological $M$-posets in general, there is a
particular class of such posets which motivates us — a class associated with
tropical phased matroids888This section can be skipped by readers only
interested in the proof of Theorem 1.3..

###  2.1. Tropical phase hyperfield

Let $T\Phi$ denote the poset $\\{0\\}\cup S^{1}\subseteq\mathbb{C}$, where the
partial ordering is fully defined by $0<x$ for all $x\in S^{1}$. This is an
$M$-poset over $\\{0,1\\}$, where the mirror $\mu$ takes $0$ to $0$ and all of
$S^{1}$ to $1$. The Openness Property holds, and with the Up topology this is
easily seen to be a geometric poset. $T\Phi$ also carries the structure of a
hyperfield. A hyperfield is similar in definition to a field (associative,
commutative etc.) except that the addition operation is allowed to be
multivalued. In the case of $T\Phi$, multiplication is inherited from the
complex plane and the sum $\boxplus$ is defined by

  * •

$x\boxplus 0=\\{x\\}$,

  * •

$x\boxplus-x=\\{0\\}\cup S^{1}$ whenever $x\neq 0$,

  * •

$x\boxplus y$ is the smallest closed arc in $S^{1}$ joining the points $x$ and
$y$ when $y\neq-x$,

  * •

For subsets $A$ and $B$, $A\boxplus B=\bigcup_{a\in A,b\in B}a\boxplus b$.

$T\Phi$ is the tropical phase hyperfield introduced by Viro in [Vir10]999Not
to be confused with the phase hyperfield where $x\boxplus-x=\\{0,x,-x\\}$ and
“closed arc” is replaced by“open arc”..

######  Remark.

The hyperfield $T\Phi$ is the complex analog of the “sign hyperfield”
${\mathbb{S}}:=T\Phi\bigcap{\mathbb{R}}$. Just as $T\Phi$ will lead us to a
discussion of tropical phased matroids, $\mathbb{S}$ would lead to an
analogous discussion of oriented matroids. However, unlike $T\Phi$, the
original topology on $\mathbb{S}$ is discrete, so the analog of the
topological posets discussed here would be discrete posets, and everything
would be much simpler.

###  2.2. Relevant $M$-posets

The product $T\Phi^{n}$ is an $M$-poset, where the comparison $<$ is defined
coordinate-wise. This has a minimal element $\mathbf{0}$ and is therefore
contractible. More interesting is the subset $T\Phi^{n}-\\{\mathbf{0}\\}$. The
mirror map $\mu:T\Phi^{n}-\\{\mathbf{0}\\}\to[n]$ is defined by
$\mu(\mathbf{v})=|\text{supp}(\mathbf{v})|$. This is given the Up topology. It
is polyhedral because the pre-image of each $i$ is homeomorphic to a disjoint
union of tori. That it has the Polyhedral Diagonal Property is then clear.

######  Proposition 2.1.

This polyhedral $M$-poset is geometric.

###### Proof.

The Additional Properties A1 and A2 are clear. With
${\mathcal{X}}:=T\Phi^{n}-\\{\mathbf{0}\\}$, recall that Property A3 refers to
our convenient choice of basis. In $T\Phi$ this consists of the interiors of
closed intervals in $S^{1}$, together with the set $T\Phi$ itself, and in
$T\Phi^{n}-\\{\mathbf{0}\\}$ the basis consists of products of those sets. Let
$i<j$ and let $U$ be a basic open set in $X_{i}$. If
${\mathbf{y}}\in{\overline{U}}^{j}$, the set
$\\{{\mathbf{x}}\in{\overline{U}}\mid{\mathbf{x}}<{\mathbf{y}}\\}$ is a
singleton, obtainable by selecting $i$ entries in the vector ${\mathbf{y}}$. ∎

For an $n$-vector $\mathbf{v}=(v_{1},\ldots,v_{n})\in
T\Phi^{n}-\\{\mathbf{0}\\}$, we define

$\mathbf{v}^{\perp}\coloneq\\{(x_{1},\ldots,x_{n})\in
T\Phi^{n}-\\{\mathbf{0}\\}\mid 0\in v_{1}x_{1}\boxplus\ldots\boxplus
v_{n}x_{n}\\}.$

More generally, given $m$ such $n$-vectors $\mathbf{v}_{i}\in
T\Phi^{n}-\\{\mathbf{0}\\}$, define

$\\{\mathbf{v}_{1},...,\mathbf{v}_{m}\\}^{\perp}:=\bigcap_{i}\mathbf{v}_{i}^{\perp}.$

Since $\\{\mathbf{v}_{1},...,\mathbf{v}_{m}\\}^{\perp}$ is a subset of the
$M$-poset $T\Phi^{n}-\\{\mathbf{0}\\}$, it is also an $M$-poset.

######  Theorem 2.2.

The $M$-poset, $\\{\mathbf{v}_{1},...,\mathbf{v}_{m}\\}^{\perp}$ is geometric.

###### Proof.

We first prove polyhedrality. It is enough to prove it for
$\mathbf{v}^{\perp}$. We work by induction on $n$. This is trivial when $n=1$
and is assumed true for vectors of length $<n$. We may assume each
$v_{i}x_{i}$ is non-zero, and that the $v_{i}x_{i}$ are distinct; otherwise
the claim would be covered by the induction hypothesis. Then $0\in
v_{1}x_{1}\boxplus\ldots\boxplus v_{n}x_{n}$ if and only if either (i) some
entry is the negative of another, or (ii) no open semicircle contains all of
the entries. The set of $n$-tuples $(v_{1}x_{1},\ldots,v_{n}x_{n})$ satisfying
(i) is a union of $\frac{n(n-1)}{2}$ tori of dimension $n-1$. The set
satisfying (ii) is also easily shown to be polyhedral. By Remark 1.2 it is a
polyhedral $M$ poset. The Additional Properties A1 through A3 hold for the
same reasons as in the proof of Proposition 2.1. ∎

######  Remark.

In [Alv], the first-named author explores the conditions under which the order
complex $\Delta(\\{{\mathbf{v}_{1}},\dots,{\mathbf{v}_{m}}\\}^{\perp})$ is
homeomorphic to the sphere $S^{2(n-m)-1}$. When this holds, Theorem 1.3
implies that, the singular homology of
$\\{\mathbf{v}_{1},...,\mathbf{v}_{m}\\}^{\perp}$ with the Up topology, is
that of $S^{2(n-m)-1}$.

In the case of just one vector ${\mathbf{v}}$, there is a unique rank $(n-1)$
“tropical phased matroid” whose $T\Phi$ circuit set is101010Here and below,
$T\Phi-\\{0\\}$ is usefully shortened to $S^{1}$. $S^{1}{\mathbf{v}}$ and
whose covector set is ${\mathbf{v}}^{\perp}$. Similarly, a unique tropical
phased matroid is defined by $\\{\mathbf{v}_{1},\ldots,\mathbf{v}_{m}\\}$ when
$m>1$, but then $S^{1}\mathbf{v}_{1}\bigcup\ldots\bigcup S^{1}\mathbf{v}_{m}$
must satisfy certain“circuit conditions”. The rank in that case may not be
$(n-1)$. See [BB].

###  2.3. Strong matroids over hyperfields

The only hyperfield occurring in this paper is $T\Phi$. The general definition
of a hyperfield can be found, for example, in [BB] or [BB19]. To simplify
notation in this subsection it is convenient to state things in terms of an
arbitrary hyperfield $F$, keeping in mind that, for us, $F$ will always be
$T\Phi$. Given a non-empty finite set $E$, a hyperfield $F$, and a positive
integer $r$, a strong Grassmann-Plücker function of rank $r$ on $E$ with
coefficients in $F$ is a function $\varphi:E^{r}\to F$ such that

  1. (1)

$\varphi$ is not identically zero;

  2. (2)

$\varphi$ is alternating, i.e. $\varphi(x_{1},\dots,{x}_{i},\dots,x_{j},\dots
x_{r})=-\varphi(x_{1},\dots,{x}_{j},\dots,x_{i},\dots x_{r})$;

  3. (3)

$\varphi$ satisfies the strong Grassman-Plücker relations: When
$\\{x_{1},\dots,x_{r+1}\\}\subseteq E$ and $\\{y_{1},\dots,y_{r-1}\\}\subseteq
E$ then

$0\in\boxplus_{k=1}^{r+1}(-1)^{k}\varphi(x_{1},\dots,{\hat{x}}_{k},\dots
x_{r+1}).\varphi(x_{k},y_{1},\dots,y_{r-1}).$

Two such functions $\varphi_{1}$ and $\varphi_{2}$ are equivalent if
$\varphi_{1}=t.\varphi_{2}$ where $t\in F$, i.e. one is a scalar multiple of
the other. Strong $F$-matroids are defined in [BB]. For our purposes we will
instead use the following Theorem 3.13 of [BB] as our definition of “weak
$F$-matroid”.

######  Theorem 2.3.

There is a natural bijection between the set of equivalence classes of strong
Grassman-Plücker functions of rank $r$ on $E$ with coefficients in $F$ and
strong $F$-matroids of rank $r$ on $E$.

###  2.4. Srong tropical phased Grassmannians

We apply the previous subsection with $E=[n]$ and $F=T\Phi$. A map
$\varphi:E^{r}\to T\Phi$ assigns an element $v(i_{1},\ldots.i_{r})\in T\Phi$
to each $r$-tuple $(i_{1},\ldots.i_{r})$ of elements of $[n]$. So $\varphi$ is
encoded by an element of $T\Phi^{n^{r}}$. The space of all maps
$\varphi:E^{r}\to T\Phi$ is thus homeomorphic to the product space
$T\Phi^{n^{r}}$. We wish to restrict attention to
$T\Phi^{n^{r}}-{\mathbf{0}}$, the space of non-zero $r$-tuples. This is an
$M$-poset over $[r]$ whose mirror map takes each $r$-tuple to the size of its
support. As before (compare Proposition 2.1) the poset
$T\Phi^{n^{r}}-\\{\mathbf{0}\\}$ with the Up topology is geometric. The
quotient space $(T\Phi^{n^{r}}-\\{\mathbf{0}\\})/S^{1}$ can be identified with
the subspace of $T\Phi^{n^{r}}-\\{\mathbf{0}\\}$ consisting of normalized
$r$-tuples, where the $r$th entry is either $1$ or $0$. This is a geometric
$M$-poset whose mirror map is denoted by
$\mu:T\Phi^{n^{r}}-\\{\mathbf{0}\\}\to[r-1]$. Each $\mu^{-1}(i)$ is a disjoint
union of tori. The strong Grassmannian, $\text{Gr}_{s}(r,T\Phi^{n})$, is
defined in the literature111111See [AD19] for more details. to be the
topological poset of all strong tropical phased matroids of rank $r$ on the
set $[n]$ with coefficients in $T\Phi$. In view of the previous paragraph,
Theorem 2.3 identifies this set with the subset of
$T\Phi^{n^{r}}-\\{\mathbf{0}\\}$ consisting of normalized non-zero anti-
symmetric functions $\varphi$ satisfying the strong Grassman-Plücker
relations. The set of $r$-tuples of elements of $[n]$ is $[n]^{r}$. A
transposition of an $r$-tuple is an $r$-tuple obtained by interchanging two
entries. In particular, every $r$-tuple containing the same element in two
different positions is a transposition of itself. We need a combinatorial
lemma:

######  Lemma 2.4.

There is a subset $T$ of $[n]^{r}$ with the following properties.

  * •

No $r$-tuple having two equal entries is in $T$;

  * •

Every $r$-tuple whose entries are all different is obtainable from an element
of $T$ by a transposition;

  * •

No transposition of an element of $T$ is in $T$.

###### Proof.

Define $S_{1}:=\\{s\in[n]^{r}\mid\text{ all entries in }s\text{ are
different}\\}$. Choose a total ordering of $S_{1}$. Let $m_{1}$ be the least
element of $S_{1}$. Define

$S_{2}:=\\{s\in S_{1}\mid s>m_{1}\text{ and }s\text{ is not obtainable from
}m_{1}\text{ by a transposition}\\}.$

If $S_{2}$ is non-empty let $m_{2}$ denote its minimal element; in particular,
$m_{2}>m_{1}$. Let

$S_{3}:=\\{s\in S_{2}\mid s>m_{2}\text{ and }s\text{ is not obtainable from
}m_{2}\text{ by a transposition}\\}.$

Proceed in this way until reaching $d$ such that $S_{d+1}$ is empty. The
resulting set $T:=\\{m_{1},\ldots,m_{d}\\}\subseteq[n]^{r}$ has the required
properties, and $|T|=d$. ∎

######  Theorem 2.5.

As a topological poset $\text{Gr}_{s}(r,T\Phi^{n})$ is geometric.

###### Proof.

It follows from Lemma 2.4 that the space of alternating functions $\varphi$ is
homeomorphic to the product space $T\Phi^{d}$. This is because once $d$ such
functions are chosen using the Lemma then the others are determined. Thus the
space of non-zero antisymmetric functions $[n]^{r}\to T\Phi$ is polyhedral.
The subspace consisting of those functions which satisfy the strong Grassman-
Plücker relations is therefore polyhedral for the reasons explained in the
proof of Theorem 2.2; see in particular Remark 1.2. The Additional Properties
A1 through A3 hold for the same reasons as in the proof of Proposition 2.1. ∎

The ordinary complex Grassmannian $\text{Gr}(r,{\mathbb{C}}^{n})$ is the space
of $r$-dimensional linear subspaces of ${\mathbb{C}}^{n}$; here $1\leq r\leq
n-1$. But it can also be viewed in the above terms, since every field is a
hyperfield (in which case $a+b=\\{a+b\\}$). Thus it can be identified with the
space of normalized strong Grassmann-Plücker functions of rank $r$ on
$[n]^{r}$ with coefficients in $\mathbb{C}$. The canonical map
${\mathbb{C}}\to T\Phi$ sending non-zero $z$ to $\frac{z}{|z|}$ and $0$ to $0$
is continuous with respect to the Up topology. It induces the map
$\rho_{s}:\text{Gr}(r,{\mathbb{C}}^{n})\to\text{Gr}_{s}(r,T\Phi^{n})$, which
is also continuous. The following is the tropical phased analog of a well-
known conjecture for oriented matroids: Question: Is the map
$\rho_{s}:\text{\rm Gr}(r,{\mathbb{C}}^{n})\to\text{\rm Gr}_{s}(r,T\Phi^{n})$
a weak homotopy equivalence?

######  Remarks.

  1. (i)

There is a homotopy commutative diagram

$\textstyle{\Delta(\text{Gr}_{s}(r,T\Phi^{n}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\text{Gr}(r,{\mathbb{C}}^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\Delta(\rho)}$$\textstyle{\text{Gr}_{s}(r,T\Phi^{n})}$

and, in view of Theorem 1.3, the map $\rho$ is a weak homotopy equivalence if
and only if $\Delta(\rho)$ is.

  2. (ii)

The analogous “MacPhersonian Conjecture” for oriented matroids (still open
having been incorrectly dealt with in [Bis03]) says that the analogous “real”
map
$\Delta(\rho):\text{Gr}(r,{\mathbb{R}}^{n})\to\Delta(\text{Gr}(r,{\mathbb{S}}^{n}))$
is a weak homtopy equivalence. Since $\text{Gr}(r,{\mathbb{S}}^{n}))$ is a
finite discrete $M$-poset, the relevant analog of Theorem 1.3 was proved long
ago by McCord (see Section 8), and it was already pointed out in [AD19] that
the MacPhersonian Conjecture is the same whether worded for $\rho$ or for
$\Delta(\rho)$.

Summary: The point of this section is to bring out how interest in Theorem 1.3
has arisen in connection with matroids. Theorems 2.2 and 2.5 are instances
where one can travel back and forth between the order complex and the actual
topological poset equipped with the Up topology. The rest of this paper is
concerned with the proof of Theorem 1.3.

##  3\. Preliminaries

###  3.1. Polyhedra.

This is for reference. We need to be precise about our polyhedral terminology.
A rectilinear simplicial complex consists of a locally finite, finite-
dimensional, abstract simplicial complex $K$, whose geometric realization is
denoted by $|K|$, and a closed proper topological embedding of $|K|$ in a
euclidean space $E$ which restricts to an affine isometric embedding of each
simplex of $|K|$. The choice of $E$ and of the embedding will be suppressed,
and from now on we will denote its image simply by $K$, omitting the vertical
bars. Each simplex of $K$ has a euclidean metric. The “length metric” on $K$
is the metric which measures the distance between points $p$ and $q$ as the
inf of lengths of piecewise linear paths joining $p$ to $q$, where each linear
part of the path lies in (and is measured in) a simplex of $K$. The
corresponding metric topology on $K$ is the same as both the weak topology and
the topology inherited from the euclidean space $E$. A locally compact space
$A$ is a polyhedron if it is equipped with a piecewise linear (PL) structure.
This means that the space $A$ has been identified, via a homeomorphism chosen
once and for all, with a rectilinear simplicial complex $K$, and the PL
structure consists of all rectilinear simplicial complexes $L$ occupying the
same space as $K$, such that $L$ and $K$ have a common rectilinear simplicial
subdivision121212References for polyhedra are [RS72], [Hud69] and Section 3.1
of [Spa95]. Here, we use nothing of that subject beyond basic definitions and
elementary properties.. We say that any such complex $L$ triangulates $A$. A
subpolyhedron of $A$ is a closed subset $B$ such that some triangulation of
$A$ in the PL structure has a subcomplex which triangulates $B$. We will often
describe a space $A$ as polyhedral when we mean that $A$ is a polyhedron in
the above sense; and when $B$ is a subpolyhedron of $A$ we simply say that $B$
is “polyhedral”.

As an extension of this, we may consider an open subset $V$ of the polyhedron
$A$. It is well-known that $V$ has a polyhedral structure compatible with that
of $A$; this means: $V$ can be triangulated in such a way that each finite
subcomplex is a subpolyhedron of $A$, and the diameters of simplexes approach
zero as the simplexes approach the frontier of $V$.

The polyhedron $A$ has many length metrics, one for each triangulation $L$. In
particular, small metric balls in polyhedra are cones on their frontiers. Once
a length metric is chosen then, for small enough $\eta>0$, two maps from a
compact domain into $A$ which are distant at most $\eta$ apart pointwise are
$\eta$-homotopic131313Maps $a,b\colon A\to B$ are $\eta$-homotopic if there is
a homotopy between $a$ and $b$ such that for each point $x\in A$ the diameter
of the image of $x\times I$ is $<\eta$..

######  Remarks.

  1. 1.

The class of polyhedra is closed under coproducts and finite products.

  2. 2.

If $B$ is a compact polyhedral subset of a finite product $A$ of polyhedra,
then projections of $B$ to sub-products of $A$ are also compact polyhedral.

  3. 3.

The join of two compact polyhedra is a compact polyhedron.

  4. 4.

Smooth manifolds are homeomorphic to polyhedra.

  5. 5.

The topological join of two locally compact spaces need not be locally
compact, so if they are polyhedra, their join does not satisfy our definition
of “polyhedron”. However, the join of two abstract simplicial complexes has an
obvious meaning even when they are not locally finite. Therefore the
topological join of two (hence of finitely many) polyhedra can be triangulated
by a simplicial complex which might not be locally finite. Occasionally, we
will allow ourselves to extend the word “polyhedral” to include this case.

###  3.2. Some Consequences of the Additional Properties

######  Proposition 3.1.

When $r<s$ and $A$ is a compact subset of $X_{r}$ then $A^{(s)}$ is a compact
subset of $X_{s}$.

For this we need notation and a lemma.

######  Notation.

When $(M,d)$ is a metric space and $S\subseteq cM$, we define
$\bigcup(S)\coloneqq\\{m\in M\mid m\in s\text{ for some }s\in S\\}$, i.e.
$\bigcup(S)$ is the union of the sets in $S$.

######  Lemma 3.2.

Let $C$ be a compact subset of $c{\mathbb{R}}^{n}$. Then $\bigcup(C)$ is a
compact subset of $\mathbb{R}^{n}$.

###### Proof.

$C$ is totally bounded in the Hausdorff metric, so $\bigcup(C)$ is a bounded
subset of ${\mathbb{R}}^{n}$. If $x$ is a limit point of $\bigcup(C)$ there
exists $(x_{n})$, a sequence in $\bigcup(C)$, converging to $x$. Let $x_{n}\in
c_{n}$ where $c_{n}\in C$. Without loss of generality, assume $\\{c_{n}\\}$
converges to $c\in C$ in the Hausdorff metric. So $x$ must lie in $c$. Thus
$\bigcup(C)$ is a closed and bounded subset of ${\mathbb{R}}^{n}$. ∎

###### Proof of Proposition 3.1.

By the definition of a geometric poset the set $\pi_{r,s}(A)$ is a compact
subset of $cX_{s}$. The polyhedron $X_{s}$ can be regarded as a subset of a
euclidean space, so Lemma 3.2 implies $A^{(s)}=\bigcup(\pi_{r,s}(A))$ is
compact. ∎

######  Proposition 3.3.

Let $A$ be a compact subset of $X_{r}$ and let $r\leq s$. If $A$ is polyhedral
then $A^{(s)}$ is polyhedral.

###### Proof.

Define $B\coloneqq(A\times X_{s})\bigcap P$. Then $B=\\{(a,x)\mid
a<x\\}\subseteq A\times A^{(s)}$ is polyhedral. The $X_{s}$ projection of $B$
is $\\{x\mid a<x\text{ for some }a\in A\\}$. This is $A^{(s)}$, and is compact
by Proposition 3.1. There are compact polyhedra $R\subseteq X_{r}$ and
$S\subseteq X_{s}$ such that $A\times A^{(s)}\subseteq R\times S$. Since $B$
is polyhedral and is a subset of $R\times S$, $B$ is compact polyhedral. Thus
the projection of $B$, namely $A^{(s)}$, is polyhedral. ∎

###  3.3. Tools from geometric topology

Here we state four theorems, well-known in geometric topology but perhaps less
well-known in other mathematical communities. Recall that a space $Z$ is
locally $n$-connected if for each $z\in Z$ and each neighborhood $U$ of $z$
there is a neighborhood $V$ of $z$ such that, for all $k\leq n$, every map
from the $k$-sphere $S^{k}$ into $V$ extends to a map of the $(k+1)$-ball
$B^{k+1}$ into $U$. For us, the important point is that polyhedra are locally
$n$-connected for all $n$.

######  Theorem 3.4.

Let $Z$ be a locally $n$-connected metrizable space. Every open cover
$\mathcal{A}$ of $Z$ has an open refinement $\mathcal{B}$ such that whenever
$K$ is a polyhedron of dimension $\leq n$, any two maps $g_{1},g_{2}:K\to Z$
which agree on the closed subset $C\subseteq K$ and are $\mathcal{B}$-close,
are $\mathcal{A}$-homotopic rel $C$.

###### Proof.

This is Theorem V.5.1 of [Hu65]. The references to $C$ are not in the
statement, but they follow from the proof. ∎

When $K$ is a simplicial complex and $L$ is a subcomplex containing all the
vertices of $K$, a partial realization of $K$ in a space $Z$ relative to a
cover $\mathcal{A}$ of $Z$ is a map $\psi:L\to Z$ such that, for every simplex
$\sigma$ of $K$, there is a set $A\in{\mathcal{A}}$ with $\psi(\sigma\bigcap
L)\subseteq A$. When $L=K$ this is a full realization.

######  Theorem 3.5.

Let $Z$ be a locally $(n-1)$-connected metrizable space, Every open cover
$\mathcal{A}$ of $Z$ has an open refinement $\mathcal{B}$ such that every
partial realization $j_{0}:L\to Z$ relative to $\mathcal{B}$ extends to a full
realization $j:K\to Z$ relative to $\mathcal{A}$.

###### Proof.

This is Theorem V.4.1 of [Hu65]. ∎

Recall that a map is proper if the pre-image of every compact subset of its
codomain is compact. A proper map $q:A\to B$ between locally compact polyhedra
is cell-like if $q^{-1}(b)$ has trivial shape141414See Section 1.5. for every
$b\in B$.

######  Theorem 3.6.

Let $A$ and $B$ be locally compact metric spaces, $K$ a finite-dimensional,
locally finite simplicial complex, $L$ a subcomplex of K, $\theta:A\to B$ a
(proper) cell-like map, $\varphi:K\to B$ a proper map, $\psi:L\to A$ a proper
map such that $\theta\circ\psi=\varphi\mid L$, and $\epsilon:B\to(0,\infty)$ a
function. Then there exists a proper map ${\widebar{\varphi}}:K\to A$ such
that $d(\theta\circ{\widebar{\varphi}}(x),\varphi(x))\leq\epsilon(\varphi(x))$
for every $x\in K$.

###### Proof.

This is Lemma 2.3 of [Lac69]. ∎

######  Theorem 3.7.

A cell-like map between locally compact polyhedra is a proper homotopy
equivalence.

###### Proof.

This is Theorem 1.5 of [Lac69]. ∎

##  4\. The special case ${\mathcal{R}}=[2]$

######  Notation.

In this section, when $A\subseteq X_{1}$ we write $A^{\prime}$ rather than
$A^{(2)}$.

Nearly all the ideas needed for our proof of Theorem 1.3 occur when dealing
with two special cases, ${\mathcal{R}}=[2]$ and ${\mathcal{R}}=[3]$, where
notation is less cluttered. We treat ${\mathcal{R}}=[2]$ here, and
${\mathcal{R}}=[3]$ in the following section. When ${\mathcal{R}}=[2]$, the
geometric $M$-poset is ${\mathcal{X}}=X_{1}\coprod X_{2}$; $\mu(X_{1})=1$ and
$\mu(X_{2})=2$. In this case, the join $X_{1}*X_{2}$ consists of a copy of
$X_{1}$, a copy of $X_{2}$, and for each $x\in X_{1}$ and $y\in X_{2}$ a
linearly parametrized path $\omega$ with $\omega(0)=x$ and $\omega(1)=y$. We
call such a path (or its image) a segment from $x$ to $y$. The point
$\omega(t)$ on a segment from $x$ to $y$ is often more conveniently denoted by
$(1-t)x+ty$ where $0\leq t\leq 1$. The map $q\colon X_{1}\times X_{2}\times
I\to X_{1}*X_{2}$ taking $(x,y,t)$ to $(1-t)x+ty$ defines the quotient
topology on $X_{1}*X_{2}$.

###  4.1. Weak contractibility of basis elements

Basis elements lying in $X_{2}$ are contractible. Here we prove that when $U$
is a (contractible) basis element in $X_{1}$ then $\operatorname{\uparrow}U$
is weakly contractible; equivalently, that for each $n$,
$\pi_{n}(\operatorname{\uparrow}U,U)$ is trivial.

######  Definition.

Let $Z$ be a space, and let $A\subseteq X_{1}$. A map $s^{\prime}\colon Z\to
A^{\prime}$ covers a map $s\colon Z\to A$ if $s(z)<s^{\prime}(z)$ for every
$z\in Z$.

Fundamental to understanding homotopies in this context is the following

######  Proposition 4.1.

If $s^{\prime}$ covers $s$ then, as maps $Z\to\operatorname{\uparrow}A$,
$s^{\prime}$ and $s$ are homotopic.

###### Proof.

Define $F\colon Z\times I\to\operatorname{\uparrow}A$ by

  1. (i)

$F=s^{\prime}$ on $Z\times[0,1)$;

  2. (ii)

$F=s$ on $Z\times\\{1\\}$.

We check continuity of $F$. When $W$ is open in $A^{\prime}$ then
$F^{-1}(W)=s^{\prime-1}(W)\times[0,1)$ which is open in $Z\times I$. When $V$
is open in $A$ then

$\displaystyle F^{-1}(V\cup V^{\prime})$ $\displaystyle=F^{-1}(V)\cup
F^{-1}(V^{\prime})$
$\displaystyle=[s^{-1}(V)\times\\{1\\}]\cup[s^{\prime-1}(V^{\prime})\times[0,1)]$

Since $s^{-1}(V)\subseteq s^{\prime-1}(V^{\prime})$ we conclude that
$F^{-1}(\operatorname{\uparrow}V)$ is open in $Z\times I$. ∎

We abbreviate the $n$-ball $B^{n}$ to $B$, and we consider an arbitrary map

$g\colon(B,\partial B)\to(\operatorname{\uparrow}U,U).$

We write $C\coloneqq g^{-1}U$, a closed subset of $B$ containing $\partial B$.
We want to produce maps $h^{\prime}$ and $h$ from $B$ to
$\operatorname{\uparrow}U$ satisfying:

  1. (i)

$h^{\prime}=h=g$ on $C$;

  2. (ii)

$h^{\prime}$ maps $B-C$ into $U^{\prime}$;

  3. (iii)

$h$ maps $B$ into $U$;

  4. (iv)

on $B-C$, $h^{\prime}$ covers $h$;

  5. (v)

$h^{\prime}$ is homotopic to $g$ rel $C$.

Proposition 4.1 will then imply that $h^{\prime}$ is homotopic to $h$ rel $C$
and hence that $g$ represents the trivial element of
$\pi_{n}(\operatorname{\uparrow}U,U)$. We need some preparatory lemmas.

######  Lemma 4.2.

If $\\{V_{m}\\}_{m=1}^{\infty}$ is a sequence of neighborhoods of $x\in X_{1}$
such that $\bigcap\limits_{m=1}^{\infty}V_{m}=\\{x\\}$, then
$\bigcap\limits_{m=1}^{\infty}V_{m}^{\prime}=x^{\prime}$.

###### Proof.

It is clear that
$x^{\prime}\subseteq\bigcap\limits_{m=1}^{\infty}V_{m}^{\prime}$. To show
containment in the other direction, let
$y\in\bigcap\limits_{m=1}^{\infty}V_{m}^{\prime}$. It follows that for each
$m$, there exists $x_{m}\in V_{m}$ such that $x_{m}<y$ in $\mathcal{X}$. Since
$\bigcap\limits_{m=1}^{\infty}V_{m}=\\{x\\}$, $x_{m}$ converges to $x$ as
$m\to\infty$. Thus, $(x_{m},y)$ converges to $(x,y)$. Since
$\\{(x_{m},y)\\}\subseteq P$ and the order relation $P$ is closed in
$\mathcal{X}^{2}$, $(x,y)\in P$. ∎

######  Lemma 4.3.

Let $x\in{\overline{U}}$. Given $\epsilon>0$, there exists $\eta(x)>0$ such
that whenever151515$B_{\delta}$ denotes the ball of radius $\delta$;
$N_{\epsilon}$ denotes the $\epsilon$-neighborhood. $y$ and $z$ lie in
$B_{\eta}(x)$ then $z^{\prime}\subseteq N_{\epsilon}(y^{\prime})$.

###### Proof.

This is an immediate consequence of Property A2. ∎

######  Lemma 4.4.

Given $\epsilon>0$ there exists $\delta>0$ such that whenever $p\in C$ and
$q\in B_{\delta}(p)-C$ then $g(q)\in N_{\epsilon}(g(p)^{\prime})$.

###### Proof.

We write ${\mathcal{C}}\coloneqq\\{\operatorname{\uparrow}B_{\eta(x)}(x)\mid
x\in{\overline{U}}\\}$, where $\eta(x)$ is as in Lemma 4.3. Then
$g^{-1}({\mathcal{C}})$ covers the compact set $C$. Let $\delta$ be a Lebesgue
number for this cover. For some $x\in{\overline{U}}$,
$g(B_{\delta}(p))\subseteq\operatorname{\uparrow}B_{\eta(x)}(x)$. This means
that for some $y\in B_{\eta(x)}(x)$, $g(q)\in y^{\prime}$. By Lemma 4.3,
$g(q)\in N_{\epsilon}(g(p)^{\prime})$. ∎

It turns out to be easier to work with the closure, ${\overline{U}}$, than
with $U$, because ${\overline{U}}$ is compact. At the end, we will show how to
get the desired result for $U$. Let
$Q:=({\overline{U}}\times{\overline{U}}^{\prime})\bigcap P$. By hypothesis,
$Q$ is a compact polyhedron. The metric on
${\overline{U}}\times{\overline{U}}^{\prime}$ is taken to be the sum of the
separate metrics on $\overline{U}$ and ${\overline{U}}^{\prime}$. Let
${\mathcal{A}}_{i}$ be an open cover of $Q$ by sets of diameter
$<\frac{1}{i}$, and let ${\mathcal{B}}_{i}$ be the refinement given by Theorem
3.5. Let $4\lambda_{i}$ be a Lebesgue number for ${\mathcal{B}}_{i}$.

For each $x\in B-C$ we choose a point $r(x)\in C$ realizing $nd(x,C)$. Define
$k_{0}(x):=gr(x)\in{\overline{U}}$. Since $g$ is continuous, Lemma 4.4 implies
that there exists $\gamma_{i}$, decreasing to $0$ as $i\to\infty$, such that
when $x\in I_{i}:=\\{x\in B-C\mid\gamma_{i+1}\leq d(x,C)\leq\gamma_{i}\\}$
then, for some point $k_{0}^{\prime}(x)$ in the compact non-empty set
$k_{0}(x)^{\prime}\subseteq{\overline{U}}^{\prime}$,
$d(k_{0}^{\prime}(x),g(x))\leq\lambda_{i}$. Then $k_{0}(x)<k_{0}^{\prime}(x)$
in the poset. We write 161616It is not claimed that $j_{0}$ is continuous. We
will only be interested in its restriction to the (discrete) $0$-skeleton of a
chosen triangulation. $j_{0}:=(k_{0},k_{0}^{\prime}):B-C\to Q$.

The map $g$ is uniformly continuous on the compact set $I_{i}$, so, there
exists $\theta_{i}$ such that when $S$ is a subset of $I_{i}$ of diameter
$<\theta_{i}$ then $g(S)$ has diameter $<\lambda_{i}$. We may assume the
sequences $\\{\theta_{i}\\}$ ais decreasing and converges to $0$. Assume that
each $I_{i}$ is a subpolyhedron171717This makes the exposition less cluttered.
If $I_{i}$ is not polyhedral one covers it by a slightly larger polyhedral
set. We omit the details. of $B-C$. Let $K$ be a triangulation of $B-C$ such
that each simplex lies in some $I_{i}$, and simplexes in $I_{i}$ have diameter
$<\theta_{i}$. When a simplex $\sigma$ lies in $I_{i}$, the diameter of
$k_{0}^{\prime}({\sigma}^{0})$ is $<3\lambda_{i}$. The restriction
$g\mid:C\to{\overline{U}}$ is uniformly continuous, so there exists
$\rho(\lambda_{i})$ such that when $r(\sigma^{0})$ has diameter
$<\rho(\lambda_{i})$ then $gr(\sigma^{0})$ has diameter $<\lambda_{i}$. We may
assume the sequence $\\{\rho(\lambda_{i})\\}$ is decreasing and converging to
$0$. For any integer $\ell$ and simplex $\sigma$ lying in $I_{\ell}$,
$r(\sigma^{0})$ has diameter $<2\gamma_{\ell}+\theta_{\ell}$. We choose an
increasing sequence $\\{\ell(i)\\}$ such that
$2\gamma_{\ell(i)}+\theta_{\ell(i)}<\rho(\lambda_{i})$. Then, for $\sigma$
lying in $I_{\ell(i)}$, $k_{0}(\sigma^{0})$ has diameter
$<\lambda_{\ell(i)}\leq\lambda_{i}$. Summarizing:

######  Proposition 4.5.

When $\sigma$ is a simplex of $I_{\ell(i)}$, then $k_{0}(\sigma^{0})$ has
diameter $<\lambda_{i}$, and $k_{0}^{\prime}(\sigma^{0})$ has diameter
$<3\lambda_{i}$.∎

Let $J_{\ell(i)}:=\bigcup_{\ell(i)\leq m\leq\ell(i+1)}I_{m}$. Then for every
simplex $\sigma$ of $J_{\ell(i)}$ the set $j_{0}(\sigma^{0})$ has diameter
$<4\lambda_{i}$ and thus lies in an element of the cover ${\mathcal{B}}_{i}$
of $Q$. The sequence $\\{\ell(i)\\}$ starts with $i=0$ where we interpret
$\frac{1}{0}=\infty$. The cover ${\mathcal{A}}_{0}$ is the singleton
$\\{Q\\}$, and ${\mathcal{B}}_{0}$ has Lebesgue number $4\lambda_{0}$. Let
$J:=\bigcup_{i\geq 0}J_{\ell(i)}$, regarded as a subcomplex of $K$. We wish to
extend $j_{0}\mid:J^{0}\to Q$ to a map $j:J\to Q$. We do this first on
$J_{\ell(0)}$ where $j_{0}$ maps the $0$-skeleton of each simplex to a set of
diameter $<4\lambda_{0}$. Theorem 3.5 gives an extension $j$ on $J_{\ell(0)}$.
Next, we consider $j_{0}^{+}$ on $J_{\ell(1)}$ where $j_{0}^{+}=j_{0}$ on the
vertices of $J_{\ell(1)}$ and $j_{0}^{+}$ agrees with the (previously defined)
$j$ on $J_{\ell(0)}\bigcap J_{\ell(1)}$. We subdivide this intersection so
that the $j$-image of each simplex in it has diameter $<\lambda_{1}$, and we
extend this subdivision to $J_{\ell(0)}\bigcup J_{\ell(1)}$ without adding
further vertices. Then we have a partial realization of $J_{\ell(1)}$ in $Q$
relative to ${\mathcal{B}}_{1}$, and Theorem 3.5 gives an extension of $j$ to
all of $J_{\ell(1)}$ extending the map $j$ previously defined on
$J_{\ell(0)}$. The $j$-image of each simplex in $J_{\ell(1)}$ lies in an
element of ${\mathcal{A}}_{1}$. We continue in this way, so that on
$J_{\ell(i)}-\text{\rm int}J_{\ell(i-1)}$ the $j$-image of the $0$-skeleton of
each simplex has diameter $<4\lambda_{i}$, extending using Theorem 3.5. Let
$N:=J\bigcup C$. This is a neighborhood of $C$ in $B$. The map
$j=(k,k^{\prime})$; i.e. its components are $k:J\to{\overline{U}}$ and
$k^{\prime}:J\to{\overline{U}}^{\prime}$. We extend both of these
maps181818Previously, we defined $k_{0}$ and $k^{\prime}_{0}$ on all of $K$.
We now confine the domains of $k_{0}$ and $k^{\prime}_{0}$ to $J$. In what
follows we need to be free to extend $j$, i.e. $(k,k^{\prime})$, to all of
$K\bigcup C$ in a different way. (without change of label) to the rest of $N$,
agreeing with $g$ on $C$.

######  Proposition 4.6.

The extended functions $k:N\to Q$ and
$k^{\prime}:N\to{\overline{U}}\coprod{\overline{U}^{\prime}}$ are continuous.

###### Proof.

The only issue is continuity at any point $p\in C$. Let
${\operatorname{\uparrow}T}$ be a basic (open) neighborhood of $g(p)$. Since
$g$ is continuous $g^{-1}({\uparrow T})$ is an open neighborhood of $p$. It is
enough to find a smaller neighborhood, $W$, of $p$ such that
$k^{\prime}(W)\subseteq T^{\prime}$ and $k(W)\subseteq T$. If
$\\{{\overline{V}}_{m}\\}$ is a basic sequence of compact neighborhoods of
$g(p)$ in $U$, then $\bigcap\\{{\overline{V}}_{m}^{\prime}\\}=g(p)^{\prime}$
by Lemma 4.2. By Property A1, $g(p)^{\prime}$ is a compact non-empty subset of
the open set $T^{\prime}$, so for sufficiently large $m$ the compact sets
${\overline{V}}_{m}^{\prime}$ and the (closed) frontier of $T^{\prime}$ are
disjoint. Choose $\epsilon$ so that
$2\epsilon=d(g(p)^{\prime},\text{fr}(T^{\prime}))$. For this choice, let
$\delta$ be as in Lemma 4.4. The following shows that $B_{\delta}(p)$ is the
required neighborhood $W$. When $x$ is a point of $K$ such that
$d(x,p)<\delta$ then $d(x,r(x))<\delta$. By Property A1, $g(r(x))^{\prime}$ is
compact and non-empty. Since $d(x,r(x))<\delta$,
$d(g(x),g(r(x))^{\prime})<\epsilon$, hence $d(g(x),k^{\prime}(x))<\epsilon$.
And since $d(x,p)<\delta$, $d(g(x),g(p)^{\prime})<\epsilon$. Hence
$d(k^{\prime}(x),g(p)^{\prime})<2\epsilon$. This implies $k^{\prime}(x)\in
T^{\prime}$. Continuity of $k$ at $p$ follows from the fact that $g$ is
continuous at $p$: given $\epsilon>0$ let $\delta>0$ be such that $g$ maps the
$\delta$-neighborhood of $p$ into the $\epsilon$-neighborhood of $g(p)$ in
$U$. If $d(p,x)<\frac{\delta}{2}$ then $d(p,r(x))<\delta$, so
$d(g(p),gr(x))<\epsilon$; i.e. $d(k(p),k(x))<\epsilon$. Thus $k$ is continuous
at $p$. ∎

The required maps $h$ and $h^{\prime}$ are defined to agree with $k$ and
$k^{\prime}$ respectively on $N$. Then $h=h^{\prime}=g$ on $C$, and
$h^{\prime}$ covers $h$ (on $N$).

######  Remark.

Theorem 3.4 implies that, with appropriate care, we may arrange that
$h^{\prime}|N$ and $g|N$ are homotopic rel $C$. The details are left to the
reader.

It remains to extend $h$ and $h^{\prime}$ to the rest of $K$ maintaining the
properties (i)-(v).  Recall that
$Q:=({\overline{U}}\times{\overline{U}}^{\prime})\bigcap P$. The projection
map
$\pi:({\overline{U}}\times{\overline{U}}^{\prime})\to{\overline{U}}^{\prime}$
is easily seen to restrict to a proper map $Q\to{\overline{U}}^{\prime}$, and,
by Property A3, this map is cell-like. Let $L$ be the finite simplicial
complement of $N$ in $B$; i.e. $L\bigcup J=K$, and $L$ is disjoint from the
interior of the neighborhood $N$. We write ${\overset{\bullet}{L}}=L\bigcap
N$, the frontier of $N$ in $B$. We have a map $(L\times
0)\bigcup({\overset{\bullet}{L}}\times I)\to{\overline{U}}^{\prime}$ agreeing
with $g$ on $L\times 0$ and being a homotopy between
$g|{\overset{\bullet}{L}}$ and $h^{\prime}|{\overset{\bullet}{L}}$ on
${\overset{\bullet}{L}}\times I$. The Homotopy Extension Property gives a map
${\widebar{g}}:L\to{\overline{U}}^{\prime}$ homotopic to $g$ rel $C$ and
agreeing with $h^{\prime}$ on ${\overset{\bullet}{L}}$. Now,
$h^{\prime}|:{\overset{\bullet}{L}}\to{\overline{U}}^{\prime}$ lifts, through
$\pi$, to a map ${\overset{\bullet}{L}}\to Q$. Theorem 3.6 then gives an
“almost lift” $j$ of ${\widebar{g}}$ through $\pi$, continuously extending
that map to all of $L$.

$\textstyle{{\overset{\bullet}{L}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$inclusion$\scriptstyle{j\mid}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi\mid}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widebar{g}}$$\scriptstyle{j}$$\textstyle{{\overline{U}}^{\prime}}$

Where the domains overlap, this $j$ agrees with that previously defined on
$N$, so, gluing the two maps together, we have $j:K\to Q$. The components of
$j$ are the required maps $h$ and $h^{\prime}$. They extend the previously
defined $h$ and $h^{\prime}$. By Theorem 3.6, the component
$h^{\prime}(=\pi\circ j)$ can be as close as we please to $\widebar{g}$, and
hence is homotopic to it. Thus $g$ is homotopic, rel $\partial B$, to a map
whose image misses ${\overline{U}}^{\prime}$. In other words:

######  Theorem 4.7.

For all $n$,
$\pi_{n}(\operatorname{\uparrow}{\overline{U}},{\overline{U}})=0$; i.e. the
set $\operatorname{\uparrow}{\overline{U}}$ is weakly contractible.

######  Corollary 4.8.

The set $\operatorname{\uparrow}{U}$ is weakly contractible

###### Proof.

Each basis element $U$ contains basis elements $V_{k}$ such that
${\overline{V}_{k}}\subseteq V_{k+1}$, and $\bigcup_{k}{\overline{V}_{k}}=U$.
It follows that

${\operatorname{\uparrow}U}=\bigcup_{k}{\operatorname{\uparrow}{\overline{V}_{k}}}=\bigcup_{k}{\operatorname{\uparrow}{V_{k}}}.$

Every singular sphere $\varphi$ in $\operatorname{\uparrow}U$ has compact
image, and so lies in some ${\operatorname{\uparrow}{V_{k}}}$, Thus, by
Theorem 4.7, $\varphi$ represents a trivial element of homotopy in
$\operatorname{\uparrow}U$. ∎

###  4.2. Pre-images of compact polyhedral sets

Here we prove Theorem 4.13, relating $\Delta({\operatorname{\uparrow}A})$ to
$f^{-1}(\operatorname{\uparrow}A)$ when $A$ is a compact polyhedral subset of
$X_{1}$.

Let $d$ be a length metric on $X_{1}$. Define $\varphi\colon
X_{1}\times(0,1]\to I$ by
$\varphi(x,u)=\text{min}\left\\{\frac{d(x,A)}{u},1\right\\}$. Writing
$\varphi_{u}$ for $\varphi(\cdot,u)$ we note

  1. (i)

$\varphi_{u}=0$ on $A$;

  2. (ii)

$\varphi_{u}=1$ on $X_{1}-N_{u}(A)$;

  3. (iii)

$0<\varphi_{u}<1$ elsewhere.

Let $B$ be a compact polyhedral subset of $X_{2}$. For each $u\in(0,1]$,
define $\psi_{u}\colon X_{1}\times B\times I\to X_{1}\times B\times I$ by

$\psi_{u}(x,y,t)=\begin{cases}(x,y,\varphi_{u}(x))&\text{ if
}t\leq\varphi_{u}(x)\\\ (x,y,t)&\text{ if }\varphi_{u}(x)\leq t.\end{cases}$

We note that $\psi_{u}$ fixes $A\times B\times I$ and that
$\psi_{u}(x,y,t)=(x,y,1)$ when $x\in X_{1}-N_{u}(A)$ and $y\in B$. We also
note the homotopy between $\psi_{u}$ and the identity map $h_{u}\colon
X_{1}\times B\times I\times[0,1]\to X_{1}\times B\times I$ defined by

$h_{u}(x,y,t,v)=\begin{cases}(x,y,v\varphi(x))&\text{ if }t\leq
v\varphi_{u}(x)\\\ (x,y,t)&\text{ if }v\varphi_{u}(x)\leq t\end{cases}$

and that the image of $\psi_{u}$ is the subset $\\{(x,y,t)\in X_{1}\times
B\times I\mid\varphi_{u}(x)\leq t\leq 1\\}$. Factoring by the canonical
quotient $X_{1}\times B\times I\to X_{1}\ast B$ we see that, while $\psi_{u}$
does not induce a well-defined map $X_{1}\ast B\to X_{1}\ast B$, it does
induce a map $\Psi_{u}\colon(X_{1}\ast B)-(X_{1}-A)\to X_{1}\ast B$.
Similarly, the homotopy $h_{u}$ induces a homotopy $H_{u}\colon(X_{1}\ast
B)-(X_{1}-A))\times[0,1]\to X_{1}\ast B$. The image of $\Psi_{u}$ is the
compact set $I_{u}\coloneqq\\{(1-t)x+ty\in X_{1}\ast B\mid
t\geq\varphi_{u}(x)\\}$.

######  Proposition 4.9.

Given $u$ in $(0,1]$ and a neighborhood $M$ of $N_{u}(A)\ast B$ in $X_{1}\ast
X_{2}$, we can deform $(X_{1}\ast B)-(X_{1}-A)$ within itself onto the compact
set $I_{u}\subseteq M$ by a deformation that is continuous in the variable
$u$.

By careful choice of the metric $d$ it can be arranged that for
$u=\frac{1}{n}$ the above functions are PL, and in particular that
$I_{\frac{1}{n}}$ is polyhedral. Thus we get:

######  Proposition 4.10.

Given $u=\frac{1}{n}$ and a neighborhood $M$ of $N_{u}(A)\ast B$ in $X_{1}\ast
X_{2}$, we can deform $(X_{1}\ast B)-(X_{1}-A)$ within itself onto the compact
polyhedral set $I_{\frac{1}{n}}\subseteq M$.

Since the order relation $P$ is polyhedral, this proposition can be restricted
as follows, where we now take $B$ to be $A^{\prime}$ (compact and polyhedral
by Proposition 3.3):

######  Corollary 4.11.

Given $u=\frac{1}{n}$ and a neighborhood $M$ of
$\Delta(\operatorname{\uparrow}A)$ in $\Delta({\mathcal{X}})$,
$f^{-1}(\operatorname{\uparrow}A)$ can be deformed within itself onto a
compact polyhedral set $J_{n}\subseteq M$.

###### Proof.

The set $f^{-1}(\operatorname{\uparrow}A)$ consists of
$\Delta(\operatorname{\uparrow}A)$ together with line segments ending in
$A^{\prime}$ whose intersections with $X_{1}$ have been deleted. Here, we
write $J_{n}$ for the relevant part of $I_{\frac{1}{n}}$. ∎

######  Lemma 4.12.

The set $\Delta(\operatorname{\uparrow}A)$ is a compact polyhedral subset of
$X_{1}*X_{2}$.

###### Proof.

By Lemmas 3.1 and 3.3, $A^{\prime}$ is compact and polyhedral. Thus
$C\coloneqq((A\times A^{\prime})\bigcap P)\times I$ is compact and polyhedral.
The projection maps $A\times A^{\prime}\to A$ and $A\times A^{\prime}\to
A^{\prime}$ define PL maps $\pi_{A}\colon((A\times A^{\prime})\bigcap
P)\times\\{0\\}\to A$ and $\pi_{A^{\prime}}\colon((A\times A^{\prime})\bigcap
P)\times\\{1\\}\to A^{\prime}$. The space $\Delta(\operatorname{\uparrow}A)$
is clearly decomposable as the union of three compact polyhedra: $C$ and the
mapping cylinders of $\pi_{A}$ and $\pi_{A^{\prime}}$, where the mapping
cylinders are glued to the $0$\- and $1$\- ends of $C$ in the obvious way.
Since mapping cylinders of PL maps between compact polyhedra are compact
polyhedra, and the union of compact polyhedra glued along compact polyhedral
subsets is again a compact polyhedron, this completes the proof. ∎

######  Theorem 4.13.

When $A$ is a compact polyhedral subset of $X_{1}$, the inclusion map
$\Delta(\operatorname{\uparrow}A)\hookrightarrow
f^{-1}(\operatorname{\uparrow}A)$ is a homotopy equivalence.

###### Proof.

$\Delta(\operatorname{\uparrow}A)$ is the intersection of the nested sequence
of polyhedra $J_{n}$, each of which is a strong deformation retract of
$f^{-1}(\operatorname{\uparrow}A)$ by Corollary 4.11. Since the inclusion maps
$J_{n+1}\to J_{n}$ are homotopy equivalences, shape theory (see Section 7 for
details) implies that $\Delta(\operatorname{\uparrow}A)$ is shape equivalent
to each of the polyhedra $J_{n}$. Moreover, since
$\Delta(\operatorname{\uparrow}A)$ is itself polyhedral by Lemma 4.12, this
means that $\Delta(\operatorname{\uparrow}A)$ is actually homotopy equivalent
to each of the $J_{n}$, hence also to $f^{-1}(\operatorname{\uparrow}A)$. ∎

###  4.3. Contractibility of pre-images of basis elements

The order complex is the subset $\Delta({\mathcal{X}})\subseteq X_{1}*X_{2}$
which consists of $X_{1}$, $X_{2}$, and the union of all segments joining $x$
to $y$ such that $x<y$. The Comparison Map
$f\colon\Delta({\mathcal{X}})\to{\mathcal{X}}$ is defined by $f(x)=x$ when
$x\in X_{1}$ (i.e. when $t=0$) and $f((1-t)x+ty)=y$ when $t>0$.

As before, we consider the two kinds of basis elements in the Up topology:
$U\in{\mathcal{U}}_{2}$ and $\operatorname{\uparrow}U$ where
$U\in{\mathcal{U}}_{1}$. To apply Theorem 1.1 we show that the pre-image under
$f$ of each of these is weakly contractible. When $U\in{\mathcal{U}}_{2}$, the
set $f^{-1}(U)$ consists of the contractible set $U\subseteq X_{2}\subseteq
X_{1}*X_{2}$, together with half-open segments ending in $U$, so $U$ is a
strong deformation retract of $f^{-1}(U)$ which is therefore contractible. Now
we consider the case where $U\in{\mathcal{U}}_{1}$. It is convenient to first
deal with $f^{-1}(\operatorname{\uparrow}{\overline{U}})$, which Theorem 4.13
tells us is homotopy equivalent to
$\Delta(\operatorname{\uparrow}{\overline{U}})$.

######  Notation.

We write ${\mathcal{C}}A$ for the topological cone on the space $A$, i.e.
$A\times I/A\times\\{0\\}$.

######  Proposition 4.14.

$\Delta(\operatorname{\uparrow}{\overline{U}})$ is homotopy equivalent to
${\mathcal{C}}\overline{U}^{\prime}$, and is therefore contractible.

###### Proof.

Let
$g\colon\Delta(\operatorname{\uparrow}{\overline{U}})\to{\mathcal{C}}\overline{U}^{\prime}$
be the map which takes $\overline{U}$ to the cone point $p$, is the identity
on $\overline{U}^{\prime}$, and maps the point $(1-t)x+ty$ to the segment
$(1-t)p+ty$ in the cone. By Lemma 4.12 this is a map between compact
polyhedra. Consider $g^{-1}((1-t)p+ty)$. When $t=0$ this pre-image is the
contractible set $\overline{U}$; when $t=1$ this pre-image is a single point;
when $0<t<1$ this pre-image is homeomorphic to $\\{x\in{\overline{U}}\mid
x<y\\}$ which has trivial shape (actually, is contractible) by Property A3.
Thus $g$ is a cell-like map, and is therefore a homotopy equivalence by
Theorem 3.7. ∎

######  Corollary 4.15.

$f^{-1}(\operatorname{\uparrow}{\overline{U}})$ is contractible.

######  Corollary 4.16.

For basis elements $U$, $f^{-1}(\operatorname{\uparrow}{U})$ is contractible.

###### Proof.

The proof is similar to that of Corollary 4.8. It shows that
$f^{-1}(\operatorname{\uparrow}{U})$ is weakly contractible. Since it is an
open subset of a polyhedron, it is actually contractible. ∎

This completes the proof of Theorem 1.3 for the case ${\mathcal{R}}=[2]$.

##  5\. The special case ${\mathcal{R}}=[3]$

######  Notation.

In this section, when $A\subseteq X_{1}$ we write $A^{\prime}$ rather than
$A^{(2)}$ and $A^{\prime\prime}$ rather than $A^{(3)}$.

###  5.1. Weak contractibility of basis elements

Here ${\mathcal{X}}=X_{1}\coprod X_{2}\coprod X_{3}$ and $U\subseteq X_{i}$ is
a basis element. We are to show that $\uparrow U$ is weakly contractible. If
$U\subseteq X_{3}$ this is trivial, and if $U\subseteq X_{2}$ this follows
from what was done in Section 4. So we assume $U\subseteq X_{1}$. As before,
it is easier to begin with $\overline{U}$. We consider an arbitrary map

$g\colon(B,\partial
B)\to(\operatorname{\uparrow}{\overline{U}},{\overline{U}}\coprod{\overline{U}}^{\prime}).$

where $B$ abbreviates $B^{n}$. We write
$C:=g^{-1}({\overline{U}}\coprod{\overline{U}}^{\prime})$, a closed subset of
$B$ containing $\partial B\subseteq C$. We wish to produce maps
$h^{\prime\prime}$ and $h$ from $B$ to $\operatorname{\uparrow}{\overline{U}}$
satisfying:

  1. (i)

$h^{\prime\prime}=h=g$ on $C$;

  2. (ii)

$h^{\prime\prime}$ maps $B-C$ into ${\overline{U}}^{\prime\prime}$;

  3. (iii)

$h$ maps $B$ into ${\overline{U}}\coprod{\overline{U}}^{\prime}$;

  4. (iv)

on $B-C$, $h^{\prime\prime}$ covers $h$;

  5. (v)

$h^{\prime\prime}$ is homotopic to $g$ rel $C$.

Proposition 4.1 will then imply that $h^{\prime\prime}$ is homotopic to $h$
rel $C$ and hence that the map $g$ represents the trivial element of
$\pi_{n}(\operatorname{\uparrow}{\overline{U}},{\overline{U}}\coprod{\overline{U}^{\prime}})$;
hence that group is trivial. By Theorem 4.7 it will follow that
$\pi_{n}(\operatorname{\uparrow}{\overline{U}},{\overline{U}})$ is trivial. As
before, we can deduce the same when ${\overline{U}}$ is replaced by $U$. The
argument here is more complicated than in Section 4 because the obvious
replacement for ${\overline{U}}\times{\overline{U}}^{\prime}$ in the
definition of $Q$, namely
$({\overline{U}}\coprod{\overline{U}}^{\prime})\times{\overline{U}}^{\prime\prime}$,
is not polyhedral (with the Up topology). For this reason we must construct
the map $h^{\prime\prime}$ in two stages, one for ${\overline{U}}$, the other
for ${\overline{U}}^{\prime}$. Let
$Q_{1}:=({\overline{U}}\times{\overline{U}}^{\prime\prime})\bigcap P$. Let
${\mathcal{A}}_{i}$ be an open cover of $Q_{1}$ by sets of diameter
$<\frac{1}{i}$, and let ${\mathcal{B}}_{i}$ be the refinement given by Theorem
3.5. Let $4\lambda_{i}$ be a Lebesgue number for ${\mathcal{B}}_{i}$. We write
$C_{1}:=g^{-1}({\overline{U}})$; this is a closed subset of $C$. For each
$x\in B-C$ we choose a point $r(x)\in C$ realizing $d(x,C)$. Define
$Y:=\\{x\in B-C\mid d(x,C)=d(x,C_{1})\\}$, a closed subset of $B-C$. We now
proceed as in Section 4.1 replacing $B-C$ by $Y$ and $C$ by $C_{1}$. The
triangulation $K$ and its subcomplex $J$ are replaced by a
triangulation191919We assumed, for simplicity, in Section 4.1 that each
$I_{i}$ is a subpolyhedron of $B-C$. Similarly, here we assume that $Y$ is a
polyhedral subset of $B-C$. As in the previous case, if it is not polyhedral,
one covers it by a slightly larger polyhedral set. Again, we omit the details.
$K_{1}$ of $Y$ and its subcomplex $J_{1}$. We define $N_{1}:=J_{1}\bigcup
C_{1}$, a neighborhood of $C_{1}$ in $Y$. We get a map
$j_{1}:=(k_{1},k_{1}^{\prime\prime})$; with components
$k_{1}:J_{1}\to{\overline{U}}$ and
$k_{1}^{\prime\prime}:J_{1}\to{\overline{U}}^{\prime\prime}$. Both of these
maps are extended to agree with $g$ on $C_{1}$. As before, the extended
function $j_{1}:N_{1}\to Q_{1}$ is continuous. The required maps $h$ and
$h^{\prime\prime}$ are defined to agree with $k_{1}$ and
$k_{1}^{\prime\prime}$ respectively on $N_{1}$. Then $h^{\prime\prime}$ covers
$h$ and $h=h^{\prime\prime}=g$ on $C_{1}$. As in Section 4.1 we may assume
$h^{\prime\prime}|N$ and $g|N$ are homotopic rel $C_{1}$. The extension of $h$
and $h^{\prime\prime}$ to the rest of $K_{1}$ maintaining the properties
(i)-(v) is achieved as in that section. At this point, the Homotopy Extension
Property implies that $g:B\to\operatorname{\uparrow}{\overline{U}}$ is
homotopic rel $C$ to a map $g_{1}$, where $g_{1}|Y$ has the desired property
of covering a map into ${\overline{U}}$. Now comes the part that is different
to Section 4.1. We wish to alter $g_{1}$ rel $Y\bigcup C$ to have the desired
covering property on the rest of $B-C$. The set $Z:=B-(C\bigcup Y)$ is an open
polyhedral subset of $B$ whose frontier is the compact set
${\overset{\bullet}{Z}}:=\text{\rm fr}_{B}(Z)$. We note that
$C-C_{1}\subseteq{\overset{\bullet}{Z}}$.  Basically, we repeat what has been
described above, with $K_{2}$ (triangulating $Z$), ${\overset{\bullet}{Z}}$,
$g_{1}$, and $Q_{2}:=(U^{\prime}\times U^{\prime\prime})\bigcap P$ playing the
respective roles202020The item to note is that ${\overset{\bullet}{Z}}$ is
playing the role previously played by $C_{1}$. of $K_{1}$, $C_{1}$, $g$ and
$Q_{1}$. There is a subcomplex $J_{2}$ of $K_{2}$, and a neighborhood $N_{2}$
of ${\overset{\bullet}{Z}}$ in $\overline{Z}$ playing the role previously
played by $J_{1}$ and $N_{1}$. We get a map
$j_{2}=(k_{2}^{\prime},k_{2}^{\prime\prime})$ with components
$k_{2}^{\prime}:J_{2}\to U^{\prime}$ and $k_{2}^{\prime\prime}:J_{2}\to
U^{\prime\prime}$. The construction (imitating what is given in detail in
Section 4.1) means that this map $j_{2}$ extends continuously to agree with
$j_{1}$ on ${\overset{\bullet}{Z}}$. The extension to the rest of $Z$ works as
before. Thus we get maps $h^{\prime\prime}$ and $h$ as required. This shows
that $g$ is homotopic, rel $\partial B$, to a map whose image misses
${\overline{U}}^{\prime\prime}$. Summarizing:

######  Theorem 5.1.

For all $n$,
$\pi_{n}(\operatorname{\uparrow}{\overline{U}},{\overline{U}}\coprod{\overline{U}^{\prime}})$
is trivial.

As explained above, this implies that
$\pi_{n}(\operatorname{\uparrow}{\overline{U}},{\overline{U}})$ is trivial,
hence also $\pi_{n}(\operatorname{\uparrow}{\overline{U}})$. Just as in
Section 4.1 this leads to:

######  Corollary 5.2.

The set $\operatorname{\uparrow}{U}$ is weakly contractible.

######  Remark.

It was important to handle ${\overline{U}}$ before ${\overline{U}}^{\prime}$
since we needed $C_{1}$ to be compact. This is a simple instance of what is a
more serious issue in the general case, below, where we need a careful
ordering of $\mathcal{R}$ in order to ensure that at each stage we are dealing
with a compact set.

###  5.2. Contractibility of pre-images of basis elements

As a special case of what was defined in Section 1.2, recall that the order
complex is the set $\Delta({\mathcal{X}})\subseteq X_{1}*X_{2}*X_{3}$
consisting of all points $z=\sum\limits_{i=1}^{3}t_{i}x_{i}$ such that when
$i,j\in\text{supp }z$ then $x_{i}<x_{j}$ in $\mathcal{X}$. In particular,
every $x_{i}$ lies in this set. The Comparison Map
$f\colon\Delta({\mathcal{X}})\to{\mathcal{X}}$ is defined by $f(z)=\text{max
}\\{x_{i}\mid i\in\text{supp }(z)\\}$.

We first consider $U\subseteq X_{1}$. We describe the set
$f^{-1}(\operatorname{\uparrow}{\overline{U}})\subseteq X_{1}*X_{2}*X_{3}$ in
some detail. Certainly it includes
$\Delta(\operatorname{\uparrow}{\overline{U}})$. Imitating what was done in
Section 4.3, we let $R^{o}$ be the union of a certain collection of half-open
segments, where the missing point of each segment is its initial point as
measured by the poset ${\mathcal{R}}=[3]$. These segments are described as
follows. In every case it is to be understood that $x_{1}<x_{2}<x_{3}$
whenever these comparisons make sense212121Notation: $[x,y]$ stands for all
the points $(1-t)x+ty$ on the segment joining $x$ to $y$.:

  1. (i)

$(x_{1},x_{2}]$ where
$x_{1}\notin{\overline{U}},x_{2}\in{\overline{U}}^{(2)}$, and $x_{2}$ is not
comparable to any member of ${\overline{U}}^{(3)}$;

  2. (ii)

$(x_{1},x_{3}]$ where
$x_{1}\notin{\overline{U}},x_{3}\in{\overline{U}}^{(3)}$, and there is no
$x_{2}\in{\overline{U}}^{(2)}$ with $x_{1}<x_{2}<x_{3}$;

  3. (iii)

$(x_{1},[x_{2},x_{3}]]$ where
$x_{1}\notin{\overline{U}},x_{2}\in{\overline{U}}^{(2)}$, and
$x_{3}\in{\overline{U}}^{(3)}$;

  4. (iv)

$([x_{1},x_{2}],x_{3}]$ where $x_{1}\notin{\overline{U}}$,
$x_{2}\notin{\overline{U}}^{(2)}$, and $x_{3}\in{\overline{U}}^{(3)}$;

  5. (v)

$(x_{2},x_{3}]$ where
$x_{2}\notin{\overline{U}}^{(2)},x_{3}\in{\overline{U}}^{(3)}$, and there is
no $x_{1}\in{\overline{U}}$ with $x_{1}<x_{2}$.

Then

$f^{-1}(\operatorname{\uparrow}{\overline{U}})=\Delta(\operatorname{\uparrow}{\overline{U}})\cup
R^{o}.$

The set $X_{2}\coprod X_{3}$ is a sub-poset of $\mathcal{X}$, and we denote
its order complex by $\Delta_{23}$. Recall that $\Delta_{23}$ consists of
copies of $X_{2}$ and $X_{3}$ together with a segment222222In all such
expressions, the sum of coefficients is understood to be $1$, and a segment
such as this will require an obvious adjustment when $t_{2}=0$ or $1$.
$t_{2}x_{2}+t_{3}x_{3}$ joining $x_{2}$ to $x_{3}$ whenever $x_{2}<x_{3}$.
This gives rise to another poset ${\mathcal{Y}}\coloneqq
X_{1}\coprod\Delta_{23}$ with the partial ordering
$x_{1}<t_{2}x_{2}+t_{3}x_{3}$ when $x_{1}<x_{2}<x_{3}$; $\mathcal{Y}$ is given
the Up topology. We denote the corresponding Comparison Map by
$g\colon\Delta({\mathcal{Y}})\to{\mathcal{Y}}$. Then, as subsets of $X_{1}\ast
X_{2}\ast X_{3}$, we have

$f^{-1}(\operatorname{\uparrow}{\overline{U}})=g^{-1}(\operatorname{\uparrow}{\overline{U}}).$

Since $g^{-1}(\operatorname{\uparrow}{\overline{U}})$ is contractible by
Proposition 4.15, we conclude that
$f^{-1}(\operatorname{\uparrow}{\overline{U}})$ is contractible. Next we
consider the case where $U\subseteq X_{2}$. We have the Comparison Map
$f_{23}\colon\Delta_{23}\to X_{2}\coprod X_{3}$. By Proposition 4.15,
$f_{23}^{-1}({\operatorname{\uparrow}{\overline{U}}})$ is contractible. But
$f^{-1}({\operatorname{\uparrow}{\overline{U}}})$ is larger in general, as it
includes deleted segments and $2$-simplexes having a vertex in
$f_{23}^{-1}({\operatorname{\uparrow}{\overline{U}}})$, the deletion being the
(non-empty) part lying in $X_{1}$. However, as before,
$f_{23}^{-1}({\operatorname{\uparrow}{\overline{U}}})$ is a strong deformation
retract of $f^{-1}({\operatorname{\uparrow}{\overline{U}}})$, so the latter is
contractible. Finally, there is the case where $U\subseteq X_{3}$. In that
case, just as in the previous paragraph,
$\operatorname{\uparrow}{\overline{U}}$ is a strong deformation retract of
$f^{-1}({\operatorname{\uparrow}{\overline{U}}})$, and therefore the latter is
contractible. As in Corollary 4.16 we conclude:

######  Proposition 5.3.

For basis elements $U$, the space $f^{-1}(\operatorname{\uparrow}{U})$ is
contractible.

##  6\. The General Case

We are now dealing with the geometric $M$-poset $\mathcal{X}$ equipped with a
mirror map $\mu:{\mathcal{X}}\to{\mathcal{R}}$, where ${\mathcal{R}}$ is a
finite poset.

###  6.1. Weak contractibility of basis elements

In Section 5.1 we showed that when ${\overline{U}}^{(1)}$ is the closure of a
basis element of $X_{1}$ then $\uparrow{\overline{U}}^{(1)}$ is weakly
contractible. The method was to consider a map $g\colon(B,\partial
B)\to(\operatorname{\uparrow}{\overline{U}^{(1)}},{\overline{U}^{(1)}}\coprod{\overline{U}^{(2)}})$,
define $C=g^{-1}({\overline{U}}^{(1)}\coprod{\overline{U}}^{(2)})$ and
$C_{1}=g^{-1}({\overline{U}}^{(1)})$. We picked a closed subset $Y$ of $B-C$
associated with $C_{1}$, and first adjusted $g$ on $Y$. Then we made a further
adjustment on $Z=B-(C\bigcup Y)$. If we were dealing with the case
${\mathcal{R}}=[4]$ we would do the same thing with one extra step: $C_{1}$
would be $g^{-1}({\overline{U}}^{(1)})$, $C_{2}$ would be
$g^{-1}({\overline{U}}^{(1)}\coprod{\overline{U}}^{(2)})$ and $C$ would be
$g^{-1}({\overline{U}}^{(1)}\coprod{\overline{U}}^{(2)}\coprod{\overline{U}}^{(3)})$.
There would be subspaces $Y$ and $Z$ associated with $C_{1}$ and $C_{2}$. The
adjustment of $g$ would be made first on $Y$, then on $Z$ and finally on the
rest of $B-C$. Thus $g$ would be homotopic, rel $\partial B$, to a map whose
image misses ${\overline{U}}^{(4)}$. Enough has been said here to indicate
that the same is true for ${\mathcal{R}}=[m]$; namely, if $U^{(1)}$ is a basis
element of $X_{1}$ then $\uparrow U^{(1)}$ is weakly contractible. To get the
same result for an arbitrary finite poset $\mathcal{R}$, we need to order its
elements with care. Let

${\mathcal{R}}_{0}:=\\{r\in{\mathcal{R}}\mid\text{there is no
}s\in{\mathcal{R}}\text{ with }s<r\\}.$

For $i>0$ let

${\mathcal{R}}_{i}:=\\{r\in{\mathcal{R}}\mid\text{the longest chain connecting
}r\text{ to an element of }{\mathcal{R}}_{0}\text{ has length }i\\}.$

We choose a total ordering of $\mathcal{R}$ so that, for all $i$, every
element of ${\mathcal{R}}_{i}$ comes before any element of
${\mathcal{R}}_{i+1}$. We call this the useful ordering to distinguish it from
the given partial ordering of $\mathcal{R}$. The useful ordering ensures a key
point: that for any $r$, $\coprod\\{X_{s}\mid s\text{ precedes }r\text{ in the
useful ordering}\\}$ is closed in the Up topology on ${\mathcal{X}}$. We are
to show that, for any basis element $U^{(s)}$ in $X_{s}$,
$\operatorname{\uparrow}{\overline{U}}^{(s)}$ is weakly contractible. There is
no loss of generality in assuming $s=m$ where $m$ is the minimal element in
the useful ordering. Let $M$ be the maximal element in the useful ordering. We
consider a map $g\colon(B,\partial
B)\to(\operatorname{\uparrow}{\overline{U}}^{(m)},\operatorname{\uparrow}{\overline{U}}^{(m)}-{\overline{U}}^{(M)})$.
If the elements of the useful ordering are taken in ascending order, then an
obvious generalization of the above discussion for ${\mathcal{R}}=[3]$ and
${\mathcal{R}}=[4]$ adjusts $g$, rel $\partial B$, to a map whose image misses
${\overline{U}}^{(M)}$. Proceeding by induction we get:

######  Theorem 6.1.

For all $n$,
$\pi_{n}(\operatorname{\uparrow}{\overline{U}}^{(m)},{\overline{U}}^{(m)})=0$.

Just as in Section 4.1 this leads to:

######  Corollary 6.2.

The set $\operatorname{\uparrow}{U^{(m)}}$ is weakly contractible.

###  6.2. Contractibility of pre-images of basis elements

The Comparison Map $f:\Delta({\mathcal{X}})\to{\mathcal{X}}$ is, as before,
$f(z)=\text{max}\\{x_{i}\mid i\in\text{ supp}(z)\\}$. Let $U\subseteq X_{r}$
where $r\in{\mathcal{R}}$.

######  Proposition 6.3.

$f^{-1}(\uparrow{\overline{U}})$ is contractible.

###### Proof.

We imitate Section 5, inducting on the cardinality of $\mathcal{R}$. When
$\mathcal{R}$ is a singleton the proposition is trivially true. When
$\mathcal{R}$ has two elements the proposition has been proved. We assume
$\mathcal{R}$ has at least three elements. There are two cases First assume
$r$ is a minimal element of $\mathcal{R}$. Then

$f^{-1}(\uparrow{\overline{U}})=\Delta(\uparrow{\overline{U}})\bigcup R^{o}$

where $R^{o}$ is the union of deleted simplexes, each having some of its
vertices outside $\uparrow{\overline{U}}$ and the remaining vertices (at least
one) in $\uparrow{\overline{U}}$, the deletions being the faces lying outside
$\uparrow{\overline{U}}$. Now, ${\mathcal{Z}}:={\mathcal{X}}-X_{r}$ is a sub-
poset. From it we can construct a new poset
${\mathcal{Y}}:=X_{r}\coprod\Delta({\mathcal{Z}})$ where the details imitate
the construction given explicitly in Section 5.2. The Comparison Map is
$g:\Delta({\mathcal{Y}})\to{\mathcal{Y}}$. Then

$f^{-1}(\uparrow{\overline{U}})=g^{-1}(\uparrow{\overline{U}}).$

By induction and Corollary 4.15, $g^{-1}(\uparrow{\overline{U}})$ is
contractible, so we conclude that $f^{-1}(\uparrow{\overline{U}})$ is
contractible.

In the other case, $r$ is not minimal. Then there is a proper sub-$M$-poset
$\uparrow X_{r}$ indexed by a proper subset of $\mathcal{R}$. By induction on
the cardinality of $\mathcal{R}$, $g^{-1}(\uparrow{\overline{U}})$ is
contractible, where $g$ denotes the Comparison Map for $\uparrow X_{r}$. Then

$f^{-1}(\uparrow{\overline{U}})=g^{-1}(\uparrow{\overline{U}})\bigcup R^{o}$

where $R^{o}$ is the union of deleted simplexes, each having some of its
vertices outside $\uparrow{\overline{U}}$ and at least one vertex in
$\uparrow{\overline{U}}$, the deletions being the faces lying outside
$\uparrow{\overline{U}}$. These deleted simplexes deform to
$g^{-1}(\uparrow{\overline{U}})$. So $f^{-1}(\uparrow{\overline{U}})$ is
contractible. ∎

As in Corollary 4.16 we conclude:

######  Corollary 6.4.

For basis elements $U$, $f^{-1}(\uparrow U)$ is contractible

This completes the proof of Theorem 1.3.

##  7\. Appendix on the shape theory used in this paper.

Shape theory is a variant of homotopy theory which works better than regular
homotopy theory when the spaces are not locally nice. Here we give a brief
review with the sole purpose of amplifying our use of the theory in Section
4.3.

Shape theory considers a compact metrizable space $X$ by dealing instead with
an arbitrary inverse sequence of compact polyhedra and maps whose inverse
limit is homeomorphic to $X$.

$\textstyle{X_{1}}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}}$$\textstyle{X_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{2}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\cdots}}$$\scriptstyle{h_{3}}$

In the following infinite diagram of space and maps, the spaces are compact
polyhedra. We denote the inverse limit of the top line [resp. bottom line] by
$X$ [resp.$Y$]. The outer squares are assumed to be homotopy commutative. If
there exist diagonal maps making the entire diagram homotopy commutative then
$X$ and $Y$ are said to be shape equivalent232323This is not the general
definition but is enough for our purposes. For more details on what is
discussed in this Appendix, see [Gui16].

$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\cdots}}$$\textstyle{Y_{1}}$$\textstyle{Y_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots}$

Now consider the special case in which all the maps $h_{i}$ are homotopy
equivalences. Then we have the following homotopy commutative diagram, where
$h_{i}^{-1}$ stands for a homotopy inverse of $h_{i}$:

$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\scriptstyle{h_{1}^{-1}}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\scriptstyle{(h_{1}h_{2})^{-1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\cdots}}$$\scriptstyle{id}$$\textstyle{X_{1}}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}}$$\scriptstyle{h_{1}}$$\textstyle{X_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{2}}$$\scriptstyle{h_{1}h_{2}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\cdots}}$$\scriptstyle{h_{3}}$

We conclude that in this case $X$ is shape equivalent to the compact
polyhedron $X_{1}$. Moreover, in the application in Section 4.3, $X$ itself
known to be a compact polyhedron. It is a basic theorem of shape theory that
two polyhedra are shape equivalent if and only if they are homotopy
equivalent.

In the context of Proposition 4.13 this discussion explains why
$\Delta(\overline{U})$ is homotopy equivalent to each $P_{n}$ and hence to
$f^{-1}(\operatorname{\uparrow}{\overline{U}})$.

##  8\. Appendix on the discrete case and McCord’s work.

A quick review of the present paper shows that the proof of the Comparison
Theorem 1.3 becomes very easy in the case when each $X_{r}$ is discrete.
Confining ourselves to the special case dealt with in Section 4, the relevant
version of Proposition 4.13 becomes almost trivial, since the issue of limit
points (which required a shape theoretic argument) is not present, and the
need for Proposition 4.14 is gone since pre-images of basic sets
$\operatorname{\uparrow}x$ are cones. The issues dealt with in Section 4.1
become trivial because Proposition 4.1 implies that each singleton $\\{x\\}$
is a strong deformation retract of the set $\operatorname{\uparrow}x$.

McCord’s Theorem $3$ in [McC66] implies our Comparison Theorem 1.3 in this
discrete case, and his proof is indicated by the contents of the previous
paragraph. It should be added, however, that his definition of
$\Delta({\mathcal{X}})$ is superficially different from ours. In his work, the
role of $\Delta({\mathcal{X}})$ is played by $|K({\mathcal{X}})|$, where
$K({\mathcal{X}})$ is the classical order complex defined by the poset
$\mathcal{X}$, namely: simplexes are finite chains in the poset. However, it
is not hard to see that $|K({\mathcal{X}})|$ and $\Delta({\mathcal{X}})$ are
the same in our setting when $\mathcal{X}$ is discrete.

McCord uses the “Down topology” rather than the “Up topology” but that is of
no consequence as the two are dual to one another and order complexes are
preserved under duality.

It should be added that McCord’s Theorem $3$ covers cases outside the setting
of the present paper.

##  9\. Appendix on homotopy equivalence

In Theorem 1.3 we showed that for geometric $M$-posets $\mathcal{X}$ the
Comparison Map $f\colon\Delta({\mathcal{X}})\to{\mathcal{X}}$ is a weak
homotopy equivalence. The Whitehead Theorem implies that $f$ is a homotopy
equivalence if and only if $\mathcal{X}$ has the homotopy type of a $CW$
complex. In this appendix we show that topological posets $\mathcal{X}$ in an
important class do not have the homotopy type of a $CW$ complex Let
$\mathcal{X}$ be a topological poset, not necessarily geometric. Define an
equivalence relation on $\mathcal{X}$ by $p\sim q$ if and only if there is a
finite sequence $(x_{i})$ in $\mathcal{X}$ with $x_{0}=p$, $x_{n}=q$, and, for
each $i\geq 0$, either $x_{i}\leq x_{i+1}$ or $x_{i}\geq x_{i+1}$. Clearly,
each equivalence class lies in a path component. We say that $\mathcal{X}$ has
discrete type if each equivalence class is an entire path component242424The
topological poset $\mathcal{X}$ has an underlying discrete poset which we
denote by ${\mathcal{X}}_{\delta}$, equipped with the Up topology. Comparing
${\mathcal{X}}_{\delta}$ with its (classical) order complex, one easily sees
that each equivalence class is a path component of ${\mathcal{X}}_{\delta}$.
This explains the term “discrete type”.

######  Example.

For each $n$ the real Grassmann Poset ${\mathcal{G}}_{n}$ has discrete type.

######  Lemma 9.1.

Each equivalence class is a closed subset of $\mathcal{X}$.

###### Proof.

The relation $P$ in the definition of topological poset is closed in
${\mathcal{X}}\times\mathcal{X}$, so the condition of not being related is an
open condition. It follows that every limit point of an equivalence class lies
in that equivalence class. ∎

For any space $Y$ the space of path components, $\pi_{0}(Y)$, occurs as a
quotient space in a natural way. When $Y$ is a $CW$ complex, this space is
discrete.

######  Theorem 9.2.

Let $\mathcal{X}$ have discrete type. Then $\mathcal{X}$ has the homotopy type
of a $CW$ complex if and only if each path component is contractible, and the
space of path components $\pi_{0}({\mathcal{X}})$ is discrete.

###### Proof.

Let $h:K\to{\mathcal{X}}$ be a homotopy equivalence, where $K$ is a $CW$
complex. Suppose there is a homotopy inverse $g$ for $h$. The fact that $g$ is
continuous shows that if $x<y$ then $g(y)=g(x)$; hence $g$ is constant on
equivalence classes (i.e. on path components). The map $g$ induces a
continuous bijection $\pi_{0}({\mathcal{X}})\to\pi_{0}(K)$, so
$\pi_{0}({\mathcal{X}})$ is discrete. And since $h\circ g$ is homotopic to the
identity map, each path component of $\mathcal{X}$ must be contractible.
Conversely, the hypotheses imply that $\mathcal{X}$ has the homtopy type of a
discrete space. ∎

######  Remark.

It follows from Theorem 9.2 that equivalence classes are closed. We include
Lemma 9.1 because this is true whether or not $\mathcal{X}$ has discrete type.
We wonder whether Theorem 9.2 holds for all topological posets.

######  Acknowledgment.

We thank Laura Anderson for telling us about this problem and for educating us
on its connection with tropical phased matroids.

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