# A generalized theory of expansions and collapses with applications to
$\mathcal{Z}$-compactification

Craig R. Guilbault  Department of Mathematical Sciences
University of Wisconsin-Milwaukee, Milwaukee, WI 53201<EMAIL_ADDRESS>and
Daniel Gulbrandsen  Department of Mathematics and Computer Science
Hampden-Sydney College, Hampden-Sydney, VA 23943<EMAIL_ADDRESS>
(Date: October 30, 2023)

###### Abstract.

We generalize the dual notions of “expansion” and “collapse” so they can be
applied to arbitrary metric spaces. We also expand the theory to allow for
infinitely many such moves. Those tools are then employed to prove a variety
of compactification theorems. We are particularly interested in
$\mathcal{Z}$-set compactifications, which play an important role in geometric
group theory and in algebraic and geometric topology.

###### Key words and phrases:

collapse, expansion, $\mathcal{Z}$-set, homotopy negligible set,
$\mathcal{Z}$-compactification, homotopy negligible compactification

This research was supported in part by Simons Foundation Grant 427244, CRG

In this paper, we expand upon the dual notions of _expansion_ and _collapse_
in ways that make them more applicable to spaces encountered in geometric
group theory. In their original forms, these are combinatorial moves. An
_elementary simplicial collapse_ removes from a simplicial complex $K$ a
simplex and one of its faces in a way that results in a homotopy equivalent
subcomplex $L$. The reverse is an _elementary simplicial expansion_. A
_collapse_ of $K$ to $L$ is a finite sequence of elementary collapses; an
_expansion_ is the same process, viewed in reverse. These definitions
generalize to polyhedral and CW complexes where they form the foundation of
_simple homotopy theory_ , which plays a central role in algebraic and
geometric topology. See, for example, [Whi39],[Whi41],[Whi49],[Whi50],[Coh73]
and [RS82].

The primary objectives of this paper are as follows:

  * i)

extend the classical notions of _expansion_ and _collapse_ to allow for
_infinite_ sequences of such moves,

  * ii)

generalize the definition of an _elementary expansion/collapse_ so that it can
be applied to a broader class of topological spaces, and

  * iii)

use i) and ii) to obtain compactification theorems which allow us to add “nice
boundaries” to a variety of spaces.

We are particularly interested in obtaining $\mathcal{Z}$-compactifications,
which play an important role in geometric and algebraic topology and geometric
group theory. Here is a sample theorem.

######  Theorem 0.1.

Every (infinitely) collapsible locally finite simplicial, cubical, or CW
complex is $\mathcal{Z}$-compactifiable.

A concrete application can be found in the second author’s dissertation
[Gul23], where it is shown that every locally finite CAT(0) cube complex $X$
is collapsible. The resulting $\mathcal{Z}$-boundary (referred to as the
_cubical boundary_) shares properties with the visual boundary, but retains
some of the combinatorial structure of $X$. It can be viewed as a hybrid of
the visual and Roller boundaries.

None of the goals listed above are entirely new. Allowing an _infinite_
sequence expansions or collapses is a fairly obvious idea, but its appearance
in the literature is surprisingly rare. One exception is a recent arXiv
posting by Adiprasito and Funar and [AF21]. A different approach, was
developed by Siebenmann [Sie70], under which only finitely many moves are
permitted, but a move can consist of an infinite discrete collection of
_simultaneous_ elementary expansions or collapses. The resulting theory, known
as _infinite simple homotopy theory_ , is important but different from what we
do here. For example, the move just described can only be performed on a
noncompact space. By contrast, our approach can _end_ with a noncompact space,
but at each step along the way the spaces are typically compact.

To accomplish the second goal, we leave the world of combinatorially
constructed spaces to define a “compact topological collapse” (or simply
_compact collapse_) of one space onto another. These compact collapses are the
same as Ferry’s _contractible retractions_ [Fer80]. We will comment more on
that work in Section 4.

Our main compactification theorems generalize some well known constructions.
One of those is the $\mathcal{Z}$-compactification of the mapping telescope of
an inverse sequence of finite polyhedra, which played a key role in work by
Chapman and Siebenmann [CS76, §4]. Another is the addition of the visual
boundary to a proper CAT(0) space $X$, as described by Bridson and Haefliger
[BH99, §II.8.5]. They do not mention $\mathcal{Z}$-sets; that connection, and
its usefulness, was noted by Bestvina [Bes96, Example 1.2(ii)]. Yet another is
the proof, as presented by Jeremy Brazus in his blog _Wild Topology_ , of a
theorem by Borsuk, showing that all _dendrites_ are contractible. Each of
these is a special case of results presented in Sections 5 and 2.

The existence of an infinite sequence of compact expansions $C_{0}\swarrow
C_{1}\swarrow C_{3}\swarrow C_{4}\swarrow\cdots$, beginning with a compactum
$C_{0}$, leads to a inverse sequence of retractions

$C_{0}\overset{r_{1}}{\longleftarrow}C_{1}\overset{r_{2}}{\longleftarrow}C_{2}\overset{r_{3}}{\longleftarrow}\cdots$

a direct sequence of inclusions

$C_{0}\overset{s_{0}}{\longrightarrow}C_{1}\overset{s_{1}}{\longrightarrow}C_{2}\overset{s_{2}}{\longrightarrow}\cdots$

and a set $X=\cup_{i=0}^{\infty}C_{i}$, whose topology is ambiguous. Our
compactification strategy relies on the inverse limit construction, which
determines the correct (but not always the obvious) topology for $X$. This
realization led to an unexpected, but interesting, fourth major theme of this
article:

  * iv)

explore the topology of inverse sequences of retractions and corresponding
direct sequences of inclusion maps.

This topic provides another connection to the literature. In [MP15], Marsh and
Prajs studied inverse sequences of retractions from a different perspective.
Whereas we use these sequences for constructing and compactifying noncompact
spaces, they _begin_ with a compact space $Y$ and look for situations when $Y$
can be realized as the inverse limit of compact subspaces
$C_{0},C_{1},C_{2},\cdots$ where the bonding maps are retractions.

The layout of this paper is as follows. Section 1 is general topology. We
identify topologies $\mathcal{T}_{I}$ and $\mathcal{T}_{D}$ on $X$—one is
strongly associated with the inverse sequence; the other with the direct
sequence. We prove a useful theorem that shows when they agree; then we use
the inverse limit construction to define a canonical compactification of
$\left(X,\mathcal{T}_{I}\right)$ that is used throughout the paper. In Section
2, we bring these abstractions to life with a variety of examples. In Section
3 we review the classical theory of expansions and collapses, and develop
terminology and notation that allows for infinitely many such moves. In
Section 4 we develop generalizations of the concepts in Section 3. In Section
5 we apply all of this to prove a variety of compactification theorems; of
particular interest are $\mathcal{Z}$-compactifications. We conclude the paper
with a variety of applications.

##  1\. The general setting

In the most general setting, we are interested in spaces that can be expressed
as monotone unions of compact metric spaces $C_{0}\subseteq C_{1}\subseteq
C_{2}\subseteq\cdots$ where, for each $i>0$, there is a retraction
$r_{i}:C_{i}\rightarrow C_{i-1}$. This situation is concisely described with
an an inverse sequence

(1.1)
$C_{0}\overset{r_{1}}{\longleftarrow}C_{1}\overset{r_{2}}{\longleftarrow}C_{2}\overset{r_{3}}{\longleftarrow}\cdots$

There is a corresponding direct sequence

(1.2)
$C_{0}\overset{s_{0}}{\longrightarrow}C_{1}\overset{s_{1}}{\longrightarrow}C_{2}\overset{s_{2}}{\longrightarrow}\cdots$

where the $s_{i}$ are inclusions. There is also a set
$X=\cup_{i=0}^{\infty}C_{i}$. Sequences (1.1) and (1.2) suggest preferred
topologies for $X$, but those topologies may not agree. We will return to that
issue after establishing some notation.

For $i\leq j$, let $r_{ij}:C_{i}\leftarrow C_{j}$ denote the obvious
composition, with special cases $r_{ii}=\operatorname*{id}_{C_{i}}$ and
$r_{(i-1)i}=r_{i}$. Note that: each $r_{ij}$ is a retraction; $r_{ij}\circ
r_{jk}=r_{ik}$; and $\left.r_{ik}\right|_{C_{j}}=r_{ij}$ whenever $i\leq j\leq
k$. Thus, we may define retractions (on the level of sets)
$r_{i\infty}:C_{i}\leftarrow X$ by letting
$\left.r_{i\infty}\right|_{C_{j}}=r_{ij}$. Similarly, let
$s_{ij}:C_{i}\rightarrow C_{j}$ denote the inclusion map or, equivalently, the
composition of maps from (1.2). Let $s_{i\infty}:C_{i}\rightarrow X$ also
denote inclusion.

One natural way to topologize $X$ is with the _coherent topology_ with respect
to the collection $\left\\{C_{i}\right\\}_{i=0}^{\infty}$. Under that
topology, $U\subseteq X$ is open if and only if $U\cap C_{i}$ is open in
$C_{i}$ for all $i$. This is equivalent to the _final topology_ on $X$ with
respect to the collection $\left\\{s_{i\infty}:C_{i}\rightarrow
X\right\\}_{i=0}^{\infty}$ which, by definition, is the finest topology on $X$
under which all of these functions are continuous. We denote this topology by
$\mathcal{T}_{D}$ since it is closely related to the _direct_ sequence (1.2).

An equally natural topology for $X$ is with the _initial topology_ with
respect to the collection $\left\\{r_{i\infty}:X\rightarrow
C_{i}\right\\}_{i=0}^{\infty}$. By definition, this is the coarsest topology
under which these functions are all continuous, i.e., it is the topology
generated by the subbasis consisting of sets of the form
$r_{i\infty}^{-1}\left(U\right)$, where $U$ is an open subset of $C_{i}$ for
some $i$. We denote this topology by $\mathcal{T}_{I}$ since it is most
naturally associated with the _inverse_ sequence (1.1). In the special cases
under consideration here, this subbasis is actually a basis.

######  Lemma 1.1.

Given the above setup, the defining subbasis

$\mathcal{S}_{I}=\left\\{r_{i\infty}^{-1}\left(U\right)\mid U\text{ is an open
subset of }C_{i}\text{ for some }i\right\\}$

for $\mathcal{T}_{I}$ is a basis for $\mathcal{T}_{I}$.

###### Proof.

It suffices to show that if $U$ and $V$ are open subsets of $C_{i}$ and
$C_{j}$ respectively, then $r_{i\infty}^{-1}\left(U\right)\cap
r_{j\infty}^{-1}\left(V\right)\in\mathcal{S}_{I}$. First note that, if $i=j$,
then $r_{i\infty}^{-1}\left(U\right)\cap
r_{i\infty}^{-1}\left(V\right)=r_{i\infty}^{-1}\left(U\cap
V\right)\in\mathcal{S}$. If $i<j$, then
$r_{i\infty}^{-1}\left(U\right)=r_{j\infty}^{-1}\left(r_{ij}^{-1}\left(U\right)\right)$,
so the desired conclusion follows from the previous case. ∎

######  Remark 1.2.

Since the inclusion maps in sequence (1.2) do not depend upon the bonding maps
in (1.1), $\mathcal{T}_{D}$ is essentially well-defined by the collection
$\left\\{C_{i}\right\\}_{i=0}^{\infty}$. By contrast, the topology
$\mathcal{T}_{I}$ depends on the bonding maps in (1.1). Example of this
phenomenon can be found in Examples 2.1-2.3 of the following section..

A few more definitions and elementary observations will be useful as we
proceed.

######  Lemma 1.3.

Given the above setup,

  1. (1)

$\mathcal{T}_{D}$ is as fine as or finer than $\mathcal{T}_{I}$; i.e.,
$\mathcal{T}_{I}\subseteq\mathcal{T}_{D}$, and

  2. (2)

in both $\left(X,\mathcal{T}_{D}\right)$ and $\left(X,\mathcal{T}_{I}\right)$,
each $C_{i}$ is a subspace of $X$.

###### Proof.

To verify 1), let $U$ be an open subset of $C_{i}$ and
$r_{i\infty}^{-1}\left(U\right)$ be the corresponding basic open set in
$\mathcal{T}_{I}$. Since
$\left.r_{i\infty}\right|_{C_{i}}=\operatorname*{id}_{C_{i}}$, it follows that
$r_{i\infty}^{-1}\left(U\right)\cap C_{i}=U$. So, for all $j\leq i$,
$r_{i\infty}^{-1}\left(U\right)\cap C_{j}=U\cap C_{j}$ is open in $C_{j}$. For
$j>i$, $r_{i\infty}^{-1}\left(U\right)\cap C_{j}=r_{ij}^{-1}\left(U\right)$,
which is open in $C_{j}$ since $r_{ij}$ is continuous. Hence
$r_{i\infty}^{-1}\left(U\right)\in\mathcal{T}_{D}$.

For 2), let $\mathcal{T}_{C_{i}}$ denote the original topology on $C_{i}$,
$\mathcal{T}_{C_{i}}^{\prime}$ the subspace topology on $C_{i}$ induced by
$\mathcal{T}_{D}$, and $\mathcal{T}_{C_{i}}^{{}^{\prime\prime}}$ the subspace
topology on $C_{i}$ induced by $\mathcal{T}_{I}$.

If $U^{\prime}\in\mathcal{T}_{C_{i}}^{\prime}$ there exists
$U\in\mathcal{T}_{D}$ such that $U^{\prime}=U\cap C_{i}$, so by definition of
$\mathcal{T}_{D}$, $U^{\prime}\in\mathcal{T}_{C_{i}}$. Conversely, if
$U^{\prime}\in\mathcal{T}_{C_{i}}$, then
$r_{i\infty}^{-1}\left(U^{\prime}\right)\in\mathcal{T}_{I}$ so, by 1),
$r_{i\infty}^{-1}\left(U^{\prime}\right)\in\mathcal{T}_{D}$. Since
$r_{i\infty}^{-1}\left(U^{\prime}\right)\cap C_{i}=U^{\prime}$,
$U^{\prime}\in\mathcal{T}_{C_{i}}^{\prime}$.

The latter half of the above argument assures that
$\mathcal{T}_{C_{i}}\subseteq\mathcal{T}_{C_{i}}^{{}^{\prime\prime}}$. For the
reverse inclusion, let $r_{j\infty}^{-1}\left(U\right)$ be a basic open subset
in $\mathcal{T}_{I}$ where $U\in\mathcal{T}_{C_{j}}$ and
$U^{\prime\prime}=r_{j\infty}^{-1}\left(U\right)\cap
C_{i}\in\mathcal{T}_{C_{i}}^{{}^{\prime\prime}}$. By again following the
argument in 1), we know that $r_{j\infty}^{-1}\left(U\right)\cap C_{i}$ is
either $U\cap C_{i}$ or $r_{ij}^{-1}\left(U\right)$, depending on whether
$i\leq j$ or $i>j$. In both cases we have an element of $\mathcal{T}_{C_{i}}$.
∎

######  Notation 1.

Going forward, we will often simplify notation by letting $X_{D}$ and $X_{I}$
denote $\left(X,\mathcal{T}_{D}\right)$ and $\left(X,\mathcal{T}_{I}\right)$.
Similarly, when $A\subseteq X$, we will let $\operatorname*{Int}_{D}A$ and
$\operatorname*{Int}_{I}A$ to denote the interiors of $A$ in those spaces.
Lemma 1.3 allows us to view $C_{i}$ as a subspace of $X_{D}$ or $X_{I}$,
without need for specification.

We say that $A\subseteq X$ is $j$_-stationary_(with respect to the sequence
$\left\\{C_{i},r_{i}\right\\}$) if $A\subseteq C_{j}$ and
$r_{j\infty}^{-1}\left(A\right)=A$. The following is immediate.

######  Lemma 1.4.

Given the above setup,

  1. (1)

every subset of a $j$-stationary set is $j$-stationary,

  2. (2)

every union of $j$-stationary sets is $j$-stationary, and

  3. (3)

if $A\subseteq X$ is $j$-stationary, then $r_{jk}^{-1}\left(A\right)=A$ and
$r_{k\infty}^{-1}\left(A\right)=A$ for all $k\geq j$. In particular, $A$ is
$k$-stationary, for all $k\geq j$.

We say that $A\subseteq X$ is $j$-_insulated_ if there exists a $j$-stationary
open neighborhood of $A$ in $C_{j}$. Lemma 1.4 gives the following.

######  Lemma 1.5.

If $A\subseteq X$ is $j$-insulated, then $A$ is $k$-insulated for all $k\geq
j$.

We will say that $A\subseteq X$ is _stationary_ [_insulated_] if there is some
$j\in\mathbb{N}$ for which $A$ is $j$-stationary [$j$-insulated]. If a set
$\left\\{x\right\\}$ has one of these properties, we attribute that property
to the point $x$.

######  Lemma 1.6.

If each $x\in C_{i}$ is insulated, then $C_{i}$ is insulated.

###### Proof.

By hypothesis, for each $x\in C_{i}$, there exists a $j_{x}\geq 0$ and a
$j_{x}$-stable open neighborhood $U_{j_{x}}$ of $x$ in $C_{j_{x}}$. When
$j_{x}\leq i$, Lemma 1.4 and continuity imply that $U_{j_{x}}$ is open in
$C_{i}$ whereas, when $j_{x}>i$, $U_{j_{x}}\cap C_{i}$ is open in $C_{i}$. So,
by compactness, there exists a finite sequence $j_{x_{1}}\leq\cdots\leq
j_{x_{k}}$ such that the sets $U_{j_{x_{1}}},\cdots,U_{j_{x_{k}}}$ cover
$C_{i}$. If $j_{x_{k}}\leq j$, then $C_{i}$ is a union of $j$-stationary open
subsets $U_{j_{x_{1}}},\cdots,U_{j_{x_{k}}}$, and is therefore $j$-insulated.
If $j_{x_{k}}>i$, then each of $U_{j_{x_{1}}},\cdots,U_{j_{x_{r}}}$ is
$j_{x_{k}}$-stationary and open in $C_{j_{x_{k}}}$. Since
$C_{i}\subseteq\cup_{r=1}^{k}U_{j_{x_{r}}}$ it follows that $C_{i}$ is
$j_{x_{k}}$-insulated. ∎

######  Theorem 1.7.

Given the above setup, the following are equivalent.

  1. (1)

Each point of $X$ is insulated.

  2. (2)

$X=\cup_{i=0}^{\infty}\operatorname*{Int}_{I}C_{i}$.

  3. (3)

$\mathcal{T}_{I}=\mathcal{T}_{D}$.

###### Proof.

We begin by proving 1)$\Leftrightarrow$2). For the forward implication, note
that Lemma 1.6 guarantees that for each $i$, there exists an integer $k>i$ and
an open subset $U_{k}\subseteq C_{k}$ that contains $C_{i}$ and for which
$r_{k\infty}^{-1}\left(U_{k}\right)=U_{k}$. This implies that $U_{k}$ is a
basic open set in $\mathcal{T}_{I}$, so
$C_{i}\subseteq\operatorname*{Int}_{I}C_{k}$. The desired conclusion follows.

For the reverse implication, let $x\in X$. Then there exists $j\geq 0$ such
that $x\in\operatorname*{Int}_{I}C_{j}$, so there exists a basic open
neighborhood $r_{k\infty}^{-1}\left(U_{k}\right)$ of $x$ contained in $C_{j}$.
If $k\geq j$, then $r_{k\infty}^{-1}\left(U_{k}\right)\subseteq C_{k}$, so $x$
is $k$-insulated. If $k<j$, then $r_{kj}^{-1}\left(U_{k}\right)$ is open in
$C_{j}$ and
$r_{j\infty}^{-1}\left(r_{kj}^{-1}\left(U_{k}\right)\right)=r_{k\infty}^{-1}\left(U_{k}\right)\subseteq
C_{j}$, so $x$ is $j$-insulated.

Next we show 2)$\Rightarrow$3). We already know
$\mathcal{T}_{I}\subseteq\mathcal{T}_{D}$, so let $V\in\mathcal{T}_{D}$ and
$x\in V$. By hypothesis, there is a $j\geq 0$ and an open neighborhood $U_{j}$
of $x$ in $C_{j}$ such that $r_{j\infty}^{-1}\left(U_{j}\right)=U_{j}$. Since
$V\in\mathcal{T}_{D}$, $V\cap C_{j}$ is open in $C_{j}$, so $V_{j}:=V\cap
U_{j}$ is open in $C_{j}$. Moreover,
$r_{j\infty}^{-1}\left(V_{j}\right)=V_{j}$, so $V_{j}$ is a basic open set in
$\mathcal{T}_{I}$. Since $x\in V_{j}\subseteq V$, $V\in\mathcal{T}_{I}$.

Lastly, we use a contrapositive argument to show 3)$\Rightarrow$1). Suppose
$x\in X$ is not insulated, and let $i_{0}$ be the minimum integer for which
$x\in C_{i_{0}}$. For each $k\geq i_{0}$, let
$\mathcal{B}_{k}=\\{B_{k,j}\\}_{j=k}^{\infty}$ be a countable neighborhood
basis at $x$ of open subsets in $C_{k}$, and for each $k\geq i_{0}$ and $j\geq
k$, let $B_{k,j}^{\prime}=r_{k\infty}^{-1}\left(B_{k,j}\right)$ be the
corresponding basic open neighborhood of $x$ in
$\left(X,\mathcal{T}_{I}\right)$. If $\mathcal{B}_{k}^{\prime}$ denotes
$\\{B_{k,j}^{\prime}\\}_{j=k}^{\infty}$ then
$\cup_{k=i_{0}}^{\infty}\mathcal{B}_{k}^{\prime}$ is a neighborhood basis at
$x$ in $\left(X,\mathcal{T}_{I}\right)$. Since $x$ is not insulated, for each
$k\geq i_{0}$ and $j\geq k$, we may choose $x_{kj}\in B_{k,j}^{\prime}-C_{j}$.
Let $A$ be the set of all such $x_{k,j}$ and notice that $x\notin A$ and
$A\cap C_{i}$ is finite, for all $i\geq 0$. As such, $A$ is closed in
$\left(X,\mathcal{T}_{D}\right)$. But, since every element of
$\cup_{k=i_{0}}^{\infty}\mathcal{B}_{k}^{\prime}$ intersects $A$, $A$ is not
closed in $\left(X,\mathcal{T}_{I}\right)$. ∎

######  Corollary 1.8.

If any of the conditions in Theorem 1.7 hold, then
$X=\cup_{i=0}^{\infty}\operatorname*{int}_{D}C_{i}$. The converse of this
assertion is false.

###### Proof.

The assertion is immediate from condition 3), while part iii) of Example 2.3
in Section 2 is a counterexample to the converse. ∎

When the conditions in Theorem 1.7 hold, we say that
$\left\\{C_{i},r_{i}\right\\}$ is _fully insulated_. Example 2.5 from Section
2 shows that, for this to be true, it is not enough to assume that each
$C_{i}$ is stationary.

###  1.1. Inverse and direct limits associated with $X$

We now provide further justification for associating the topologies
$\mathcal{T}_{D}$ and $\mathcal{T}_{I}$ with the direct and inverse sequences
(1.2) and (1.1). In particular, we look at the roles played by the limits of
those sequences.

The _direct limit_ of (1.2) can be defined by

$\underrightarrow{\lim}\left\\{C_{i},s_{i}\right\\}:=(\sqcup_{i=0}^{\infty}C_{i}\times\left\\{i\right\\})/\sim$

where $\left(x.i\right)\sim\left(y,j\right)$ if there exist
$k\geq\max\left\\{i,j\right\\}$ such that
$s_{ik}\left(x\right)=s_{jk}\left(y\right)$. The _direct limit topology_ is
the quotient topology induced by

$q:\sqcup_{i=0}^{\infty}C_{i}\times\left\\{i\right\\}\rightarrow(\sqcup_{i=0}^{\infty}C_{i}\times\left\\{i\right\\})/\sim$

where the left-hand side is given the disjoint union topology. In the case at
hand, there is a bijection
$f:X\rightarrow\underrightarrow{\lim}\left\\{C_{i},s_{i}\right\\}$ taking
$x\in X$ to the equivalence class $\left[\left(x,i\right)\right]$, where $i$
is chosen sufficiently large that $x\in C_{i}$. If we use $f$ to pull back the
direct limit topology, we obtain a topology on $X$ under which $U\subseteq X$
open if and only if $U\cap C_{i}$ is open in $C_{i}$ for all $i$. In other
words, we arrive at $\mathcal{T}_{D}$, which we think of, informally, as the
direct limit topology on $X$.

There is a similar, but more subtle, relationship between $\mathcal{T}_{I}$
and the inverse limit of (1.1). Recall that

$\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}:=\left\\{\left(x_{0},x_{1},x_{2},\cdots\right)\mid
r_{i}\left(x_{i}\right)=x_{i-1}\text{ for all }i>0\right\\}$

is topologized as a subspace of ${\textstyle\prod_{i=0}^{\infty}}C_{i}$, where
the latter is given the product topology. There is a natural injection
$e:X\rightarrow\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}$ defined by
$e\left(x\right)=\left(r_{i\infty}\left(x\right)\right)_{i=0}^{\infty}$. Under
both $\mathcal{T}_{D}$ and $\mathcal{T}_{I}$, the coordinate function are
continuous, so $e$ is a continuous injection.

######  Theorem 1.9.

The function $e:X\rightarrow\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}$
is an embedding if and only if $X$ is given the topology $\mathcal{T}_{I}$;
$e$ need not be surjective, but $e\left(X\right)$ is dense in
$\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}$.

###### Proof.

Clearly, there is a unique topology on $X$ under which $e$ is an embedding. We
show that $\mathcal{T}_{I}$ is that topology.

Let $U=r_{j\infty}^{-1}\left(V_{j}\right)$ be a basic open set in
$\mathcal{T}_{I}$, where $V_{j}$ is an open subset of $C_{j}$, and let
$\pi_{j}:{\textstyle\prod_{i=0}^{\infty}}C_{i}\rightarrow C_{j}$ be the
projection map. Then

$\displaystyle e\left(U\right)$ $\displaystyle=\left\\{e\left(x\right)\mid
x\in X\text{ and }r_{j\infty}\left(x\right)\in V_{j}\right\\}$
$\displaystyle=e\left(X\right)\cap\pi_{j}^{-1}\left(V_{j}\right)$
$\displaystyle=e\left(X\right)\cap\left(\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}\cap\pi_{i}^{-1}\left(V_{i}\right)\right)$

Since $\pi_{i}^{-1}\left(V_{i}\right)$ is open in the product topology, then
$\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}\cap\pi_{i}^{-1}\left(V_{i}\right)$
is open in $\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}$, so
$e\left(U\right)$ is open in $e\left(X\right)$. Therefore, the corestriction
of $e$ to $e\left(X\right)$ is an open map, hence a homeomorphism.

To see that $e\left(X\right)$ is dense in
$\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}$, notice that if $x\in
C_{i}$ then $r_{j\infty}\left(x\right)=x$ for all $j\geq i$. As such,
$e\left(X\right)$ is the set of all eventually constant sequences
$\left(x_{0},\cdots,x_{k-1},c,c,c,\cdots\right)$ such that $c\in C_{k}$,
$x_{k-1}=r_{k}\left(c\right)$, and $x_{i}=r_{i+1}\left(x_{i+1}\right)$ for all
$i<k-1$. Clearly, every element of
$\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}$ is the limit of a sequence
of such points.

Non-surjectivity of $e$ occurs in most of the examples presented in Section 2.
∎

In most cases of interest to us, $X_{I}$ (and therefore $X_{D}$) will be
noncompact. We have assumed that each $C_{i}$ is a compact metric space, so
${\textstyle\prod_{i=0}^{\infty}}C_{i}$ is compact and metrizable and
$\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}$ is a closed subset of
${\textstyle\prod_{i=0}^{\infty}}C_{i}$. See, for example, [Dug78]. If we
adopt a standard convention, that a _compactification_ of a space $S$ is a
compact Hausdorff space that contains $S$ as a dense subspace, then
$\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}$ is a compactification of
$e\left(X\right)$. Letting
$R:=\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}-e\left(X\right)$, we
have

$\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}=\overline{e\left(X\right)}=e\left(X\right)\sqcup
R$

By following a standard procedure we arrive at:

######  Corollary 1.10.

The space $X_{I}$ is separable and metrizable and admits a compactification
$\overline{X_{I}}=X_{I}\sqcup R$ which is homeomorphic to
$\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}$.

###### Proof.

As a subspace of the compact metrizable space
${\textstyle\prod_{i=0}^{\infty}}C_{i}$, $e\left(X\right)$ is separable and
metrizable—so $X_{I}$ is separable and metrizable. To obtain
$\overline{X_{I}}=X_{I}\sqcup R$, extend $e:X_{I}\rightarrow e\left(X\right)$
to the function
$\overline{e}:\overline{X_{I}}\rightarrow\overline{e\left(X\right)}$ which is
the identity on $R$. Then topologize $\overline{X}$ by declaring
$U\subseteq\overline{X_{I}}$ to be open if and only if
$\overline{e}\left(U\right)$ is open in $\overline{e\left(X\right)}$. ∎

Under the most general of the circumstances described so far, $X_{I}$ and
$\overline{X_{I}}$ can be quite unusual. Since we are primarily interested in
locally compact metric spaces and their compactifications, the following
observation is relevant.

######  Proposition 1.11.

For an arbitrary compactification $\overline{X}$ of a Hausdorff space $X$, $X$
is open in $\overline{X}$ if and only if it is locally compact. For the
compactifications $\overline{e\left(X\right)}=e\left(X\right)\sqcup R$ and
$\overline{X_{I}}=X_{I}\sqcup R$ described above, these conditions hold
whenever the conditions in Theorem 1.7 are satisfied.

###### Proof.

For the forward implication of the initial assertion, note that $\overline{X}$
is locally compact and Hausdorff. Hence, local compactness is truly a local
property [Mun00, Th.29.2], so every open subset of $\overline{X}$ is locally
compact. The reverse implication is well known but less obvious; see [Dug78,
XI.8.3].

To prove the latter assertion, let $x\in X_{I}$ be arbitrary. By hypothesis,
there exists a $j\geq 0$ and an open neighborhood $U_{j}$ of $x$ in $C_{j}$
such that $r_{j\infty}^{-1}\left(U_{j}\right)=U_{j}$. Since $C_{j}$ is
compact, it is locally compact, so $U_{j}$ contains a compact neighborhood
$V_{j}$ of $x$ in the space $C_{j}$. But
$r_{j\infty}^{-1}\left(V_{j}\right)=V_{j}$, so $V_{j}$ is also a neighborhood
of $x$ in $X_{I}$. ∎

######  Remark 1.12.

Example 2.2 shows that the converse of the second sentence of Proposition 1.11
is false. One might ask whether local compactness of
$\left(X,\mathcal{T}_{I}\right)$ implies that
$\mathcal{T}_{I}=\mathcal{T}_{D}$. The same example shows that the answer is
“no”.

##  2\. Examples

In this section, we present a collection examples that fit the above setup and
exhibit a variety of properties. As additional topics are developed, we will
often refer back to these examples.

######  Example 2.1.

Let $P=\left\\{p_{1},p_{2},\cdots\right\\}$ be a sequence of points with the
discrete topology. For $i\geq 1$, let $C_{i}$ be the cone on
$\left\\{p_{1},\cdots,p_{i}\right\\}$ with cone point $q$ and let
$C_{0}=\left\\{q\right\\}$. Then, as a set, $X=\cup_{i=0}^{\infty}C_{i}$ is
the cone on $P$, and we have a direct system of inclusion maps

$C_{0}\overset{s_{0}}{\longrightarrow}C_{1}\overset{s_{1}}{\longrightarrow}C_{2}\overset{s_{2}}{\longrightarrow}\cdots$

For each $i>0$, let $r_{i}:C_{i-1}\leftarrow C_{i}$ be the retraction which
maps segment segment $\overline{qp}_{i}$ linearly onto $\overline{qp}_{i-1}$
by sending $p_{i}$ to $p_{i-1}$. This gives an inverse sequence

$C_{0}\overset{r_{1}}{\longleftarrow}C_{1}\overset{r_{2}}{\longleftarrow}C_{2}\overset{r_{3}}{\longleftarrow}\cdots$

With a little effort, one can show that $X_{D}$ does not have a countable
local basis at $q$, and hence is nonmetrizable. By contrast, $X_{I}$ is not
only metrizable, but planar. In fact, $\overline{X_{I}}=X_{I}\sqcup R$ is
homeomorphic to the left-hand compactum shown in Figure 1 with $R\approx(0,1]$
corresponding to the limiting arc minus the point $q$.

######  Example 2.2.

Take the same collection $\left\\{C_{i}\right\\}$ as in the previous example,
but now define $r_{i}^{\prime}:C_{i-1}\leftarrow C_{i}$ to take the entire
segment $\overline{qp}_{i}$ to the point $q$. The direct system and $X_{D}$
remain the same, but with these bonding maps, $X_{I}$ is homeomorphic to the
planar continuum on the right in Figure 1. In this case, $R$ is the empty set.

Figure 1. $\overline{X_{I}}=X_{I}\sqcup R$ in Examples 2.1 (left) and 2.2
(right)

######  Example 2.3.

For each $i\geq 0$, let $C_{i}$ be the closed interval $\left[0,i\right]$,
subdivided into $i$ subintervals of length $1$. Define three different
sequences of retractions of $C_{i}$ onto $C_{i-1}$.

  * i)

Let $r_{i}:C_{i-1}\leftarrow C_{i}$ be the retraction that maps
$\left[i-1,i\right]$ to its left endpoint.

  * ii)

Let $r_{i}^{\prime}:C_{i-1}\leftarrow C_{i}$ be the retraction that maps
$\left[i-1,i\right]$ linearly onto $\left[i-2,i-1\right]$ with
$r_{i}\left(i\right)=i-1$; and

  * iii)

let $r_{i}^{\prime\prime}:C_{i-1}\leftarrow C_{i}$ be the retraction that maps
$\left[i-1,i\right]$ linearly onto $\left[0,i-1\right]$ with
$r_{i}\left(i\right)=0$.

In all three cases, $X_{D}$ is homeomorphic to the standard ray $[0,\infty)$.
In case i), $X_{I}$ is also homeomorphic to $[0,\infty)$, and
$\overline{X_{I}}$ is the one-point compactification $\left[0,\infty\right]$.
In case ii), $X_{I}$ is again homeomorphic to $[0,\infty)$, but
$\overline{X_{I}}$ is a version of the “topologist’s $\sin\left(1/x\right)$
curve” with $R$ corresponding to the limit arc.

In case iii), $\overline{X_{I}}$ is homeomorphic to a famous indecomposable
continuum $B$, known as the _bucket handle_ or the _Brouwer-Janiszewski-
Knaster continuum_. Here $R$ and $X_{I}$ are both dense in $\overline{X_{I}}$,
and the topology $\mathcal{T}_{I}$ on $[0,\infty)$ is not the standard one.
Open sets are most easily seen by taking preimages of open subsets of $B$
under a rather exotic injection $e:[0,\infty)\rightarrow B$ under which the
sequence $\left\\{e\left(n\right)\right\\}_{n\in\mathbb{Z}^{+}}$ converges to
$e\left(0\right)$.111We are abusing notion slightly by calling this map $e$.
As such, every neighborhood of $0$ in $\mathcal{T}_{I}$ contains an unbounded
sequence of open intervals. Other points in
$\left([0,\infty),\mathcal{T}_{I}\right)$ have similar neighborhoods. See
Figure 2. One takeaway from this example is that, even when $X_{D}$ is
metrizable—or even a locally finite simplicial complex—the space $X_{I}$ can
be quite different. The example also illustrates the significance of the
insulation conditions; sequences i) and ii) are fully insulated while sequence
iii) is not.

Figure 2. $\overline{X_{I}}=X_{I}\sqcup R$ in Example 2.3

######  Example 2.4.

For each integer $i\geq 0$, let $I^{i}$ be the standard $i$-cube

$\begin{array}[c]{ccc}\underbrace{\left[0,1\right]\times\cdots\left[0,1\right]}&\times&0\times
0\times\cdots\\\ i\text{-times}&&\end{array}$

where the additional $0$-coordinates are added for notational convenience, and
so that $\left\\{I^{i}\right\\}_{i=0}^{\infty}$ forms a nested sequence. For
$i>0$, let $r_{i}:I^{i-1}\leftarrow I^{i}$ be the projection map. We may view
$X=\cup_{i=0}^{\infty}I^{i}$ as a subset $I_{0}^{\omega}$ of the countable
infinite product $I^{\omega}$; in particular, $I_{0}^{\omega}$ consists of all
sequences of elements of $\left[0,1\right]$ that end in a sequence of $0$’s.
An argument similar to the one alluded to in Example 2.1 shows that, under the
$\mathcal{T}_{D}$ (which is commonly used when $I_{0}^{\omega}$ is viewed as a
CW complex), $I_{0}^{\omega}$ is nonmetrizable. By contrast, $\mathcal{T}_{I}$
is the same as the subspace topology induced by the product topology on
$I^{\omega}$. Additionally, $\overline{I_{0}^{\omega}}$ (as defined in Section
1.1) is equivalent to taking the closure of $I_{0}^{\omega}$ in $I^{\omega}$.
These observation require some work since, by definition, $e\left(X\right)$
and $\underleftarrow{\lim}\left\\{I^{i},r_{i}\right\\}$ are somewhat thin
subsets of $I^{0}\times I^{1}\times I^{2}\times\cdots$.

For readers interested in non-locally finite _cube complexes_ ,
$I_{0}^{\omega}$ is a beautifully simple example. In that context, it is
common to topologize $I_{0}^{\omega}$ using the path length metric $d$ induced
by the Euclidean metric on each $I^{i}$. Interestingly, that topology is
neither $\mathcal{T}_{D}$ nor $\mathcal{T}_{I}$. Being metric, it it is
clearly not $\mathcal{T}_{D}$. On the other hand, if $\mathbf{e}_{i}$ denotes
the point with all coordinates $0$ except for $1$ as its
$\left(i+1\right)^{\text{st}}$ coordinate, then
$\left\\{\mathbf{e}_{i}\right\\}$ converges to $\left(0,0,0,\cdots\right)$
under $\mathcal{T}_{I}$, but has no limit points under $d$ since
$d\left(\mathbf{e}_{i},\mathbf{e}_{j}\right)=\sqrt{2}$ for all $i\neq j$.

######  Example 2.5.

Let $C_{0}$ be the origin $\mathbf{0}$ in $\mathbb{R}^{2}$, $C_{j}$ the closed
circular disk of radius $j$ centered at $\left(0,j\right)$, and
$r_{j}:C_{j-1}\leftarrow C_{j}$ the map taking $x$ to the first point where
the ray $\overrightarrow{x\mathbf{0}}$ intersects $C_{j-1}$. Then $X_{I}$ is
the open upper half-plane plus the point $\mathbf{0}$ with the subspace
topology, and $\overline{X_{I}}$ can be seen as the result of adding an open
interval $\left(0,\pi\right)$ to $X_{I}$, with the one-point for each ray in
$X_{I}$ emanating from $\mathbf{0}$. Another way to envision this example is
to embed the construction in the unit disk with $\left(0,-1\right)$ as the
point of tangency. In this approach, $\overline{X_{I}}$ can be realized as the
unit disk and the remainder $R$ as $\mathbb{S}^{1}-\\{\left(0,-1\right)\\}$.

In this example, each $C_{i}$ is stationary but not insulated. As a
consequence, Theorem 1.7 cannot be applied, and we end up with a space $X_{I}$
that is not locally compact and not an open subset of $\overline{X_{I}}$. The
reader is invited to compare $\mathcal{T}_{I}$ and $\mathcal{T}_{D}$.

######  Example 2.6.

Let $C_{0}$ be origin and, for $j>0$, let $C_{j}$ be the convex hull of

$A_{j}=\left\\{\exp\left(\frac{k\pi i}{2^{j-1}}\right)\mid
k=0,1,2,\cdots,2^{j}-1\right\\}$

in $\mathbb{R}^{2}$, and $r_{j}:C_{j}\rightarrow C_{j-1}$ the radial
projection of $C_{j}$ onto $C_{j-1}$. Then $X_{I}$ is the open unit disk
together with the countable dense set $\cup_{j\geq 1}A_{j}$ of points on the
unit circle (with the subspace topology); and $\overline{X_{I}}$ is
homeomorphic to the closed disk. As in the previous example, $X_{I}$ is not
locally compact and is not open in $\overline{X_{I}}$.

######  Example 2.7.

Let $\left(Y,d\right)$ be a proper CAT(0) space and $y_{0}$ an arbitrary base
point. Let $C_{0}=\left\\{y_{0}\right\\}$ and, for each $i>0$,
$C_{i}=B_{d}\left[y_{0},i\right]$, the closed ball of radius $i$. Let
$r_{i}:C_{i-1}\leftarrow C_{i}$ be the geodesic retraction taking $y\in C_{i}$
to the nearest point in $C_{i-1}$. As a set, $X=Y$, and since each $C_{i}$ is
insulated, $\mathcal{T}_{I}=\mathcal{T}_{D}$. With a little effort, one sees
that these topologies agree with original metric topology and that
$\overline{X_{I}}$ is homeomorphic to the compactification
$\overline{Y}:=Y\cup\partial_{\infty}Y$ obtained by adjoining the visual
sphere at infinity to $Y$ via the “cone topology”.

Note also that the only points of $C_{i}=B_{d}\left[y_{0},i\right]$ with
(possibly) non-singleton preimages in $C_{i+1}=B_{d}\left[y_{0},i+1\right]$
lie in the metric sphere $S_{d}[y_{0},i]$. It follows that

$R=\overline{X_{I}}-X_{I}=\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}-e\left(X\right)$

is equal to the inverse limit of the sequence

$\left\\{y_{0}\right\\}\overset{f_{1}}{\longleftarrow}S_{d}[y_{0},1]\overset{f_{2}}{\longleftarrow}S_{d}[y_{0},2]\overset{f_{3}}{\longleftarrow}\cdots$

where $f_{i}$ is the restriction of $r_{i}$. See [BH99, §II.8] for more
details related to this construction. We will revisit to this example in
Section 6.2 and prove some additional properties.

######  Example 2.8.

Let

(2.1)
$K_{0}\overset{f_{1}}{\longleftarrow}K_{1}\overset{f_{2}}{\longleftarrow}K_{2}\overset{f_{3}}{\longleftarrow}\cdots$

be an inverse sequence of compact metric spaces and, for each integer $i>0$,
let

$\operatorname*{Map}\left(f_{i}\right):=K_{i}\times\left[i-1,i\right]\sqcup
K_{i-1}/\sim$

(the _mapping cylinder_) be the quotient space, where $\sim$ is generated by
the rule $\left(x,i-1\right)\sim f_{i}\left(x\right)$. By identifying
$K_{i-1}$ with its image in $\operatorname*{Map}\left(f_{i}\right)$ and
$K_{i}$ with the image of $K_{i}\times\left\\{i\right\\}$, we view $K_{i-1}$
and $K_{i}$ as subsets of $\operatorname*{Map}\left(f_{i}\right)$ and refer to
them as the _range end_ and the _domain end_ of
$\operatorname*{Map}\left(f_{i}\right)$. There is a retraction of __
$\operatorname*{Map}\left(f_{i}\right)$ onto $K_{i-1}$ (often called the
_mapping cylinder collapse_) which sends $\left(x,t\right)$ to
$f\left(x\right)$.

Let $C_{0}=K_{0}$ and for each $i>0$, let

$C_{i}=\operatorname*{Map}\left(f_{1}\right)\cup_{K_{1}}\operatorname*{Map}\left(f_{2}\right)\cup_{K_{2}}\cdots\cup_{K_{i-1}}\operatorname*{Map}\left(f_{i}\right)$

By applying the mapping cylinder collapses, we get retractions
$r_{i}:C_{i-1}\leftarrow C_{i}$ which can be assembled into an inverse
sequence of compact metrizable spaces

$C_{0}\overset{r_{1}}{\longleftarrow}C_{1}\overset{r_{2}}{\longleftarrow}C_{2}\overset{r_{3}}{\longleftarrow}\cdots$

In this case

$X=\cup_{i=0}^{\infty}C_{i}=\operatorname*{Map}\left(f_{1}\right)\cup_{K_{1}}\operatorname*{Map}\left(f_{2}\right)\cup_{K_{2}}\cdots$

is called the _inverse mapping telescope_ of sequence (2.1).

By reasoning similar to the previous example,
$\mathcal{T}_{I}=\mathcal{T}_{D}$ and
$R=\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}-e\left(X\right)$ is the
inverse limit of sequence (2.1). We will return to this compactification in
Section 2.

##  3\. Expansions and collapses in the simplicial category

For our favorite applications of the above setting, we will need to place
conditions on the retraction maps in our inverse sequences. Necessity will
direct us to the notion of a _compact collapse_ —a generalization of a
_simplicial collapse_. We will also need to all for infinitely many such moves
(or their inverses). Before proceeding to that level of generality, we spend
some time reviewing the classical theory and adjusting it to allow for
infinitely many moves.

###  3.1. Simplicial complexes: preliminaries

An _abstract simplicial complex_ $K$ is a pair $\left(V,\mathcal{S}\right)$
where $V$ is a set and $\mathcal{S}$ is a collection of nonempty finite
subsets of $V$ containing all singletons and satisfying the rule: if
$\sigma\in\mathcal{S}$ and $\varnothing\neq\tau\subseteq\sigma$, then
$\tau\in\mathcal{S}$. The elements of $V$ are called _vertices_ and the
elements of $\mathcal{S}$ the _simplices_ of $K$. (Typically, we do not
distinguish between a vertex $v$ and the simplex $\left\\{v\right\\}$.) If
$\tau\subseteq\sigma\in\mathcal{S}$, we call $\tau$ is a _face_ of $\sigma$;
if $\tau\subsetneq\sigma$, it is called a _proper face_. If $\sigma$ contains
exactly $n+1$ vertices, it is called an $n$_-dimensional simplex_ (or simply
an $n$_-simplex_), and we sometimes denote it by $\sigma^{n}$. A _subcomplex_
of $K$ is a simplicial complex $L$, each of whose simplices is a simplex of
$K$; this will be denoted $L\leq K$. As a key example, if $\sigma$ is a
simplex of $K$ then $\partial\sigma\leq K$ is made up of all proper faces of
$\sigma$. A simplicial complex is _finite_ if its vertex set is finite; it is
_locally finite_ if each vertex belongs to only finitely many simplices.

We define the _geometric realization_ $\left|K\right|$ of
$K=\left(V,\mathcal{S}\right)$ (also called the _polyhedron corresponding to_
$K$) to be a subset of the real vector space $W_{V}$ with basis $V$. By
endowing $W_{V}$ with the standard inner product metric $d_{V}$, each subset
$F\subseteq V$ containing $k<\infty$ elements, is the basis for a subspace
$W_{F}$ that is isometric to $\mathbb{R}^{k}$ with the Euclidean metric. For
$\sigma=\left\\{v_{0},\cdots,v_{n}\right\\}\in\mathcal{S}$, the _geometric
realization_ $\left|\sigma\right|$ is the convex hull of
$\left\\{v_{0},\cdots,v_{n}\right\\}$ in $W_{\sigma}$ with the subspace
metric. Now define $\left|K\right|\subseteq W_{v}$ to be the union of the
geometric realizations of its simplices.

Note that

$\left|\sigma\right|=\left\\{\left.\sum\nolimits_{i=0}^{n}x_{i}v_{i}\right|0\leq
x_{i}\leq 1\text{ and }\sum\nolimits_{i=0}^{n}x_{i}=1\right\\}$

If $\mathbf{x}=\sum\nolimits_{i=0}^{n}x_{i}v_{i}\in\left|\sigma\right|$, the
$x_{i}$ are called the _barycentric coordinates_ of $\mathbf{x}$. The
_barycenter_ of $\left|\sigma\right|$ is the point
$\widehat{\sigma}=\sum\nolimits_{i=0}^{n}\frac{1}{n+1}v_{i}$. We will refer to
any space isometric to $\left|\sigma\right|$ as a _standard Euclidean_
$n$_-simplex_ , in which case the distance between any pair of distinct
vertices is $\sqrt{2}$.222Some authors rescale the metric so that the distance
between pairs of vertices in a standard Euclidean $n$-simplex is $1$. For our
purposes, that is unnecessary.

There are multiple ways to topologize $\left|K\right|$. One can give
$\left|K\right|$ the initial topology with respect to the individual
simplices; in other words, $U\subseteq\left|K\right|$ is open if and only if
$U\cap\left|\sigma\right|$ is open for every $\sigma\in\mathcal{S}$. This is
often called the _weak topology_ on $\left|K\right|$, so we denote it
$\mathcal{T}_{W}$. Alternatively, one can use the inner product metric $d_{V}$
on $W_{V}$ to induce the subspace topology on $\left|K\right|$. We call this
the _barycentric topology_ and denote it $\mathcal{T}_{B}$. It is a classical
fact that $\mathcal{T}_{W}=\mathcal{T}_{B}$ if and only if $K$ is locally
finite; otherwise, $\mathcal{T}_{W}\supsetneq\mathcal{T}_{B}$ and the identity
map
$\operatorname*{id}_{\left|K\right|}:\left(\left|K\right|,\mathcal{T}_{W}\right)\rightarrow\left(\left|K\right|,\mathcal{T}_{B}\right)$
is a homotopy equivalence. See [Dow52].

######  Remark 3.1.

For the purposes of metric geometry and geometric group theory, the _path
length metric on_ $\left|K\right|$ (also called the _intrinsic metric_ in
[BH99, Chap.I.7]) is often more useful than the barycentic metric. For
example, the latter is always bounded and thus ignores the large-scale
geometry of $\left|K\right|$. Nevertheless, for each
$\mathbf{x}\in\left|K\right|$, there is an $\varepsilon>0$ such that balls
centered at $\mathbf{x}$ and having radius $<\varepsilon$ are identical under
the two metrics. As such they generate the same topologies.

Abstract simplicial complexes $K_{1}$ and $K_{2}$ are _isomorphic_ , denoted
$K_{1}\cong K_{2}$, if there exists a bijection $f:V_{1}\rightarrow V_{2}$
between vertex sets that induces a bijection between the sets of simplices.
Clearly $f$ induces a corresponding homeomorphism
$\left|f\right|:\left|K_{1}\right|\rightarrow\left|K_{2}\right|$ provided
$\left|K_{1}\right|$ and $\left|K_{2}\right|$ are topologized in the same
manner. Polyhedra $\left|L_{1}\right|$ and $\left|L_{2}\right|$ are called
_piecewise-linearly (PL) homeomorphic_ if there exist isomorphic subdivisions
$L_{1}^{\prime}$ and $L_{2}^{\prime}$ of $L_{1}$ and $L_{2}$.

A _triangulation_ of a topological space $X$ is a homeomorphism
$h:\left|K\right|\rightarrow X$ for some simplicial complex $K$. In that case,
we say $X$ is _triangulated_ by $K$.

###  3.2. Simplicial expansions and collapses

An _elementary combinatorial collapse_ $K\overset{e}{\searrow}L$ of $K$ onto a
subcomplex $L$ can be performed when $K$ contains an $n$-simplex $\sigma^{n}$
that has a _free_ face $\tau^{n-1}$ (meaning that $\tau^{n-1}$ is a face of no
other simplices of $K$). The subcomplex $L$ is obtained by deleting
$\sigma^{n}$ and $\tau^{n-1}$ from $K$. This combinatorial move can be
realized topologically by a particularly nice type of deformation retraction
of $\left|K\right|$ onto $\left|L\right|$. Due to its importance in this
paper, we will carefully describe one such deformation retraction in the
following paragraphs. Before that, we note that the reverse of an elementary
combinatorial collapse is called an _elementary combinatorial expansion_ and
denoted $L\overset{e}{\nearrow}K$.

To define the deformation retraction alluded to above, label the vertices of
$\sigma^{n}$ by $\left\\{v_{0},\cdots,v_{n}\right\\}$ such that
$\left\\{v_{0},\cdots,v_{n-1}\right\\}$ are the vertices of $\tau^{n-1}$, and
(abusing notation slightly) choose a standard Euclidean realization of
$\left|\sigma^{n}\right|$ in $\mathbb{R}^{n}$ with
$\left|\tau^{n-1}\right|\subseteq\mathbb{R}^{n-1}\times\left\\{0\right\\}$
having barycenter at the origin and so that
$v_{n}=\left(0,\cdots,0,\sqrt{\frac{n+1}{n}}\right)$. Let $L^{\prime}$ be the
simplicial complex $\partial\sigma^{n}-\tau^{n-1}$. For each
$\mathbf{x\in}\left|\sigma^{n}\right|$, define $r\left(\mathbf{x}\right)$ be
the unique point where the ray $\overrightarrow{-v_{n}\mathbf{x}}$ intersects
$\left|L^{\prime}\right|$. Then define
$J:\left|\sigma^{n}\right|\times[0,1]\longrightarrow\left|\sigma^{n}\right|$
to be the map for which $\left.J\right|_{\mathbf{x\times}\left[0,1\right]}$ is
the constant speed linear path from $\mathbf{x}$ to
$r\left(\mathbf{x}\right)$. In particular, $J$ is fixed on
$\left|L^{\prime}\right|$,
$J_{0}=\operatorname*{id}_{\left|\sigma^{n}\right|}$, and $J_{1}=r$ is a
retraction of $\left|\sigma^{n}\right|$ onto $\left|L^{\prime}\right|$. Now
define $H:\left|K\right|\times[0,1]\longrightarrow\left|L\right|$ to be $J$ on
$\left|\sigma^{n}\right|\times[0,1]$ while keeping all points of
$\left|L\right|$ fixed throughout. By a mild abuse of notation, we also let
$r$ denote the map $H_{1}:\left|K\right|\rightarrow\left|L\right|$ and call it
an _elementary simplicial collapse_. The corresponding inclusion map
$s:\left|L\right|\hookrightarrow\left|K\right|$ is called an _elementary
simplicial expansion_. It is easy to see that $H$ is continuous under all of
the above-mentioned topologies on $\left|K\right|$.

Now define a _combinatorial collapse_ of $K$ to a subcomplex $L_{0}$ to be a
finite sequence of elementary combinatorial simplicial collapses
$K=L_{n}\overset{e}{\searrow}L_{n-1}\overset{e}{\searrow}\cdots\overset{e}{\searrow}L_{0}$,
a process denoted loosely by $K\searrow L_{0}$. The concatenation of the
corresponding deformation retractions, gives a deformation retraction of
$\left|K\right|$ onto $\left|L_{0}\right|$ whose end result
$r:\left|K\right|\rightarrow\left|L_{0}\right|$ is a retraction that we call a
_simplicial collapse_. Note that $r$ depends on the specific sequence of
elementary collapses used to take $K$ onto $L_{0}$ but, regardless of the
sequence chosen, it is a homotopy equivalence, with homotopy inverse
$\left|L_{0}\right|\hookrightarrow\left|K\right|$. Under these circumstances
we also write
$L_{0}\overset{e}{\nearrow}L_{1}\overset{e}{\nearrow}\cdots\overset{e}{\nearrow}K$,
or simply $L_{0}\nearrow K$, and say that $L_{0}$ _expands_ _combinatorially_
to $K$ and that $\left|L_{0}\right|$ _expands simplicially_ to
$\left|K\right|$.

######  Example 3.2.

It is a standard (and useful) fact that, for every finite simplicial complex
$K$, there exists a triangulation of $\left|K\right|\times\left[0,1\right]$ by
a simplicial complex $L$ with subcomplexes $K\times\left\\{i\right\\}$
corresponding to $\left|K\right|\times\left\\{i\right\\}$ for $i=0,1$, and
such that $\left|L\right|$ collapses simplicially to
$\left|K\times\left\\{i\right\\}\right|$ for $i=0,1$.

A simplicial complex $K$ is _combinatorially collapsible_ if it contains a
vertex $v$ such that $K\searrow\left\\{v\right\\}$ or, equivalently, there
exists a sequence of elementary combinatorial expansions
$\left\\{v\right\\}=L_{0}\overset{e}{\nearrow}L_{1}\overset{e}{\nearrow}\cdots\overset{e}{\nearrow}L_{n}=K$
where each $L_{i}\leq K$. In this case, we say that $\left|K\right|$ is
_simplicially collapsible_.

######  Remark 3.3.

The geometric realization of a collapsible simplicial complex is clearly
contractible. In fact, collapsibility is often viewed as the combinatorial
analog of contractibility. Nevertheless, there are well known examples, such
as the _topologist’s dunce hat_ and _Bing’s house with two rooms_ , that are
contractible but not collapsible. Also clear, from the above definition, is
that a collapsible simplicial complex is necessarily finite. We will soon
relax key definitions to eliminate that requirement.

######  Remark 3.4.

The subject of _simple homotopy theory_ defines a pair of finite simplicial
complexes $K$ and $L$ to be simple homotopy equivalent if, beginning with $K$,
there exists a finite sequence of elementary expansions and collapses ending
in $L$. As an example, the dunce hat and Bing’s house with two rooms—despite
the fact that they are not collapsible—are simple homotopy equivalent to a
vertex. By contrast, there exist homotopy equivalent simplicial complexes that
are not simple homotopy equivalent. Siebenmann [Sie70] has developed a simple
homotopy theory for infinite complexes, but it differs substantially from what
we will do here. Simple homotopy theory does not play a significant role in
this paper.

###  3.3. Infinite simplicial expansions and collapses

We now extend the classical notions of collapses and expansions to allow for
infinitely many moves. We say that a simplicial complex $K$ _collapses
combinatorially to_ $L$, or equivalently, that $L$ _expands combinatorially
to_ $K$, if there exists a (possibly infinite) sequence of elementary
combinatorial simplicial expansions
$L=L_{0}\overset{e}{\swarrow}L_{1}\overset{e}{\swarrow}L_{2}\overset{e}{\swarrow}\cdots$
through subcomplexes of $K$, such that $\cup_{i=0}^{\infty}L_{i}=K$. In this
case, we write $K\searrow L$ or $L\nearrow K$. When extra clarity is desired
(and infinitely many moves are used) we will write
$K\overset{\infty}{\searrow}L$ or $L\overset{\infty}{\nearrow}K$. We say that
$K$ is _combinatorially collapsible_ if $\left\\{v\right\\}\nearrow K$, with
the option of specifying $\left\\{v\right\\}\overset{\infty}{\nearrow}K$, when
infinitely many expansions are required.

Although these new definitions are straightforward and natural, some oddities
result. For example, a simplicial complex can be collapsible without having a
free face. For example, the standard triangulation of $\mathbb{R}$ is
collapsible. Despite such oddities, the generalized definitions are useful. To
begin, notice that the assumption $L\nearrow K$ allows one to construct a
direct sequence of geometric realizations and inclusion maps

(3.1)
$\left|L\right|=\left|L_{0}\right|\overset{s_{1}}{\hookrightarrow}\left|L_{1}\right|\overset{s_{2}}{\hookrightarrow}\left|L_{2}\right|\overset{s_{3}}{\hookrightarrow}\cdots$

and an inverse sequence of (deformation) retractions.

(3.2)
$\left|L\right|=\left|L_{0}\right|\overset{r_{1}}{\longleftarrow}\left|L_{1}\right|\overset{r_{2}}{\longleftarrow}\left|L_{2}\right|\overset{r_{3}}{\longleftarrow}\cdots$

We say that $\left|K\right|$ _collapses simplicially to_ $\left|L\right|$ and
that $\left|L\right|$ _expands simplicially to_ $\left|K\right|$. In this
context, we always assume that the retractions used in sequence (3.2) are the
simplicial collapses defined earlier. When $\left|K\right|$ is endowed with
the barycentric or weak topology, a straightforward application of Whitehead’s
Theorem [Hat02] ensures that, $\left|L\right|\hookrightarrow\left|K\right|$ is
a homotopy equivalence—as it was in the finite case. If we opt for the
topology $\mathcal{T}_{I}$ on $\left|K\right|$, which is often the case in
this paper, proving the inclusion is a homotopy equivalence is more subtle;
that will be done in Section 5.2. For now we observe that, in a case of
primary interest, everything works out nicely.

######  Proposition 3.5.

Given the above setup, if $K$ is locally finite, then the topologies
$\mathcal{T}_{I}$, $\mathcal{T}_{D}$, $\mathcal{T}_{W}$ and $\mathcal{T}_{B}$
on $\left|K\right|$ are the same.

###### Proof.

By definition and [Dow52], we know
$\mathcal{T}_{D}=\mathcal{T}_{W}=\mathcal{T}_{B}$, so it suffices to show that
$\mathcal{T}_{D}=\mathcal{T}_{I}$. By Theorem 1.7, we need only show that
every $x\in\left|K\right|$ is insulated. Toward that end, choose $j$
sufficiently large that every simplex of $\left|K\right|$ that contains $x$ is
contained in $\left|L_{j}\right|$ and let $U$ be the open star neighborhood of
$x$ in $\left|L_{j}\right|$ (and hence in $\left|K\right|$). We claim that
$r_{j\infty}^{-1}\left(U\right)=U$. If not, let $k>j$ be the smallest integer
such that $r_{jk}^{-1}\left(U\right)$ contains a point not in $U$. Then
$\left|L_{k}\right|=\left|L_{k-1}\right|\cup\tau$ and, by definition, $r_{k}$
maps $\tau$ onto a union of its proper faces. By our assumption, one of those
faces lies in $U$, and hence contains $x$. But then, $\tau$ is a simplex of
$K$ that contains $x$ and hence lies in $U$, a contradiction. ∎

##  4\. Compact expansions and collapses

We are ready to expand the notions of expansion and collapse beyond the
simplicial category. That, in itself, is not new. There are well-known
extensions to:

  * •

polyhedral complexes, where collapses are performed on convex polyhedra;

  * •

$n$-manifolds, where topological $n$-balls are collapsed onto
$\left(n-1\right)$-balls in their boundary (a process known as _shelling_);
and

  * •

CW complexes, where $k$-cells of various dimensions are collapsed onto
potentially crumpled $\left(k-1\right)$-cells in their attaching spheres.

The CW version is often viewed as the most general context for defining
expansions and collapses. We will argue otherwise, by formulating a version
that includes all of the above as special cases. It makes room for new
examples and allows some well known constructions to be viewed as special
cases of a broader theory.

###  4.1. Track-faithful deformation retractions

Let $H:C\times\left[0,1\right]\rightarrow C$ be a self-homotopy of a space $C$
and, for each $t\in\left[0,1\right]$, let $H_{t}:C\rightarrow C$ be defined by
$H_{t}\left(x\right)=H\left(x,t\right)$. For our purposes, we are only
interested cases where $H_{0}=\operatorname*{id}_{C}$. We call $H$ a
_deformation retraction_ of $C$ onto $A$ if $A=H_{1}\left(C\right)$ and
$\left.H_{t}\right|_{A}=\operatorname*{id}_{A}$ for all $t$.333Some authors
call this a _strong_ deformation retraction.In that case, we call the
retraction of $C$ onto $A$, obtained by restricting the range of $H_{1}$, the
_culmination_ of $H$.

The _track_ of a point $x\in C$ under $H$ is the path
$\lambda_{x}:\left[0,1\right]\rightarrow C$ defined by
$\lambda_{x}\left(t\right)=H_{t}\left(x\right)$. We sometimes abuse
terminology by conflating $\lambda_{x}$ with its image. Call $H$ _track-
faithful_ if, for every $x\in C$ and $t\in\left[0,1\right]$,
$H_{1}\left(H_{t}\left(x\right)\right)=H_{1}\left(x\right)$. This condition is
equivalent to requiring that, if two tracks intersect, then they end at the
same point. We are particularly interested in track-faithful deformation
retractions.

######  Example 4.1.

If $C=X\times\left[0,1\right]$ and
$H_{t}\left(x,s\right)=\left(x,\left(1-t\right)s\right)$ then $H$ is a track-
faithful deformation retraction of $C$ onto $X\times\left\\{0\right\\}$.

######  Example 4.2.

For any map $f:X\rightarrow A$, the deformation retraction described in
Example 4.1 induces a track-faithful deformation retraction
$\operatorname*{Map}\left(f\right)$ onto $A$. By contrast, whenever $f$ is a
homotopy equivalence, $\operatorname*{Map}\left(f\right)$ deformation retracts
onto $X$, but in some cases there is no track-faithful deformation retraction.
An example with $X=A=S^{1}$ can be obtained by comparing [Gui01] with Theorem
5.7 in this paper.

######  Example 4.3.

Given a top-dimensional face $\tau^{n-1}$ of an $n$-simplex $\sigma^{n}$ and
the subcomplex $L^{\prime}=\partial\sigma^{n}-\tau^{n-1}$, the deformation
retractions of $\left|\sigma^{n}\right|$ onto $\left|L^{\prime}\right|$ and
$\left|K\right|$ onto $\left|L\right|$, described in Section 3 and used to
define an elementary simplicial collapse, are track-faithful.

######  Example 4.4.

Let $\square^{n}=\left[-1,1\right]^{n}$, $B^{n-1}\
=\square^{n-1}\times\left\\{1\right\\}$, and
$J^{n-1}=\partial\square^{n}-\operatorname*{int}B^{n-1}$. Let
$\mathbf{p}=\left(0,\cdots,0,2\right)$ and for each
$\mathbf{x}\in\square^{n}$, let $\mathbf{q}_{x}$ be the unique point
$\overrightarrow{\mathbf{px}}\cap J$. This gives a continuously varying family
of segments $\overline{\mathbf{xq}}_{\mathbf{x}}\subseteq\square^{n}$ (some of
them degenerate) connecting points of $\square^{n}$ to points of $J^{n-1}$. By
choosing constant speed parametrization
$\lambda_{\mathbf{x}}:\left[0,1\right]\rightarrow\overline{\mathbf{xq}}_{\mathbf{x}}$
of these segments we obtain a track-faithful deformation retraction of
$\square^{n}$ onto $J^{n-1}$. We will refer to culmination as an _elementary
cubical collapse_.

######  Example 4.5.

A contraction of a space $C$ to a point $p$ is track-faithful if and only if
$p$ is fixed by the contraction.

######  Example 4.6.

The retractions $r,r^{\prime}:\left[-1,1\right]\rightarrow\left[0,1\right]$
defined by $r\left(t\right)=\left|t\right|$ and

$r^{\prime}\left(t\right)=\left\\{\begin{tabular}[c]{cc}$0$&if $t\leq 0$\\\
$t$&if $t\geq 0$\end{tabular}\ \ \ \ \ \right.$

can each be realized as the culmination of a deformation retraction. But,
whereas $r^{\prime}$ is the culmination of a track-faithful deformation
retraction, $r$ is not.

######  Lemma 4.7.

Let $H_{t}$ be a track-faithful deformation retraction of $C$ onto $A$
culminating in $r:C\rightarrow A$. Then

  1. (1)

for each $x\in A$, $r^{-1}\left(x\right)$ is contractible, and

  2. (2)

if $x\in\operatorname*{Int}A$, then $r^{-1}\left(x\right)=\left\\{x\right\\}$.

###### Proof.

For the first assertion, note that if $y\in r^{-1}\left(x\right)$, then
$\lambda_{y}\subseteq r^{-1}\left(x\right)$. Therefore, the restriction of $H$
to $r^{-1}\left(x\right)\times\left[0,1\right]$ is a contraction of
$r^{-1}\left(x\right)$. For the second assertion, let
$x\in\operatorname*{Int}A$ and suppose there exists $y\in
r^{-1}\left(x\right)$ such that $y\neq x$. Then there exists
$t_{0}\in\left[0,1\right]$ such that $\lambda_{x}\left(t_{0}\right)\in A$ and
$\lambda_{x}\left(t_{0}\right)\neq x$. Let
$y_{0}=\lambda_{x}\left(t_{0}\right)$ and note that
$\lambda_{y_{0}}=\left\\{y_{0}\right\\}$. Our hypothesis ensures that
$r\left(y_{0}\right)=H_{1}\left(y_{0}\right)=H_{1}\left(H_{t_{0}}\left(y\right)\right)=x$,
but since $y_{0}\in A$, $r\left(y_{0}\right)=y_{0}$. ∎

######  Lemma 4.8.

If $H:C\times\left[0,1\right]\rightarrow C$ is a track faithful deformation
retraction of $C$ onto $A$ and $J:A\times\left[0,1\right]\rightarrow A$ is a
track faithful deformation retraction of $A$ onto $B$, then the concatenation
of $H$ and $J$ is a track faithful deformation retraction of $C$ onto $B$ with
$\left(H\ast J\right)_{1}=J_{1}\circ H_{1}$.

###### Proof.

By “concatenation of $H$ and $J$”, we mean the homotopy $H\ast
J:C\times\left[0,1\right]\rightarrow C$ defined by

$(H\ast
J)\left(x,t\right)=\left\\{\begin{tabular}[c]{cc}$H\left(x,2t\right)$&if
$0\leq t\leq\frac{1}{2}$\\\ $J(x,2t-1)$&if $\frac{1}{2}\leq t\leq
1$\end{tabular}\ \right.$

Verification of the desired properties is straightforward. ∎

######  Remark 4.9.

There are other reasonable candidates for the definition of track-faithful
homotopy. For example, one could require that the track of every point lying
on $\lambda_{x}$ be contained in $\lambda_{x}$. We have chosen the weakest
condition that is required in our proofs.

###  4.2. Compact expansions and collapses

Given a metric space $C$, a map $\rho:C\rightarrow A$ is an _compact collapse_
if

  * a)

$\rho$ is the culmination of track-faithful deformation retraction, and

  * b)

$C-A$ has compact closure in $C$.

In this case, we write $C\overset{\rho}{\searrow}A$ or
$A\overset{\rho}{\swarrow}C$; we call the inclusion map
$\theta:A\hookrightarrow C$ an _compact expansion_ and write
$A\overset{\theta}{\nearrow}C$ or $C\overset{\theta}{\nwarrow}A$.

######  Notation 2.

As an aid to the reader, we will use the letter “rho”  _only_ to denote
compact collapses and “theta” only to denote compact expansions.

######  Example 4.10.

Examples 4.1, 4.2 and 4.5 give compact collapses when the corresponding spaces
are compact. The elementary simplicial and cubical collapses in Example 4.3
and 4.4 are also compact collapses.

######  Lemma 4.11.

If $C\overset{\rho}{\searrow}A$ and the pair $\left(C,A\right)$ is
homeomorphic to $\left(C^{\prime},A^{\prime}\right)$, then
$C^{\prime}\overset{\rho^{\prime}}{\searrow}A^{\prime}$ for an appropriately
chosen retraction $\rho^{\prime}$.

######  Corollary 4.12.

Let $B^{n}$ be a topological $n$-ball and $B^{n-1}$ be an $\left(n-1\right)$
ball tamely embedded in $\partial B^{n}$. Then
$B^{n}\overset{\rho}{\searrow}B^{n-1}$.

###### Proof.

By standard methods in geometric topology
$\left(B^{n},B^{n-1}\right)\approx\left(\square^{n},J^{n-1}\right)$. So the
claim follows from Lemma 4.11 and Example 4.4. ∎

######  Lemma 4.13.

Suppose $X$ can be expressed as a union $X=C\cup Y$ of closed subspaces;
$A=C\cap Y$; and $C\overset{\rho}{\searrow}A$. Then
$X\overset{\rho}{\searrow}Y$. More generally, if $C\overset{\rho}{\searrow}A$
and $f:A\rightarrow Y$ is continuous, then
$Y\cup_{f}C\overset{\rho}{\searrow}Y$.

###### Proof.

The deformation retraction needed for the initial statement follows from the
Pasting Lemma. For the second statement, note that the track-faithful
deformation retraction of $C$ onto $A$ induces a track-faithful deformation
retraction of the induced quotient space $C/\sim$ onto
$f\left(A\right)=A/\sim$. Since points in $f\left(A\right)$ have constant
tracks, this deformation retraction extends via the identity homotopy over
$Y$. ∎

######  Corollary 4.14.

Every elementary CW collapse (in the sense of [Coh73, p.14]) of a CW complex
$K$ to a subcomplex $L$ can be realized as a compact collapse
$K\overset{\rho}{\searrow}L$.

###### Proof.

Apply Corollary 4.12 and Lemma 4.13 to the definition. ∎

The reader may have wondered why we did not include the word “elementary” in
our definition of compact collapse. The reason is that, unlike in the more
prescriptive simplicial and CW versions, any finite sequence of compact
collapses can be combined into a single compact collapse.

######  Lemma 4.15.

Suppose

$X_{n}\overset{\rho_{n}}{\searrow}X_{n-1}\overset{\rho_{n-1}}{\searrow}\cdots\overset{\rho_{1`}}{\searrow}X_{0}$

is a finite sequence of compact collapses. Then
$X_{n}\overset{\rho}{\searrow}X_{0}$, where
$\rho=\rho_{1}\circ\cdots\circ\rho_{n}$.

######  Remark 4.16.

Note that every base point preserving contraction of a compactum $X$ is a
compact collapse. If we were to allow “noncompact collapses”, then all
reasonably nice contractible spaces would be collapsible in a single move.
That extreme level of flexibility is unlikely to yield a useful theory. By
contrast, we will show that a theory built upon compact expansions and
collapses can be quite useful.

###  4.3. Infinite compact expansions and collapses

We are ready for our most general treatment of expansions and collapses—one
based on compact expansions and collapses and allowing infinitely many moves.

We say that a metric space $X$ _collapses compactly_ to a subspace $L$, or
equivalently, that $L$ _expands compactly to_ $X$, if there exists a (possibly
infinite) sequence of compact expansions
$L=L_{0}\overset{e}{\swarrow}L_{1}\overset{e}{\swarrow}L_{2}\overset{e}{\swarrow}\cdots$
through subsets of $X$, such that $\cup_{i=0}^{\infty}L_{i}=X$. In this case,
we write $X\searrow L$ or $L\nearrow X$. For extra clarity we sometimes write
$X\overset{\infty}{\searrow}L$ or $L\overset{\infty}{\nearrow}X$. We say that
$X$ is _compactly collapsible_ if $\left\\{p\right\\}\nearrow X$, for some
$p\in X$. Note that, although we can combine finite pieces of an infinite
sequence of compact expansions into single compact expansion, an infinite
sequence will remain infinite.

######  Remark 4.17.

It is worth commenting further on the work in [Fer80] mentioned in the
introduction. Ferry’s motivation for studying compact collapses (which he
called _contractible retractions_) was an attempt to develop a “simple
homotopy theory for compact metric spaces”. In analogy with the traditional
theory, a pair of compact metric spaces would be _simple homotopy equivalent_
if one could be converted to the other by a finite sequence of compact
expansions and collapses. That idea was undone by the strength of Ferry’s own
work. He proved that two compacta satisfy that version of simple homotopy
equivalent, if and only if they are homotopy equivalent—a nice theorem, but
one which nullified the utility of a new definition. Our work resurrects a
portion that program, but with different goals. By utilizing _infinite_
sequences of compact expansions and collapses, we will obtain some new
results. We are particularly interested in $\mathcal{Z}$-compactifications.
For that purposes, Ferry’s definition is a perfect fit.

##  5\. Compactifications

A primary goal of this paper is to develop general methods for obtaining nice
compactifications. We are especially interested in
$\mathcal{Z}$-compactifications of spaces arising in geometric group theory
and geometric and algebraic topology; but much the machinery can be applied
more widely to obtain various interesting compactifications. To allow for the
broadest set of applications, we begin with a flexible set of definitions. The
reader is warned that the terminology surrounding this topic—including key
definitions, such as $\mathcal{Z}$-set—is not consistent across the
literature. Rather than trying to blend with everything written previously, we
will choose definitions that works for our purposes and causes as little
confusion as possible. In many cases, conflicts in terminology vanish in the
presence of certain “niceness” assumptions on the spaces to which it applies.
For a thorough treatment of the various definitions, with proofs of
equivalences under appropriate conditions, the reader is directed to [Anc].

For this paper, there is no loss in restricting the following definitions to
cases where the ambient space $Y$ is compact and metrizable. In that case, we
can let $d$ be any metric inducing the correct topology on $Y$. For open cover
$\mathcal{U}$ of $Y$, maps $f,g:Y\rightarrow Y$ are $\mathcal{U}$_-close_ if,
for every $x\in Y$ there exists a $U\in\mathcal{U}$ such that
$f\left(x\right)$ and $g\left(x\right)$ are contained in $U$. These maps are
called $\mathcal{U}$_-homotopic_ if there exists a homotopy $H_{t}$ between
$f$ and $g$, each of whose tracks is contained in an element of $\mathcal{U}$.
In that case, $H_{t}$ is called a $\mathcal{U}$_-homotopy._ Clearly
$\mathcal{U}$-homotopic maps are $\mathcal{U}$-close.

A subset $A$ of $Y$ is called _mapping negligible_ if, for every open cover
$\mathcal{U}$ of $Y$, there is a map $f:Y\rightarrow Y$ that is
$\mathcal{U}$-close to $\operatorname*{id}_{Y}$ and for which
$f\left(Y\right)\subseteq Y-A$. We say that $A$ is _homotopy negligible_ if
there exists a homotopy $H:Y\times\left[0,1\right]\rightarrow Y$ such that
$H_{0}=\operatorname*{id}_{Y}$ and $H_{t}\left(Y\right)\subseteq Y-A$ for all
$t>0$. For an arbitrary open cover $\mathcal{U}$ of $Y$, one can arrange—by
restricting $H$ to a short interval $\left[0,\delta\right]$, then
reparametrizing—that this is accomplished by a $\mathcal{U}$-homotopy. If $A$
is closed and mapping negligible, we call it an $\Omega$_-set_ ; if it is a
closed homotopy negligible, we call it a $\mathcal{Z}$_-set_. One of the
beauties of homotopy negligible sets (and thus $\mathcal{Z}$-sets) is captured
by the following simple observation.

######  Proposition 5.1.

If $A$ is a homotopy negligible in $Y$, then $s:Y-A\hookrightarrow Y$ is a
homotopy equivalence.

###### Proof.

Let $H:Y\times\left[0,1\right]\rightarrow Y$ be a homotopy with
$H_{0}=\operatorname*{id}_{Y}$ and $H_{t}\left(Y\right)\subseteq Y-A$ for all
$t>0$. Define $f:Y\rightarrow Y-A$ by $f\left(x\right)=H_{1}\left(x\right)$.
Then $s\circ f=H_{1}$ which is homotopic to $\operatorname*{id}_{Y}$ by
assumption. Similarly, $f\circ s=\left.H_{1}\right|_{Y-A}$ can be homotoped to
$\operatorname*{id}_{Y-A}$ by applying
$\left.H\right|_{\left(Y-A\right)\times\left[0,1\right]}$. ∎

######  Example 5.2.

The homotopy negligible subsets of a compact manifold with boundary $M^{n}$
are precisely the subsets of $\partial M^{n}$; the $\mathcal{Z}$-sets are the
closed subsets of $\partial M^{n}$.

######  Example 5.3.

For the space $\overline{X}=X\sqcup R$, described in Example 2.1, $R$ is
mapping negligible but not an $\Omega$-set. It is not homotopy negligible even
though the conclusion of Proposition 5.1 holds.

######  Example 5.4.

For the space $\overline{X}=X\sqcup R$, described in Example 2.3ii), $R$ is an
$\Omega$-set but not a $\mathcal{Z}$-set, and the conclusion of Proposition
5.1 fails.

A metric compactification $\overline{X}=X\sqcup R$ of a space $X$ is called:

  * •

a _mapping negligible compactification_ if $R$ is mapping negligible in
$\overline{X}$;

  * •

an $\Omega$_-compactification_ if $R$ is an $\Omega$-set in $\overline{X}$;

  * •

a _homotopy negligible compactification_ if $R$ is homotopy negligible in
$\overline{X}$; and

  * •

a $\mathcal{Z}$_-compactification_ if $R$ is a $\mathcal{Z}$-set. in
$\overline{X}$.

All compactifications in this paper are obtained via the strategy described in
Section 1.1. The question then becomes: When does that procedure yield a
compactification satisfying one (or more) of the above definitions?

###  5.1. Mapping negligible compactifications and $\Omega$-compactifications

Our most general compactification theorem is the following. In the proof,
$\simeq$ denotes homotopic maps, and a composition of bonding maps
$r_{n-1}\circ\cdots\circ r_{m}:C_{m}\rightarrow C_{n}$ is denoted $r_{n}^{m}$.

######  Theorem 5.5.

Let
$C_{0}\overset{r_{1}}{\longleftarrow}C_{1}\overset{r_{2}}{\longleftarrow}C_{2}\overset{r_{3}}{\longleftarrow}\cdots$
be an inverse sequence of compact metric spaces and retractions and let
$X_{I}={\textstyle\bigcup_{i=0}^{\infty}}C_{i}$ endowed with the topology
$\mathcal{T}_{I}$. Adopt all other notational conventions established in
Section 1. Then

  1. (1)

$\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}$ is a mapping negligible
compactification of $e\left(X\right)$,

  2. (2)

equivalently, $\overline{X_{I}}=X_{I}\sqcup R$ is a mapping negligible
compactification of $X_{I}$,

  3. (3)

if the conditions in Theorem 1.7 hold, then these are
$\Omega$-compactifications, and

  4. (4)

if each $r_{i}$ is a homotopy equivalence, then the projection maps
$\pi_{j}:\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}\rightarrow C_{j}$
and their analogs $\pi_{j}^{\prime}:\overline{X_{I}}\rightarrow C_{j}$ are
shape equivalences.

###### Proof.

For the first assertion, apply $e\circ\pi_{i}$ for arbitrarily large $i$; the
second assertion follows immediately. The third is an application of 1.11.

The final assertion is a routine application of the inverse sequence approach
to shape theory, if the $C_{i}$ are assumed to be finite polyhedra—or even
just ANRs. In particular,
$C_{0}\overset{r_{1}}{\longleftarrow}C_{1}\overset{r_{2}}{\longleftarrow}C_{2}\overset{r_{3}}{\longleftarrow}\cdots$
then represents the shape of
$\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}$, and is pro-homotopic to
each constant inverse sequence
$C_{j}\overset{\operatorname*{id}}{\longleftarrow}C_{j}\overset{\operatorname*{id}}{\longleftarrow}\cdots$.
For a brief introduction to that approach, see [Gui16, §3.7]. If the $C_{i}$
are permitted to be arbitrary compacta, a more subtle approach is needed. The
following argument was communicated to us by Jerzy Dydak. It uses the approach
to shapes which considers sets of maps into ANRs.

By [DS78, Th.4.1.5] (definition on p.22), $\\{C_{i},r_{i}\\}$ satisfies the
following “continuity condition”: Any map
$f:\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}\rightarrow K$, where $K$
is an ANR, factors up to homotopy as $f\simeq g\circ\pi_{n}$, where
$g:C_{n}\rightarrow K$, for some $n$. And, if $g\circ\pi_{n}\simeq
h\circ\pi_{n}$, then there is $m>n$ such that $g\circ r_{n}^{m}\simeq h\circ
r_{n}^{m}$.

To show that
$\pi_{0}:\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}\rightarrow C_{0}$
is a shape equivalence, we need to show that every map
$f:\underleftarrow{\lim}\left\\{C_{i},r_{i}\right\\}\rightarrow K$ factors
uniquely, up to homotopy, via $\pi_{0}$. It factors as $g\circ\pi_{n}$ for
some $n\geq 0$, so the homotopy inverse $s:C_{0}\rightarrow C_{n}$ of
$r_{0}^{n}$ satisfies $s\circ\pi_{0}\simeq\pi_{n}$ (as
$r_{0}^{n}\circ\pi_{n}=\pi_{0}$), and therefore $f\simeq\left(g\circ
s\right)\circ\pi_{0}$.

For uniqueness, suppose $g,h:C_{0}\rightarrow K$ satisfy $g\circ\pi_{0}\simeq
h\circ\pi_{0}$. Then for some $n>1$, $g\circ r_{0}^{n}\simeq h\circ r_{0}^{n}$
and composing with $s$ gives $g\simeq h$. The same argument applies to
$\pi_{i}$ for $i>0$. ∎

######  Remark 5.6.

Section 2 contains several examples where the projection maps are homotopy
equivalences and others where they are shape equivalences but not homotopy
equivalences.

###  5.2. Homotopy negligible compactifications and
$\mathcal{Z}$-compactifications

We are ready to prove one of our main theorems.

######  Theorem 5.7.

Let
$C_{0}\overset{\rho_{1}}{\swarrow}C_{1}\overset{\rho_{2}}{\swarrow}C_{2}\overset{\rho_{3}}{\swarrow}\cdots$
is an inverse sequence of compact metric spaces and compact collapses, and let
$X_{I}={\textstyle\bigcup_{i=0}^{\infty}}C_{i}$ endowed with the topology
$\mathcal{T}_{I}$. Adopt all other notational conventions established in
Section 1. Then there is a homotopy
$H:\underleftarrow{\lim}\left\\{C_{i},\rho_{i}\right\\}\times\left[0,1\right]\rightarrow\underleftarrow{\lim}\left\\{C_{i},\rho_{i}\right\\}$
such that
$H_{t}\left(\underleftarrow{\lim}\left\\{C_{i},\rho_{i}\right\\}\right)\subseteq
e\left(X\right)$ for all $t>0$; moreover, for each $i\geq 0$,
$\left.H\right|_{\underleftarrow{\lim}\left\\{C_{i},\rho_{i}\right\\}\times\left[0,\frac{1}{2^{i}}\right]}$
is a compact collapse of
$\underleftarrow{\lim}\left\\{C_{i},\rho_{i}\right\\}$ onto
$e\left(C_{i}\right)$. By letting $J=\overline{e}^{-1}\circ
H\circ\left(\overline{e},\operatorname*{id}\right)$, it follows that

  1. (1)

$\overline{X_{I}}=X_{I}\sqcup R$ is a homotopy negligible compactification of
$X_{I}$,

  2. (2)

$\left.J\right|_{\overline{X}\times\left[0,\frac{1}{2^{i}}\right]}$ is a
compact collapse of $\overline{X_{I}}$ onto $C_{i}$ for each $i\geq 0$,

  3. (3)

$X_{I}$ deformation retracts onto $C_{i}$ for each $i\geq 0$, and

  4. (4)

if the conditions in Theorem 1.7 hold, then $\overline{X_{I}}$ is a
$\mathcal{Z}$-compactification,

###### Proof.

For each $i\geq 1$, let $\phi_{i}:C_{i}\times\left[0,1\right]\rightarrow
C_{i}$ be a track-faithful deformation retraction of $C_{i}$ onto $C_{i-1}$
culminating in $\rho_{i}$. Then define
$H_{0}:C_{0}\times\left[0,1\right]\rightarrow C_{0}$ to be the projection map
and, for $i>0$, inductively define
$H_{i}:C_{i}\times\left[0,1\right]\rightarrow C_{i}$ by

$H_{i}\left(x,t\right)=\left\\{\begin{tabular}[c]{lll}$x$&if&$0\leq t\leq
1/2^{i}$\\\ $\phi_{i}\left(x,2^{i}t-1\right)$&if&$1/2^{i}\leq t\leq
1/2^{i-1}$\\\ $H_{i-1}\left(\rho_{i}\left(x\right),t\right)$&if&$1/2^{i-1}\leq
t\leq 1$\end{tabular}\ \right.$

Using the track-faithful property of the $\phi_{i}$, one can check that the
following diagram commutes

$\begin{array}[c]{cccccccc}C_{0}\times\left[0,1\right]&\overset{\rho_{1}\times\operatorname*{id}}{\longleftarrow}&C_{1}\times\left[0,1\right]&\overset{\rho_{2}\times\operatorname*{id}}{\longleftarrow}&C_{2}\times\left[0,1\right]&\overset{\rho_{3}\times\operatorname*{id}}{\longleftarrow}&C_{3}\times\left[0,1\right]&\overset{\rho_{4}\times\operatorname*{id}}{\longleftarrow}\cdots\\\
H_{0}\downarrow&&H_{1}\downarrow&&H_{2}\downarrow&&H_{3}\downarrow&\\\
C_{0}&\overset{\rho_{1}}{\longleftarrow}&C_{1}&\overset{\rho_{2}}{\longleftarrow}&C_{2}&\overset{\rho_{3}}{\longleftarrow}&C_{3}&\overset{\rho_{4}}{\longleftarrow}\cdots\end{array}$

It therefore induces a map $H$ between the inverse limits of the top and
bottom rows. Since each bonding map in the top row is a product, the inverse
limit of that sequence factors as a product of inverse limits; and since the
second factor in each of those product maps is
$\operatorname*{id}_{\left[0,1\right]}$, the second inverse limit is
canonically homeomorphic to $\left[0,1\right]$. So we have a homotopy

$H:\underleftarrow{\lim}\left\\{C_{i},\rho_{i}\right\\}\times\left[0,1\right]\rightarrow\underleftarrow{\lim}\left\\{C_{i},\rho_{i}\right\\}$

which, by definition, is the restriction of $H_{0}\times H_{1}\times
H_{2}\times\cdots:\prod C_{i}\times\left[0,1\right]\rightarrow$ $\prod C_{i}$.

To see that $H$ satisfies the desired properties, recall that, for each $i\geq
0$, $e\left(C_{i}\right)$ is the set of all points
$\left(x_{0},x_{1},x_{2},\cdots\right)$ in
$\underleftarrow{\lim}\left\\{C_{i},\rho_{i}\right\\}$ for which $x_{j}=x_{i}$
for all $j\geq i$ (see Theorem 1.9). For $t>0$, choose $j$ such that
$1/2^{j}\leq t\leq 1/2^{j-1}$ and notice that, for $i\geq j$ and
$\left(x_{0},x_{1},x_{2},\cdots\right)\in\underleftarrow{\lim}\left\\{C_{i},\rho_{i}\right\\}$

$H_{i}\left(x_{i},t\right)=H_{i}\left(\rho_{i+1}\left(x_{i+1}\right),t\right)=H_{i+1}\left(x_{i+1},t\right)$

so $H\left(\left(x_{0},x_{1},x_{2},\cdots\right),t\right)\in
e\left(C_{i}\right)\subseteq e\left(X\right)$. Furthermore, if
$\left(x_{0},x_{1},x_{2},\cdots\right)\in e\left(C_{i}\right)$, and $t\leq
1/2^{i}$ then $H_{i}\left(x_{i},t\right)=x_{i}$, so by the same formula,
$H_{j}\left(x_{i},t\right)=x_{i}$ for all $j>i$. Thus,
$\left.H\right|_{\underleftarrow{\lim}\left\\{C_{i},\rho_{i}\right\\}\times\left[0,\frac{1}{2^{i}}\right]}$
is a deformation retract onto $e\left(C_{i}\right)$. A similar calculation
shows that this homotopy is track faithful. ∎

######  Remark 5.8.

The above proof is a generalization and refinement of an argument outlined by
Jeremy Brazus in his _Wild Topology_ blog. See [Bra].

###  5.3. Compactifications of ANRs

In the context of geometric group theory and algebraic and geometric topology,
the spaces $Y$, $X$ and $\overline{X}$ discussed at the beginning of this
section are frequently ANRs. When that is the case, several stronger
assertions and equivalences hold. It will be useful to review a few of those.

A metric space $W$ is an _absolute neighborhood retract_(ANR) if, whenever it
is embedded as a closed subset of a metric space $S$, a neighborhood of $W$ in
$S$ retracts onto $W$. We call $W$ an _absolute retract_ (AR) if whenever $W$
is embedded as a closed subset of a metric space $S$, the entire space $S$
retracts onto $W$. Equivalently, an AR is a contractible ANR. Notice that we
are requiring $W$ and $S$ to be metric spaces in these definitions; not all
treatments of ANR theory include those assumptions.

Key examples of ANRs include: manifolds; metric CW and simplicial complexes
(in particular, all locally finite complexes under the Whitehead topology);
and CAT(0) spaces. See [Hu65] and [Ont05]. The following two well known
propositions are useful for identifying larger classes of ANRs.

######  Proposition 5.9.

A finite-dimensional metric space $W$ is an ANR if and only if it is locally
contractible; in other words, for every $x\in W$ and neighborhood $U$ of $x$,
there exists a neighborhood $V$ of $x$ that contracts in $U$.

######  Proposition 5.10.

Being an ANR is a local property. More precisely, every open subset of an ANR
is and ANR, and a metric space with the property that every point has an ANR
neighborhood is an ANR.

######  Corollary 5.11.

In order for a locally compact space $X$ to admit a compactification
$\overline{X}=X\sqcup R$, in which $\overline{X}$ is an ANR, it is necessary
for $X$ to be an ANR.

Versions of the following are woven into the literature in places where
$\mathcal{Z}$-compactifications are used to define and analyze group
boundaries. For example, [Bes96].

######  Proposition 5.12.

For a closed subset $A$ of a compact ANR $Y$, the following are equivalent:

  1. (1)

$A$ is a $\mathcal{Z}$-set,

  2. (2)

$A$ is an $\Omega$-set,

  3. (3)

for every open set $U\subseteq Y$, the inclusion $U-A\hookrightarrow U$ is a
homotopy equivalence.

###### Proof.

All of these can be obtained from Toruńczyk’s Equivalence Theorem from [Anc].
See also [Tor78]. ∎

In order to apply Proposition 5.12, the following partial converse to
Proposition 5.10 and its corollary can be useful.

######  Proposition 5.13.

If $A$ is a homotopy negligible subset of a compact metric space $Y$, and
$Y-A$ is an ANR, then $Y$ is an ANR. In other words, every homotopy negligible
compactification of an ANR is an ANR.

###### Proof.

Apply Hanner’s Theorem [Han51], as described in [Tir11]. ∎

######  Corollary 5.14.

Suppose $\overline{X}=X\sqcup R$ is an $\Omega$-compactification of an ANR
[resp., AR] $X$. Then $\overline{X}$ is a $\mathcal{Z}$-compactification of
$X$ if and only if $\overline{X}$ is an ANR [resp., AR].

###### Proof.

The forward and reverse implications follow from Propositions 5.13 and 5.12,
respectively. The assertion about ARs relies on Proposition 5.1. ∎

By using essentially the same argument, one can deduce a strengthening of
Proposition 5.13, that will be useful in the next section.

######  Proposition 5.15.

Let $A$ is a homotopy negligible subset of a compact metric space $Y$ and
$H:Y\times\left[0,1\right]\rightarrow Y$ a corresponding homotopy with
$H_{0}=\operatorname*{id}_{Y}$ and $H_{t}\left(Y\right)\subseteq Y-A$ for all
$t>0$. Suppose that for each $\varepsilon>0$, there exist a
$t\in\left[0,\varepsilon\right]$ and an ANR $B_{t}\subseteq Y$ such that
$H_{t}\left(Y\right)\subseteq B_{t}$, then $Y$ is an ANR.

######  Corollary 5.16.

If $\left\\{C_{i},\rho_{i}\right\\}$ is an inverse sequence of compact ANRs
and compact collapses, and $\overline{X}=X_{I}\sqcup R$ is the corresponding
homotopy negligible compactification of $X_{I}$ (as in Theorem 5.7), then
$\overline{X}$ is an ANR.

##  6\. Applications and more examples

We conclude this article with a variety of theorems and examples that can be
proved or reinterpreted using the ideas presented here.

###  6.1. General applications to simplicial and CW complexes and mapping
telescopes

The first pair of applications are immediate consequences of Theorem 5.7 and
the discussion in Sections 1 and 3.

######  Theorem 6.1.

Let $K$ be a simplicial complex that collapses to a finite subcomplex $L$, and
let $\left|K\right|_{I}$ be the geometric realization of $K$ with the topology
$\mathcal{T}_{I}$ induced by a corresponding inverse sequence of simplicial
collapses beginning with $\left|L\right|$. Then the compactification
$\overline{\left|K\right|_{I}}=\left|K\right|_{I}\sqcup R$ is homotopy
negligible. Furthermore, there is a compact collapse of
$\overline{\left|K\right|_{I}}$ onto $\left|L\right|$ and a deformation
retraction of $\left|K\right|_{I}$ to $\left|L\right|$.

######  Corollary 6.2.

Let $K$ be a locally finite simplicial complex that collapses to a finite
subcomplex $L$, and let $\left|K\right|$ be the geometric realization of $K$
with its standard topology. Then there is a $\mathcal{Z}$-compactification
$\overline{\left|K\right|}=\left|K\right|\sqcup R$ such that
$\overline{\left|K\right|}$ is an ANR. Furthermore, there is a compact
collapse of $\overline{\left|K\right|}$ onto $\left|L\right|$ and a
deformation retraction of $\left|K\right|$ to $\left|L\right|$. If
$\left|L\right|$ is contractible, then $\left|K\right|$ and
$\overline{\left|K\right|}$ are ARs.

###### Proof.

Along with the previous theorem, apply Propositions 1.11 and 3.5 and Corollary
5.14. ∎

######  Remark 6.3.

Examples 2.2 and 2.3 show how $\mathcal{T}_{I}$ can differ from the usual
topologies on $\left|K\right|$, even when $K$ is locally finite. Examples 2.1
and 2.3 illustrate the importance of having bonding maps that are compact
collapses and not just arbitrary deformation retractions.

By combining the above argument with Corollary 4.14, we can obtain a version
of Corollary 6.2 for CW complexes.

######  Theorem 6.4.

Let $X$ be a locally finite CW complex that collapses, via elementary CW
collapses, to a finite subcomplex $L$. Then there exists a
$\mathcal{Z}$-compactification $\overline{X}=X\sqcup R$ such that
$\overline{X}$ is an ANR. Furthermore, there is a compact collapse of
$\overline{X}$ onto $L$ and a deformation retraction of $X$ to $L$. If $L$ is
contractible, then $X$ and $\overline{X}$ are ARs.

Versions of the next theorem have played a key role in earlier work on
$\mathcal{Z}$-compactifications. See [CS76].

######  Theorem 6.5.

Let
$K_{0}\overset{f_{1}}{\longleftarrow}K_{1}\overset{f_{2}}{\longleftarrow}K_{2}\overset{f_{3}}{\longleftarrow}\cdots$
be an inverse sequence of compact metric spaces. Then there is a
$\mathcal{Z}$-compactification

$\overline{\operatorname*{Tel}\left(K_{i},f_{i}\right)}=\operatorname*{Tel}\left(K_{i},f_{i}\right)\sqcup
R$

where $R=\underleftarrow{\lim}(\left\\{K_{i},f_{i}\right\\})$. The spaces
$\overline{\operatorname*{Tel}\left(K_{i},f_{i}\right)}$ and
$\operatorname*{Tel}\left(K_{i},f_{i}\right)$ are ANRs if and only if each
$K_{i}$ is an ANR.

###### Proof.

The main assertion combines Examples 2.8 and 4.2 and Theorem 5.7. For the
follow-up assertion, apply [Hu65, IV.1] and Corollary 5.16 to see that
$\operatorname*{Tel}\left(K_{i},f_{i}\right)$
and$\overline{\operatorname*{Tel}\left(K_{i},f_{i}\right)}$ are ANRs when all
$K_{i}$ are ANRs. For the converse, note that for each $i$,
$\operatorname*{Tel}\left(K_{i},f_{i}\right)$ contains an open set that
retract onto $K_{i}$. ∎

###  6.2. Applications to CAT(0) spaces

Our first theorem about CAT(0) spaces is well known and obtainable using
analytic techniques. The methods developed in this paper provide an
alternative approach.

######  Theorem 6.6.

Every proper CAT(0) space $Y$ is $\mathcal{Z}$-compactifiable by addition of
the visual sphere at infinity. Moreover,
$\overline{Y}=Y\sqcup\partial_{\infty}Y$ is an AR.

###### Proof.

The basic properties of this compactification were covered in Example 2.7. The
assertions that this is a $\mathcal{Z}$-compactification and that
$\overline{Y}$ is an AR follow from 5.7 and Propositions 5.13. ∎

######  Theorem 6.7 ([Gul23]).

Every locally finite CAT(0) cube complex is collapsible via a sequence of
elementary cubical collapses and therefore admits a
$\mathcal{Z}$-compactification $\overline{Y}=Y\sqcup\partial_{\square}Y$
corresponding to such a sequence.

######  Remark 6.8.

The $\mathcal{Z}$-compactification in Theorem 6.7 differs from the one in
Theorem 6.6. The proof in [Gul23] is combinatorial rather than analytic, and
the resulting boundary (referred to as a _cubical boundary_) retains some of
the combinatorial structure of $Y$. Further investigation of that
compactification is part of an ongoing project by the authors of this paper.

An interesting sort of partial converse to Theorem 6.7 is contained in [AF21].
There it is shown that every collapsible locally finite simplicial complex
admits a subdivision that is a CAT(0) cube complex.

###  6.3. Applications to manifolds

Finite sequences of simplicial and CW expansions and collapses play important
roles in the study of compact manifolds. See [RS82]. The generalizations
developed here can be used to obtain some analogous theorems for noncompact
manifolds; we offer a sample in this section. For the sake of brevity, we
focus our attention to contractible open manifolds.

A countable simplicial complex $K$ (or its geometric realization
$\left|K\right|$) is a _combinatorial n-manifold_ if the link of each
$k$-simplex of $K$ is PL homeomorphic to either
$\left|\partial\sigma^{n-k}\right|$ or $\left|\sigma^{n-k-1}\right|$. The
simplices with links of the latter type constitute a subcomplex $\partial
K\leq K$. Under these circumstances, $\left|K\right|$ satisfies the
topological definition of an $n$-manifold with
$\partial\left|K\right|=\left|\partial K\right|$. A _PL n-manifold_ is a space
$M^{n}$ together with a triangulation $h:\left|K\right|\rightarrow M^{n}$ by a
combinatorial $n$-manifold; we call this a _PL triangulation_ of $M^{n}$.

A homeomorphism $h:\left|K\right|\rightarrow M^{n}$, where $M^{n}$ is a
topological $n$-manifold and $K$ is _not_ a combinatorial $n$-manifold, is
called a _non-PL triangulation_ of $M^{n}$. The existence of non-PL
triangulations of manifolds is a deep fact first proved by Edwards [Edw06].
That work implies that all PL manifolds of dimension $\geq 5$ admit non-PL
triangulations. By contrast, it is known that non-PL triangulations do not
exist in dimensions $\leq 4$.

Our first theorem is an easy one.

######  Theorem 6.9.

For all $n$, $\mathbb{R}^{n}$ admits a simplicially collapsible PL
triangulation.

###### Proof.

Begin with an _n_ -simplex $\sigma^{n}$, then inductively add PL
triangulations of $\left|\partial\sigma^{n}\right|\times\left[k,k+1\right]$ of
the sort described in Example 3.2. ∎

The converse of Theorem 6.9 is more striking. It can be viewed as the
noncompact analog of an important theorem which asserts that every compact
simplicially collapsible combinatorial $n$-manifold is a PL $n$-ball [RS82,
Cor.3.28].

######  Theorem 6.10.

If a combinatorial $n$-manifold $\left|K\right|$ is simplicially collapsible,
then $\left|K\right|$ is PL homeomorphic to $\mathbb{R}^{n}$.

###### Proof.

Let
$\left\\{v\right\\}=L_{0}\overset{e}{\swarrow}L_{1}\overset{e}{\swarrow}L_{2}\overset{e}{\swarrow}\cdots$
be a corresponding sequence of elementary combinatorial expansions through
subcomplexes such that $\cup L_{i}=K$. Then each $L_{i}$ is finite and
collapsible, so a regular neighborhood $N_{i}$ of $\left|L_{i}\right|$ in
$\left|K\right|$ is a PL $n$-ball. By choosing wisely (or passing to a
subsequence) we can arrange that $N_{i}\subseteq\operatorname*{Int}N_{i+1}$
for all $i$, so by the Combinatorial Annulus Theorem [RS82, Cor.3.19],
$N_{i+1}-\operatorname*{Int}N_{i}$ is PL homeomorphic to
$S^{n-1}\times\left[i,i+1\right]$. Gluing these products together and plugging
the hole with $N_{0}$ gives a homeomorphism to $\mathbb{R}^{n}$. ∎

######  Corollary 6.11.

No triangulation of the Whitehead contractible 3-manifold is simplicially
collapsible. More generally, the only open manifolds of dimension $\leq 4$
that admit simplicially collapsible triangulations are those PL homeomorphic
to $\mathbb{R}^{n}$.

###### Proof.

Every triangulation of a manifold of dimension $\leq 4$ is PL, so the
assertion follows from Theorem 6.10. ∎

If we allow non-PL triangulations, a different picture emerges, even for the
simplest class of contractible open manifolds.

######  Theorem 6.12.

If $C^{n}$ is a compact contractible PL $n$-manifold with $n\geq 5$, then
$\operatorname*{int}C^{n}$ admits a simplicially collapsible non-PL
triangulation.

We begin this proof with a lemma.

######  Lemma 6.13.

Let $C^{n}$ be a compact contractible $n$-manifold that is not homeomorphic to
a ball. Then

  1. (1)

there does not exist a PL triangulation of $C^{n}$ that is simplicially
collapsible,

  2. (2)

there exists a non-PL triangulation of $C^{n}$ that is simplicially
collapsible if and only if $n\geq 5$ and $\partial C^{n}$ admits a PL
triangulation. (The latter condition holds whenever $n\geq 6$.)

  3. (3)

$C^{n}$ is always compactly collapsible.

###### Proof.

Assertion 1 is a consequence of regular neighborhood theory. Suppose such a
triangulation exists, and let $DC^{n}$ be the double of $C^{n}$ along its
boundary. Then, a regular neighborhood of $C^{n}$ in $DC^{n}$ is PL
homeomorphic to $C^{n}$, but a regular neighborhood of a compact collapsible
polyhedron is always a PL ball.

Assertion 2 relies on a construction described in [ADG97] which triangulates
$C^{n}$ with a simplicial complex of the form

$T=(\left\\{p_{1}\right\\}\ast K_{1})\cup_{\left\\{q\right\\}\ast
J}(\left\\{p_{2}\right\\}\ast K_{2})$

(the union of two simplicial cones on finite simplicial complexes $K_{1}$ and
$K_{2}$ along a common conical subcomplex $\left\\{q\right\\}\ast J$ of
$K_{1}$ and $K_{2}$). Since every finite simplicial cone collapses to the
subcone over any of its subcomplexes, we can collapse $T$ to the subcomplex
$\left\\{p_{1},p_{2}\right\\}\ast\left(\left\\{q\right\\}\ast J\right)$ which,
by associativity of simplicial joins is the cone
$\left\\{q\right\\}\ast(\left\\{p_{1},p_{2}\right\\}\ast J)$. From there we
can collapse to $q$.

For the parenthetic remark, note that the boundary of a compact contractible
$n$-manifold has the homology of an $\left(n-1\right)$-sphere. The Kirby-
Siebenmann obstruction [KS77] to a PL triangulation of $\partial C^{n}$ lies
in $H^{4}\left(\partial C^{n};\mathbb{Z}_{2}\right)$, and therefore vanishes
when $n\geq 6$.

For assertion 3 choose any contraction of $C^{n}$ that fixes a point. That
contraction is a compact collapse. ∎

###### Proof of Theorem 6.12.

If we attach an exterior open collar $\partial C^{n}\times[0,\infty)$ to
$C^{n}$, the resulting space is homeomorphic to $\operatorname*{int}C^{n}$.
Thus,

(6.1) $\operatorname*{int}C^{n}\approx C^{n}\cup(\partial
C^{n}\times\left[0,1\right])\cup(\partial
C^{n}\times\left[1,2\right])\cup(\partial
C^{n}\times\left[2,3\right])\cup\cdots$

By Lemma 6.13 we can choose a simplicially collapsible non-PL triangulation
$\left|K\right|$ of $C^{n}$ with a subcomplex $L\leq K$ that triangulates
$\partial C^{n}$. A sequence of applications of Example 4.10 applied to
$\left|L\right|\times\left[i,i+1\right]$ produces triangulations of $\partial
C^{n}\times\left[i,i+1\right]$ which agree on their overlap and collapse to
their left-hand boundaries. Assembling these produces the desired
triangulation. ∎

If we allow arbitrary compact collapses, a still different picture emerges.

######  Theorem 6.14.

The interior of every compact contractible $n$-manifold $C^{n}$ admits a
compact collapse.

###### Proof.

Begin with the decomposition of $\operatorname*{int}C^{n}$ provided in (6.1).
Since each $\partial C^{n}\times\left[i,i+1\right]$ compactly collapses to
$\partial C^{n}\times\left\\{i\right\\}$ (Example 4.3) and $C^{n}$ compactly
collapses to a point, the theorem follows. ∎

It is not the case that _every_ contractible open manifold is compactly
collapsible.

######  Theorem 6.15.

The Whitehead contractible 3-manifold $W^{3}$ is not compactly collapsible.

###### Proof.

By Theorem 5.7, if $W^{3}$ were compactly collapsible, it would be
$\mathcal{Z}$-compactifiable. Then, by [GT03, Th.1.2], $W^{3}$ would be
semistable at infinity, contradicting [Geo08, Prop.16.4.2]. ∎

######  Remark 6.16.

The Whitehead manifold is by no means unique. In all dimensions $\geq 3$ there
exist contractible open manifolds that are not semistable; hence they are not
$\mathcal{Z}$-compactifiable; hence they are not compactly collapsible. (Note.
For non-manifolds, compact collapsibility does not imply semistability.)

Another interesting collection of contractible open $n$-manifolds are the
exotic universal covers of closed aspherical $n$-manifolds constructed by
Davis in [Dav83]. We will take a closer look at the collapsibility properties
of those manifolds in [AGS]. For now, we mention one particularly nice class.

######  Theorem 6.17.

Let $K$ be a flag simplicial complex for which $\left|K\right|$ is a closed
$\left(n-1\right)$-manifold with the same homology as $S^{n-1}$. Then the
right-angled Coxeter group $W_{K}$ with nerve $K$ acts geometrically on a
CAT(0) cube complex $X$ whose underlying space is a contractible open
$n$-manifold that is not homeomorphic to $\mathbb{R}^{n}$. The space $X$ is
cubically collapsible.

###### Proof.

The space $X$ is constructed in [ADG97]. It is a variation on the Davis
complex $\Sigma_{K}$ (which is not a manifold). By [Gul23], $X$ is cubically
collapsible. ∎

###  6.4. Applications to 1-dimensional spaces

######  Theorem 6.18.

Every finite valence tree $T$ is simplicially collapsible and admits a
$\mathcal{Z}$-compactification $\overline{T}=T\sqcup R$ where $\overline{T}$
is an AR and $R$ is homeomorphic to the space of ends of $T$.

###### Proof.

Endow $T$ with the path length metric; choose a base vertex $v_{0}$; and for
each $i\in\mathbb{N}$, let $K_{i}$ be the closed ball of radius $i$ centered
at $v_{0}$. Then each $K_{i}$ is a finite subtree and
$\overline{K_{i+1}-K_{i}}$ is a collection of edges, each having one end point
in $K_{i-1}$. Clearly there is a simplicial collapse
$\rho_{i+1}:K_{i+1}\rightarrow K_{i}$. Now apply Corollary 6.2. The
correspondence between $R$ and the space of ends is almost immediate from the
definitions. See, for example [Gui16, §3.3]. ∎

Provided $T$ is countable, we can prove a weaker result without assuming
finite valences.

######  Theorem 6.19.

Every countable tree $T$ is a union of finite subtrees
$\left\\{v_{0}\right\\}=T_{0}\subseteq T_{1}\subseteq T_{2}\subseteq\cdots$
with the property that $\overline{T_{i+1}-T_{i}}$ is a collection of edges,
each having one end point in $T_{i-1}$; thus $T$ is simplicially collapsible.
Under the topology induced by the corresponding inverse sequence, $T_{I}$
admits a homotopy negligible compactification to an AR
$\overline{T_{I}}=T_{I}\sqcup R$.

###### Proof.

We leave it to the reader to prove the existence of the collection
$\left\\{T_{i}\right\\}$. From there, the conclusions follow from Theorem 5.7
and Corollary 5.16. ∎

######  Remark 6.20.

Example 2.2 is an illustration of Theorem 6.19. Note that $T_{I}$ is not the
usual topology on $T$. Nevertheless, this theorem provides a method for
defining and topologizing the ends of $T$. For example, the tree in Example
2.2 has no ends.

A _dendrite_ is a compact, connected, locally path connected metric space $D$
that is uniquely arc-wise connected; in other words, for any pair of distinct
points $p,q\in D$, there is a unique embedded arc with end points $p$ and $q$.
Dendrites are of central importance in the field of continuum theory. They
have also played an important role in the study of hyperbolic groups [Bow99].
A fundamental theorem from continuum theory provides a link to the previous
two theorems.

######  Theorem 6.21 ([Nad92, Th.10.27]).

Every dendrite $D$ is homeomorphic to the inverse limit of a sequence

$\left\\{v_{0}\right\\}=T_{0}\overset{\rho_{1}}{\swarrow}T_{1}\overset{\rho_{2}}{\swarrow}T_{2}\overset{\rho_{3}}{\swarrow}\cdots$

of finite trees where $\overline{T_{i+1}-T_{i}}$ is an arc with one end point
in $T_{i}$ and $\rho_{i+1}$ is the corresponding compact collapse.

In this generality, we cannot assume there are fixed simplicial structures on
the $T_{i}$ making them subcomplexes of $T_{j}$ for $j\geq i$ and making the
$\rho_{i}$. Nevertheless, the proof used for Theorem 6.19 is applicable and
gives the following important theorem about dendrites.

######  Corollary 6.22.

Every dendrite $D$ is a compact AR; in particular, $D$ is contractible.

With additional care, one can say a little more. A point $p\in D$ is an _end
point_ if it has arbitrarily small neighborhoods whose boundaries are one-
point sets. Let $D^{\left[1\right]}$ denote the set of end points of $D$.

######  Theorem 6.23.

The set $D^{\left[1\right]}$ is a homotopy negligible in $D$.

###### Proof.

One strategy for proving Theorem 6.21 is to begin with a countable dense set
$X=\left\\{x_{0},x_{1},x_{2},\cdots\right\\}\subseteq$ $D$, then inductively
define $T_{i+1}$ to be the union of $T_{i}$ with the arc $A_{i+1}$ connecting
$x_{i+1}$ to $x_{0}$. It is easy to see that $T_{i}\cap A_{i+1}$ is either
$\left\\{x_{0}\right\\}$ or a subarc of $A_{i+1}$; hence $T_{i+1}$ can be
given the structure of a finite tree. (If $T_{i}=T_{i+1}$, and we can later
omit it from the sequence.). It is a simple fact that $D^{\left[1\right]}$ is
nowhere dense in $D$, so we can choose $X$ to be disjoint from $D$. In that
case, an element of $D^{\left[1\right]}$ is never the leaf of a $T_{i}$; and
clearly, it cannot be a nonleaf of a $T_{i}$. Since or compactification is of
the form $\overline{D}=\left(\cup T_{i}\right)\sqcup R$, then
$D^{\left[1\right]}\subseteq R$. ∎

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