# Evasion Paths by Homotopy Limits

Gunnar Carlsson, Benjamin Filippenko, Wyatt Mackey

###### Contents

  1. 1 Introduction
    1. 1.1 Prior work
    2. 1.2 Paper outline
  2. 2 Holim and spaces of sections
    1. 2.1 Homotopy inverse limits
    2. 2.2 Path spaces
    3. 2.3 Homotopy type of the sequential path space
    4. 2.4 Comparing the homotopy inverse limit and the space of sections
      1. 2.4.1 0-sectional maps
  3. 3 Application: Tame functions
    1. 3.1 Local constructions
    2. 3.2 Tame functions are locally 0-sectional
  4. 4 Application: sensor ball evasion
    1. 4.1 Notation
    2. 4.2 Sensor ball covered and uncovered regions are locally 0-sectional
    3. 4.3 Sensor ball evasion with homology

##  1 Introduction

Suppose a short-staffed collection of guards is patrolling a region—can their
lack of manpower be exploited? Or can they devise a patrol pattern that
stymies any would-be intruders? This is the general topic of pursuer-evader
games. We will view this as a topological problem: at each time $t$, there is
a patrolled region $P_{t}$ and an unpatrolled region $U_{t}$. By gluing
together the copies of $U_{t}$ (with an appropriate subspace topology) we get
a map of spaces $U\to[t_{0},t_{1}]$, delineating how the unpatrolled region
changes over time. Navigating through the unpatrolled region, then, is
equivalent to finding a section to the map $U\to[t_{0},t_{1}]$.

More generally, given a map $X\xrightarrow{\pi}B$ of spaces, we would like to
be able to compute the homotopy type of the space of sections of $\pi$,
$\Gamma\pi=\\{s:B\to X\ |\ \pi\circ s=\operatorname{id}_{B}\\}$. The number of
distinct sections, as searched for in the patroller-evader games, is
$\pi_{0}(\Gamma\pi)$. These computations are well understood when $\pi$ is a
fibration, but the non-fibrant case has received comparatively little
attention.

###  1.1 Prior work

In the context of mobile sensor networks, this problem was introduced by de
Silva and Ghrist in [SG06]. They gave a necessary condition for existence of
sections based on the homology of the covered region. Adams and Carlsson
proved in [AC15] that while zig-zag homology can provide a necessary
condition, it cannot provide a sufficient condition for the existence of a
section. Arone and Jin gave an extension of this in [AJ21], and, by using
Goodwillie's calculus of functors, were able to detect the lack of a section
in Adams-Carlsson's example.

Ghrist and Krishnan introduced positive homology in [GK17] as a tool which
calculates directly the homology of the space of sections. When the map from
the uncovered region to time is tame, Carlsson and Filippenko in [CF20] gave a
complete computation of the set of connected components of the space of
sections in terms of the homotopy groups of the fibers.

###  1.2 Paper outline

In Section 2 we give a computation of the homotopy groups of $\Gamma\pi$ for
maps of spaces $X\to I$, where $I=[0,1]$ when $\pi$ satisfies some mild
lifting condition; the case $X\to S^{1}$ is very similar. The central idea is
to subdivide $X\xrightarrow{\pi}I$ into well behaved projections to
subintervals of $I$, where it is easier to compute the homotopy groups of the
space of sections, and then stitch these together. Let

$0=s_{0}<s_{1}<...<s_{n}=1$

be a sequence, so that $\\{[s_{i},s_{i+1}]\\}_{i=0,...,n-1}$ is a partition of
the interval. We will write $X_{i}:=\pi^{-1}(s_{i})$ for $0\leq i\leq n$, and
$X_{i}^{i+1}:=\pi^{-1}([s_{i},s_{i+1}])$ for $0\leq i\leq n-1$. This gives us
a diagram of spaces

$X_{0}\to X_{0}^{1}\leftarrow X_{1}\to X_{1}^{2}\leftarrow
X_{2}\to...\leftarrow X_{n},$

which we will denote $ZX$. Any section $\gamma:I\to X$ of $\pi$ gives the data
of: (1) a point in each $X_{i}$, and (2) a path in each $X_{i}^{i+1}$,
compatible with the maps in $ZX$ in the obvious way.

On the other hand, the homotopy inverse limit of this diagram is given by

  1. (1)

The choice of a point $x_{i}\in X_{i}$ for $0\leq i\leq n$,

  2. (2)

the choice of a point $x_{i}^{i+1}\in X_{i}^{i+1}$, for $0\leq i\leq n-1$,

  3. (3)

the choice of a path from the image of $x_{i}$ in $X_{i}^{i+1}$ to
$x_{i}^{i+1}$, for $0\leq i\leq n-1$,

  4. (4)

and a path from the image of $x_{i}$ in $X_{i-1}^{i}$ to $x_{i-1}^{i}$, for
$1\leq i\leq n$.

Since we are interested only up to homotopy, we may combine the data of (3)
and (4) as

  1. (3’)

the choice of a path in $X_{i}^{i+1}$ from the image of $x_{i}$ to the image
of $x_{i+1}$, for $0\leq i\leq n-1$.

There is therefore a map of spaces $\Gamma\pi\to\operatorname{holim}ZX$. In
Section 2.4, we develop conditions on the partition and the map $X\to I$ under
which this map becomes an $n$-equivalence, for $n\in\mathbb{Z}$. In such
examples, we can then compute the low homotopy groups of $\Gamma\pi$ by
studying the much more approachable $\operatorname{holim}ZX$.

In particular, the Bousfield-Kan spectral sequence for homotopy inverse limits
(Ch. XI, Section 7 of [BK72]) computes the homotopy groups of the homotopy
inverse limit of $ZX$.

######  Theorem 1.1.

Let $\\{s_{i}\\}_{i=0,...,n}$ be a partition of the interval, and
$X\xrightarrow{\pi}I$ be locally 0-sectional relative to an open cover that
contains $[s_{i},s_{i+1}]$, and $ZX$ be as above. Then there is a short exact
sequence of sets

$*\to R^{1}\lim\pi_{1}(ZX)\to\pi_{0}\Gamma\pi\to\lim\pi_{0}(ZX)\to*,$

where $R^{1}\lim$ denotes the first right derived functor of the limit
functor.

We prove this in Section 2.4.

For diagrams of the same form as $ZX$, we can describe $R^{1}\lim\pi_{1}(ZX)$
as the orbit set of a group action of $\prod_{0}^{n}\pi_{1}(X_{i})$ on
$\prod_{0}^{n-1}\pi_{1}(X_{i}^{i+1})$, despite the fact that the
$\pi_{1}(X_{i})$ need not be abelian.

######  Remark 1.2.

We are working without basepoints here, so the use of $\pi_{1}(ZX)$ demands
some explanation. Its meaning here is closer to that of the fundamental
groupoid than a true fundamental group. The ``valid" extensions associated to
each sequential path space $\gamma\in\lim\pi_{0}(ZX)$ are
$R^{1}\lim\pi_{1}(ZX,\gamma)$, where we travel along $\gamma$ to define the
isomorphism in

$\pi_{1}(X_{i},\gamma(i))\to\pi_{1}(X_{i}^{i+1},\gamma(i))\cong\pi_{1}(X_{i}^{i+1},\gamma(i+1))\leftarrow\pi_{1}(X_{i+1},\gamma(i+1).$

Unfortunately, since this level of the spectral sequence deals only with
pointed sets, the computations must be done pointwise over the elements of
$\lim\pi_{0}(ZX)$. Of course, if all we care about is whether $\pi_{0}\Gamma$
is empty, no computations of $R^{1}\lim\pi_{1}(ZX)$ are necessary.

The condition of being ``locally 0-sectional" essentially means that the
section space really is the same as the piecewise path space constructed by
the homotopy inverse limit–or at least, it is similar enough to correctly
compute $\pi_{0}$. We give a full discussion in Section 2.4.

Section 3 is devoted to tame maps of manifolds (Definition 3.2), similar to
Morse functions. We use the smooth structure to show that tame maps are
locally 0-sectional, so that their section space can be computed as in Theorem
2.9. This section is drawn from [CF20], which gives an explicit computation
using the structure of tame maps. This section was particularly due to the
second author.

Section 4 is devoted to mobile sensor networks. Sensor ball evasion has been
studied several times in the past, after being introduction by de Silva and
Ghrist in [SG06]. In this game, sensors wander through a region, observing
certain areas and noticing when they overlap with other sensors. We want to
know when there is a path through the uncovered region.

We provide a new calculation of the path components of the uncovered region,
using Theorem 1.1. This relies on the computation of unstable homotopy groups
of the fiber spaces, however a necessary condition can be extracted from just
the homology calculations: there is an easy dictionary between $\pi_{0}$ and
$H_{0}$, so we can restate $\lim\pi_{0}(ZX)$ in terms of the cohomolology
groups of the uncovered region, which Alexander duality allows us to turn into
a question about the homology groups of the covered region. This allows us to
provide a new necessary and sufficient condition for the existence of a
section, in terms of the fiberwise homology of the covered region.

##  2 Holim and spaces of sections

We suggested above that the homotopy inverse limit of diagrams is relatively
computable, and can be used as a good approximation of the space of sections
of a map $X\xrightarrow{\pi}I$. This makes it a natural candidate for helping
us understand $\Gamma\pi$. To this end, we first give a brief overview of
homotopy inverse limits in Section 2.1.

Section 2.2 details how to go from a map of spaces $X\xrightarrow{\pi}B$ to a
diagram whose homotopy inverse limit is a good approximation for $\Gamma\pi$.
There is no extra difficulty in working over general spaces $B$, so we work in
increased generality there.

Section 2.3 then records the results of the Bousfield-Kan spectral sequence
when applied to the diagrams from Section 2.2. In particular, we show it
collapses on the $E_{2}$ page for spaces $X\xrightarrow{\pi}I$ parameterized
over the interval.

Section 2.4 addresses the question: how good an approximation is this homotopy
inverse limit to the space of sections? We write down some lifting diagrams
which make the natural map $\Gamma\pi\to\operatorname{holim}ZX$ an
$n$-equivalence, and then deduce Theorem 2.9.

###  2.1 Homotopy inverse limits

There are many constructions of the homotopy inverse limit. We will follow
[BK72], which essentially writes limits as a Hom set in the category of
diagrams, and then takes a cofibrant replacement of the source. A somewhat
more common construction starts the same, but takes a fibrant replacement of
the target rather than a cofibrant replacement of the source.

More precisely, let $D$ be a small category, $F:D\to\textsc{Spaces}$ a diagram
indexed by $D$. The functor category
$\textsc{Funct}_{D}:=\operatorname{Hom}_{\textsc{Cat}}(D,\textsc{Spaces})$ is
enriched over Spaces, either via the compact open topology when Spaces is some
topological category, or by setting

$\operatorname{Maps}_{\textsc{Funct}_{D}}(F,G)_{n}:=\operatorname{Hom}(\Delta^{n}\times
F,G),$

when Spaces is the category of simplicial sets. Then one definition of the
inverse limit of the diagram $F$ is

$\lim_{\leftarrow}F:=\operatorname{Maps}_{\textsc{Funct}_{D}}(*_{D},F),$

where $*_{D}:D\to\textsc{Spaces}$ is the functor which sends every object of
$D$ to a terminal object of Spaces.

The homotopy inverse limit mimics this definition, after taking a cofibrant
replacement of $*_{D}$. For $d$ an object of $D$, we define a space $D/d$,
whose $n$-simpleces are sequences

$d_{n}\xrightarrow{\alpha_{n-1}}d_{n-1}\to...\xrightarrow{\alpha_{0}}d_{0}\xrightarrow{\alpha}d$

in $D$. Given a map $d\xrightarrow{\phi}d^{\prime}$ in $D$, there is a map
$D/d\to D/d^{\prime}$ which takes

$(d_{n}\xrightarrow{\alpha_{n-1}}d_{n-1}\to...\xrightarrow{\alpha_{0}}d_{0}\xrightarrow{\alpha}d)\mapsto(d_{n}\xrightarrow{\alpha_{n-1}}d_{n-1}\to...\xrightarrow{\alpha_{0}}d_{0}\xrightarrow{\phi\circ\alpha}d^{\prime}).$

This assembles to a functor

$D/-:D\to\textsc{Spaces},$

which will be our cofibrant replacement. We can now define the homotopy
inverse limit,

######  Definition 2.1.

The homotopy inverse limit of $F:D\to\textsc{Spaces}$ is the mapping space

$\operatorname{holim}F:=\operatorname{Maps}_{\operatorname{Funct}_{D}}(D/-,F).$

###  2.2 Path spaces

Let $X\xrightarrow{\pi}B$ be a map of simplicial sets. We want a nice way to
organize the data of $X$ into a functor that we can analyze, similar to $ZX$
in the introduction. To this end, consider the diagram $\widetilde{B}$, whose
objects are the simplices of $B$, and whose maps are inclusions of simplices.
Given a simplicial set $S$, let $H(S)$ denote its barycentric subdivision.
Then $\widetilde{B}/-$ assigns to each $n$-simplex of $B$ the barycentric
subdivision of $\Delta^{n}$, $H(\Delta^{n})$, and sends the inclusion maps
$\Delta^{m}\hookrightarrow\Delta^{n}$ to the corresponding inclusions of their
barycentric subdivisions.

Let $\widetilde{\pi}:\widetilde{B}\to\textsc{Spaces}$ denote the functor which
assigns to the each simplex $\sigma$ of $B$ the barycentric subdivision of its
preimage $H(\pi^{-1}(\sigma))\subseteq H(X)$. Then
$\operatorname{holim}_{\widetilde{B}}\widetilde{\pi}$ is a collection of path
data: we have a map $H(\Delta^{n})$ to $\pi^{-1}(\sigma)$, and the collection
of all of this path data glues together well. Thus
$\operatorname{holim}\widetilde{\pi}$ is a reasonable first approximation to
the space of sections of $\pi$.

When $B$ is not a simplicial set, we can take a simplicial decomposition of
$B$, call it $S$, and a similar procedure applies. Let $i:|S|\to B$ be the
decomposition relationship between $S$ and $B$, then we get a diagram
$\widetilde{\pi}_{S}:\widetilde{S}\to\textsc{Spaces}$, by assigning to each
simplex $\sigma$ of $S$ the space $\pi^{-1}(i(\sigma))$.

Figure 1: $X$ projects down onto $[b_{0},b_{2}]$

######  Example 2.2.

In Figure 1, let $X\xrightarrow{\pi}B$ be the projection map. The map $\pi$
clearly has no sections. However, if we used a single 1-simplex
$[b_{0},b_{2}]$ to approximate the base the homotopy limit would be the space
of paths in $X$ that map the endpoints to the preimage of $b_{0}$ and $b_{2}$,
which is certainly not empty! Subdividing $B$ further, though, yields better
results.

If we approximate $B$ as the union of two 1-simpleces, $[b_{0},b_{1}]$ and
$[b_{1},b_{2}]$, then $\operatorname{holim}\widetilde{\pi}$ contains
collections of data: a point in the preimage of $b_{0}$, $b_{1}$, and $b_{2}$,
and paths in the preimage of $[b_{0},b_{1}]$ and $[b_{1},b_{2}$ that connect
these points. But clearly no such points and paths exist, so

$\operatorname{holim}\widetilde{\pi}\cong\emptyset\cong\Gamma\pi.$

Finer simplicial approximations in general result in more accurate
approximations of the section space. The idea is that, over a smaller space,
there are fewer obstructions to a path being homotopic to a section.

We collect these observations here:

######  Proposition 2.3.

Let $X\xrightarrow{\pi}B$ be a map of topological spaces, and let $S$ and
$S^{\prime}$ be simplicial approximations of $B$.

  1. (1)

There is a natural map $\Gamma\pi\to\operatorname{holim}\widetilde{\pi}_{S}$.

  2. (2)

If $S^{\prime}$ is a finer approximation of $B$ than $S$ (for instance, if
$S^{\prime}=H(S)$), then the map from (1) factors as

$\Gamma\pi\to\operatorname{holim}\widetilde{\pi}_{S^{\prime}}\to\operatorname{holim}\widetilde{\pi}_{S}.$

  3. (3)

Let $I$ denote the partially ordered set of simplicial approximations of $B$,
ordered by fineness. Then there is a natural map

$\Gamma X\to\lim_{S\in I}\operatorname{holim}\widetilde{\pi}_{S}.$

Points (1) and (2) follow directly from our description of the homotopy
inverse limit in this scenario. (3) follows from (1) and (2).

###  2.3 Homotopy type of the sequential path space

Using the homotopy limit as an approximation to the space of sections would be
silly if the approximation didn't have some nice, computable features. This
section gives those computation results, mostly referring to the work done in
[BK72]. We will focus on the case where $B$ is an interval, as it simplifies
our eventual spectral sequence, and is the primary case of interest anyway.
When $B$ is higher dimensional, the spectral sequence still exists—it just no
longer collapses on the second page.

The Bousfield-Kan spectral sequence for homotopy inverse limits computes the
homotopy groups of a homotopy inverse limit in terms of the homotopy groups of
its constituent parts. The reader interested in the details should look to
Chapter XI of [BK72]. Given a diagram of spaces $F\in\textsc{Spaces}^{D}_{*}$,
this has

$E_{2}^{s,t}\cong R^{s}\lim\pi_{t}(F),$

where $R^{s}\lim$ denotes the $s$ right derived functor of inverse limit.

The differential $d_{r}:E_{r}^{s,t}\to E_{r}^{s+r,t+r-1}$, and the spectral
sequence converges so that

$\operatorname{gr}\pi_{i}\operatorname{holim}D\cong\oplus_{s}E_{\infty}^{s-i,s}.$

Applied to $ZX$, note that $R^{s}\lim ZX$ vanishes for $s\geq 2$ since there
are no nondegenerate 2-simplices in the underlying diagram of $ZX$ (XI.6.5 in
[BK72] and the cosimplicial Dold-Kan). Thus our spectral sequence only has two
nonzero columns, and collapses on the $E_{2}$ page.

######  Remark 2.4.

A word should be said on base points: our diagrams $ZX$ cannot have a
compatible collection of basepoints on the nose, since the image of the
inclusions $X_{i},X_{i+1}\to X_{i}^{i+1}$ do not intersect. Not all is lost,
however: fix a section $\sigma:I\to X$. This sets a base points in each of the
$X_{i}$, along with a canonical compatibility between these base points. In
this way, we should say that all homotopy groups are relative to a given
section: we get a corresponding spectral sequence for each choice.

######  Example 2.5.

If we want to compute $\pi_{0}\operatorname{holim}ZX$, it will fit into a
``short exact sequence" of sets

$*\to
R^{1}\lim\pi_{1}(ZX)\to\pi_{0}\operatorname{holim}ZX\to\lim\pi_{0}(ZX)\to*,$

by which we mean the fiber of
$\pi_{0}\operatorname{holim}ZX\to\lim\pi_{0}(ZX)$ over
$\gamma\in\lim\pi_{0}(ZX)$ is isomorphic to $R^{1}\lim\pi_{1}(ZX,\gamma)$.

Then $R^{1}\lim\pi_{1}(ZX)$ is the coequalizer of the two maps

$\displaystyle\prod_{i=0}^{n}$
$\displaystyle\pi_{1}(X_{i})\times\prod_{i=0}^{n-1}\pi_{1}(X_{i}^{i+1})\rightrightarrows$
$\displaystyle\prod_{\alpha:X_{i}\to
X_{i}^{i+1}}\pi_{1}(X_{i}^{i+1})\times\prod_{\beta:X_{i+1}\to
X_{i}^{i+1}}\pi_{1}(X_{i}^{i+1})\times\prod_{i=0}^{n}\pi_{1}(X_{i})\times\prod_{i=0}^{n-1}\pi_{1}(X_{i}^{i+1}).$

The first map is induced by
$\alpha_{*}:\pi_{1}(X_{i})\to\pi_{1}(X_{i}^{i+1})$,
$\beta_{*}:\pi_{1}(X_{i+1})\to\pi_{1}(X_{i}^{i+1})$, and identity maps on the
remaining factors. The second map is given only by identity maps on the last
two factors of the product.

###  2.4 Comparing the homotopy inverse limit and the space of sections

Now tooled with a calculation of $\pi_{*}\operatorname{holim}ZX$, we would
like to leverage the map from Proposition 2.3,

$\Gamma\pi\to\operatorname{holim}ZX,$

to gain some insight into the homotopy groups of $\Gamma\pi$. In particular:
when is this map an $n$-equivalence?

Our general strategy is to investigate the minimal hypotheses we need to get
the low dimensional parts of the long exact sequence of a fibration. The
homotopy inverse limit of

$X_{0}\xrightarrow{i}X\xleftarrow{j}X_{1}$

is the space of paths $\gamma:I\to X$, with $\gamma(0)\in
i(X_{0}),\gamma(1)\in j(X_{1})$. (In our setting, $i$ and $j$ will always be
inclusions, and will often be suppressed from the notation.) We will denote
this space by $\operatorname{Hom}((I,0,1),(X,X_{0},X_{1}))$. Let
$\operatorname{Sec}([a,b],X)$ denote the space of sections of
$X\xrightarrow{p}[a,b]$. Then for each choice of homeomorphism
$(I,0,1)\xrightarrow{f}([a,b],a,b)$, there is a pullback diagram

${\operatorname{Sec}([a,b],X)}$${\operatorname{Hom}((I,0,1)(X,X_{0},X_{1}))}$${*}$${\operatorname{Hom}((I,0,1),([a,b],a,b)).}$$\scriptstyle{f^{*}}$$\scriptstyle{p_{*}}$$\scriptstyle{*\mapsto
f}$

######  Example 2.6.

If $p$ is a fibration, then so is

$p_{*}:\operatorname{Hom}((I,0,1),(X,X_{0},X_{1}))\to\operatorname{Hom}((I,0,1),([a,b],a,b)).$

Now, $\operatorname{Hom}((I,0,1),([a,b],a,b))$ is contractible, so if $p_{*}$
is a fibration, then

$\operatorname{Sec}([a,b],X)\to\operatorname{Hom}(I,X;\partial I,\partial X)$

is a weak homotopy equivalence.

It is too much to hope in applications of interest that $f^{*}$ will always be
a weak homotopy equivalence. However, in applications we are particularly
interested in the first few homotopy groups of $\operatorname{Sec}([a,b],X)$.
In this case, we need not require that $p_{*}$ be a fibration.

######  Definition 2.7.

Let $p:X\to[a,b]$ be a continuous map. We will say that $p:X\to U$ is _$0$
-sectional_ if the map
$\operatorname{Sec}([a,b],X)\xrightarrow{i}\operatorname{Hom}(([a,b],a,b),(X,p^{-1}(a),p^{-1}(b)))$
is $0$-connected for each choice of basepoint
$\sigma\in\operatorname{Sec}([a,b],X)$.

######  Definition 2.8.

We will say $p:X\to I$ is _locally $0$-sectional_ if for every $t\in I$, there
is a neighborhood $(a,b)\ni t$ such that $p|_{[a,b]}$ is $0$-sectional. If
$\mathcal{U}=\\{U_{i}\\}_{i\in\alpha}$ is an open cover of $I$, we will say
$p$ is locally $0$-sectional subordinate to $\mathcal{U}$ if
$p|_{p^{-1}(\overline{U_{i}})}$ is $0$-sectional for each
$U_{i}\in\mathcal{U}$.

With the terminology all defined, we're now ready to prove Theorem 1.1.

###### Proof of Theorem 1.1.

By Proposition 2.3 (1), there is a natural map from the space of sections to
the homotopy inverse limit,
$\Gamma\pi\to\operatorname{holim}\widetilde{\pi}_{\cup[s_{i},s_{i+1}]}$. The
Bousfield-Kan spectral sequence gives a computation of
$\pi_{0}\operatorname{holim}\widetilde{\pi}_{\cup[s_{i},s_{i+1}]}$ which
matches that claimed in the theorem, so it suffices to show that

$\pi_{0}\Gamma\pi\to\pi_{0}\operatorname{holim}\widetilde{\pi}_{\cup[s_{i},s_{i+1}]}$

is an isomorphism.

The condition that $X\xrightarrow{\pi}I$ is locally 0-sectional subordinate to
an open cover containing $[s_{i},s_{i+1}]$ means that each of the maps

$\Gamma\pi|_{[s_{i},s_{i+1}]}\to\operatorname{Hom}(([s_{i},s_{i+1}],s_{i},s_{i+1}),(\pi^{-1}([s_{i},s_{i+1}]),\pi^{-1}(s_{i}),\pi^{-1}(s_{i+1}))$

induce isomorphisms on $\pi_{0}$.

Now elements of $\Gamma\pi$ are the same as elements of the limit

$\lim_{\leftarrow}(...\to\pi^{-1}(s_{i-1})\leftarrow\Gamma\pi|_{[s_{i-1},s_{i}]}\to\pi^{-1}(s_{i})\leftarrow\Gamma\pi|_{[s_{i},s_{i+1}]}\to\pi^{-1}(s_{i+1})\leftarrow...)$

Likewise, elements of the homotopy inverse limit over ZX are the same as
elements of the (normal) limit over the diagram

$...\to\pi^{-1}(s_{i-1})\leftarrow\operatorname{Hom}(([s_{i-1},s_{i}],s_{i-1},s_{i}),(\pi^{-1}([s_{i-1},s_{i}]),\pi^{-1}(s_{i-1}),\pi^{-1}(s_{i}))\to\pi^{-1}(s_{i})\leftarrow...$

Let
$A_{i}:=\operatorname{Hom}(([s_{i-1},s_{i}],s_{i-1},s_{i}),(\pi^{-1}([s_{i-1},s_{i}])$,
for page space. The induced map of diagrams

${...}$${\pi^{-1}(s_{i-1})}$${\Gamma\pi|_{[s_{i-1},s_{i}]}}$${\pi^{-1}(s_{i})}$${...}$${...}$${\pi^{-1}(s_{i-1})}$${A_{i}}$${\pi^{-1}(s_{i})}$${...}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

The middle map also induces an isomorphism on connected components by
hypothesis, and this property is preserved under taking limits. ∎

Let's abstract, and suppose we have

${F}$${E}$${*}$${B,}$$\scriptstyle{i}$$\scriptstyle{p}$

Note: we are _not_ assuming $p$ is a fibration.

####  2.4.1 0-sectional maps

There is a sequence

$\pi_{0}F\xrightarrow{\pi_{0}(i)}\pi_{0}E\xrightarrow{\pi_{0}(p)}\pi_{0}B.$
(1)

The map $\pi_{0}(p)$ is certainly an epi if $E\to B$ is surjective, which is
true in our case whenever $X$ contains a path connected component whose map to
$I$ is surjective. The composite of (1) is automatically 0. To be exact,
everything sent to the basepoint of $\pi_{0}B$ must be homotopic to an element
of the image of $F\to E$.

In our setting, that means that each map $I\to X$ is homotopic to a section.
(Technically we should specify that the pullback under $f$ is homotopic to a
section, but the choice of $f$ doesn't matter.) We can state this condition by
requiring that in the following diagram, there is a lift

${I}$${X}$${I\times I}$${{[a,b]}}$$\scriptstyle{\operatorname{id}\times
0}$$\scriptstyle{p}$$\scriptstyle{H}$$\scriptstyle{h}$

where $H(0,t),H(1,t)$ are constant and $h$ is any homotopy. This gives us
surjectivity of the map from sections to paths. The diagram we must require
for injectivity is similar; we summarise the results in the following theorem:

######  Theorem 2.9.

A map $p:X\to[a,b]$ is 0-sectional if

  1. 1.

The map
$\pi_{0}\operatorname{Sec}([a,b],X)\to\pi_{0}\operatorname{Hom}(([a,b],a,b),(X,p^{-1}(a),p^{-1}(b)))$
is surjective, i.e. there is a lift

${I}$${X}$${I\times I}$${{[a,b]}}$$\scriptstyle{\operatorname{id}\times
0}$$\scriptstyle{p}$$\scriptstyle{H}$$\scriptstyle{h}$

where $H(0,t),H(1,t)$ are constant whenever $h(1,s)=f^{*}$; and

  2. 2.

The map
$\pi_{0}\operatorname{Sec}([a,b],X)\to\pi_{0}\operatorname{Hom}(([a,b],a,b),(X,p^{-1}(a),p^{-1}(b)))$
is injective, i.e. there is a lift

${I\times I}$${X}$${I\times I\times
I}$${{[a,b]},}$$\scriptstyle{p}$$\scriptstyle{H}$$\scriptstyle{h}$

when $h|_{(s,t,1)}:I\times I\to[a,b]$ is equal to $f\circ\pi_{1}$, where
$\pi_{1}:I\times I\to I$ is projection onto the first factor.

We have already justified the first diagram; the second diagram just says that
sections which are homotopic are also fiberwise homotopic.

We could hope to get a nicer theorem statement by reproducing the proof of the
long exact sequence of a fibration, and seeing what diagrams are required for
the low dimensional parts of the sequence to hold. Suppose we wanted exactness
of $\pi_{1}B\to\pi_{0}F\to\pi_{0}E$, for instance. Here exactness means that
$\pi_{1}B$ acts on $\pi_{0}F$, and the quotient by this action includes into
$\pi_{0}E$.

For $\pi_{1}(B)$ to have an action on $\pi_{0}F$, we need to be able to form a
lift as indicated in the diagram

${*}$${E}$${I}$${B.}$$\scriptstyle{\sigma}$$\scriptstyle{\widetilde{\sigma}}$

The action is well defined if homotopies of paths also lift: that is, if the
dashed arrow exists as in the following diagram for any square

${I}$${E}$${I\times
I}$${B.}$$\scriptstyle{\widetilde{\sigma}}$$\scriptstyle{\operatorname{id}\times
0}$$\scriptstyle{H}$

In our setting, we can use the product/hom adjunction to get an equivalence
between the diagram

${*}$${\operatorname{Hom}((I,0,1),(X,p^{-1}(a),p^{-1}(b)))}$${I}$${\operatorname{Hom}((I,0,1),({[a,b]},a,b))}$

and the diagram

${I}$${X}$${I\times I}$${{[a,b]},}$

with the understanding that maps preserve $\partial I$ and $\partial X$.
Likewise, the action being well defined translates to existence of a dashed
arrow in the diagram

${I\times I}$${X}$${I\times I\times I}$${{[a,b]}.}$

These are very nice requirements, which unfortunately often will not hold in
the cases in which we are interested: for instance, lifts (for either diagram)
need not exist in the case of a manifold mapping to the interval by a tame
function with a unique critical point. (See Definition 3.2.)

##  3 Application: Tame functions

We now give some applications, to show that locally $n$-sectional maps really
do show up in nature. This section is devoted to tame functions, a kind of
smooth function similar to Morse functions. We construct a ``nice" vector
field on a tame function in Section 3.1, then use this to prove that tame
functions are locally 0-sectional in Section 3.2. We can then conclude from
Theorem 2.9 and Example 2.5

######  Theorem 3.1.

Given a tame map $\pi:X\to[s_{-},s_{+}]$, let $ZX$ be the diagram associated
to a partition of $[s_{-},s_{+}]$ for which there is at most one critical
point of $\pi$ in each $[s_{a},s_{a+1}]$. Then there is a short exact sequence
of sets

$*\to R^{1}\lim\pi_{1}(ZX)\to\pi_{0}\Gamma\pi\to\lim\pi_{0}(ZX)\to*.$

Moreover, $R^{1}\lim\pi_{1}(ZX)$ is isomorphic to the orbit set of a group
action of $\prod_{0}^{n}\pi_{1}(X_{i})$ on
$\prod_{0}^{n-1}\pi_{1}(X_{i}^{i+1})$. Basepoints are here suppressed, and we
mean them as in Remarks 1.2 and 2.4.

The following notion of a tame function, as well as the constructions we
perform with them in this section, is inspired by the Morse theory on
manifolds with boundary.

######  Definition 3.2.

Let $X$ be a compact cobordism of manifolds with boundary, and
$\pi:X\rightarrow[s_{-},s_{+}]$ a smooth function. Let $B=\partial
X\setminus\pi^{-1}(s_{-})\cup\pi^{-1}(s_{+})$. We say $\pi$ is a _tame
function_ if it satisfies the following conditions:

  * •

The critical points of the restriction $\pi|_{B}:B\rightarrow[s_{-},s_{+}]$
are isolated and have distinct critical values in $(s_{-},s_{+})$,

  * •

$\pi$ is submersive,

  * •

$X_{-}=\pi^{-1}(s_{-})$ and $X_{+}=\pi^{-1}(s_{+})$.

######  Remark 3.3.

Definition 3.2 does not require any nondegeneracy condition on the critical
points of $\pi|_{B}:B\rightarrow[s_{-},s_{+}]$, so $\pi|_{B}$ does not have to
be a Morse function in the sense of [Mil65, Def. 2.3]. In this way, our
definition is more general than the Morse condition. On the other hand, we do
not allow $\pi$ itself to have critical points. Note that for the projection
$\pi:X\rightarrow[s_{-},s_{+}]$ from a codimension-$0$ submanifold
$X\subset\mathbb{R}^{d}\times[s_{-},s_{+}]$, as is the situation in the
smoothed evasion path problem, the map $\pi$ is submersive.

There are two types of critical points of a tame function on the boundary.

######  Definition 3.4.

Let $\pi:X\rightarrow[s_{-},s_{+}]$ be a tame function, $p\in B$ a critical
point of $\pi|_{B}:B\rightarrow[s_{-},s_{+}]$, and $\eta\in T_{p}X$ an outward
pointing vector. Then $p$ is _type N_ if $d\pi(\eta)<0$ and _type D_ if
$d\pi(\eta)>0.$

Note that $d\pi(\eta)=0$ is impossible since it would imply that $p$ is a
critical point of $\pi$.

###  3.1 Local constructions

This section is devoted to creating a nice smooth vector field on tame
manifolds. The idea is that, by flowing along that vector field, we can find
local homotopies of paths into sections.

Our constructions are local in the sense that for a tame function
$\pi:X\rightarrow[s_{-},s_{+}]$ and $t\in[s_{-},s_{+}$], the constructions
apply in the preimage $\pi^{-1}([t-\epsilon,t+\epsilon])$ for $\epsilon>0$
small enough.

######  Proposition 3.5.

Let $\pi:X\rightarrow[s_{-},s_{+}]$ be a tame function. If $\pi|_{B}$ has only
type D critical points, then there exists a smooth vector field $\xi$ on $X$
with the following properties:

  1. 1.

$d\pi(\xi)=-1$ on all of $X$.

  2. 2.

For all111For $p\in X_{-}$, the vector field $\xi$ constructed in the proof of
Proposition 3.5 is outward pointing on $Int(X_{-})$, and on $\partial X_{-}$
it is outward pointing with respect to $X_{-}$ and inward pointing with
respect to $B$. $p\in\partial X\setminus X_{-}$, the vector $\xi_{p}\in
T_{p}X$ is inward pointing.

If $\pi|_{B}$ has only type N critical points, then there exists a smooth
vector field $\xi$ on $X$ with the following properties:

  1. 1.

$d\pi(\xi)=1$ on all of $X$.

  2. 2.

For all $p\in\partial X\setminus X_{+}$, the vector $\xi_{p}\in T_{p}X$ is
inward pointing.

Moreover, given any smooth section $\mathfrak{b}:[s_{-},s_{+}]\rightarrow X$
of $\pi$ (i.e. $\pi\circ\mathfrak{b}(t)=t$) such that $\mathfrak{b}(t)\not\in
B$ for all $t\in[s_{-},s_{+}]$, the vector field $\xi$ can be chosen such that

$\frac{d}{dt}\mathfrak{b}(t)=\begin{cases}-\xi|_{\mathfrak{b}(t)}&\text{if
}\text{type D}\\\ \xi|_{\mathfrak{b}(t)}&\text{if }\text{type N}.\end{cases}$

###### Proof.

We consider the case of type D critical points; the type N case is symmetric.

It suffices to construct $\xi$ locally in an open neighborhood of every point
$p\in X$ and then sum up these local vector fields with a partition of unity.
For the local construction, let $p\in X$ and for now assume that if $p\in B$
then $p$ is a regular point of $\pi|_{B}$. Then by the implicit function
theorem there exists an open neighborhood $U(p)\subset X$ of $p$ and a smooth
chart $\varphi:U(p)\xrightarrow{\sim}V$ such that
$\varphi(p)=(\pi(p),0,\ldots,0)\in V$ where $V$ is an open subset of one of
the following spaces depending on where $p$ sits on $X$, and in such a way
that

$\pi\circ\varphi^{-1}:V\rightarrow[s_{-},s_{+}]$

is the projection onto the first coordinate:

  * •

If $p\in Int(X)$, then $V\subset\mathbb{R}^{\dim X}$,

  * •

If $p\in Int(X_{-})$, then $V\subset[s_{-},\infty)\times\mathbb{R}^{\dim
X-1}$,

  * •

If $p\in Int(X_{+})$, then $V\subset(-\infty,s_{+}]\times\mathbb{R}^{\dim
X-1}$,

  * •

If $p\in Int(B)$, then
$V\subset\mathbb{R}\times([0,\infty)\times\mathbb{R}^{\dim X-2})$,

  * •

If $p\in\partial X_{-}$, then
$V\subset[s_{-},\infty)\times([0,\infty)\times\mathbb{R}^{\dim X-2})$,

  * •

If $p\in\partial X_{+}$, then
$V\subset(-\infty,s_{+}]\times([0,\infty)\times\mathbb{R}^{\dim X-2})$.

In all cases above, the constant vector field

$(-1,1,0,\ldots,0)$

on $V$ pulls back through the diffeomorphism $\varphi$ to a vector field $\xi$
on $U(p)$ satisfying $d\pi(\xi)=-1$, and moreover it is inward pointing along
all points $p\in\partial X\setminus X_{-}=Int(B)\cup X_{+}$, as required.

It remains to consider a critical point $p\in B$ of type $D$. By definition of
tame function, $p$ is in $Int(B)=B\setminus(\partial X_{-}\cup\partial
X_{+})$. Consider a neighborhood $\tilde{U}(p)\subset X$ of $p$ and a
coordinate chart
$\tilde{U}(p)\xrightarrow{\sim}V\subset[0,\infty)\times\mathbb{R}^{\dim X-1}$
that sends $p$ to $0$. Then the constant vector field $(1,0,\ldots,0)$ pulls
back to a vector field $\xi$ on $\tilde{U}(p)$ that is inward pointing, and
hence $d_{p}\pi(\xi_{p})<0$ since $p$ is type D. Then $d\pi(\xi)<0$ in a
smaller open neighborhood $U(p)\subset\tilde{U}(p)$. Hence the vector field
$\xi/|d\pi(\xi)|$ satisfies (i) and (ii) on $U(p)$, as required.

Consider now the final statement of the proposition where we are given a
smooth section $\mathfrak{b}$ that is disjoint from $B$. For
$p\neq\mathfrak{b}(t)$ for all $t$, choose the neighborhood $U(p)$ to be
disjoint from the image of $\mathfrak{b}$ and perform the construction as
above. Suppose $p=\mathfrak{b}(t)$ for some $t$. Choose $U(p)$ to be disjoint
from $B$. Define $\xi|_{\mathfrak{b}(t)}=-\mathfrak{b}^{\prime}(t)$ for all
$t$ such that $\mathfrak{b}(t)\in U(p)$. Then
$d\pi(\xi|_{\mathfrak{b}(t)})=-(\pi\circ\mathfrak{b})^{\prime}(t)=-1$. Extend
$\xi$ over $U(p)$ so that $d\pi(\xi)=-1$ on $U(p)$. ∎

###  3.2 Tame functions are locally 0-sectional

We now will use the vector field constructed in Proposition 3.5 to show that
tame functions are locally 0-sectional. The idea is to work in subdivisions
that contain at most one critical point, and flow our path along the
pseudogradient. This produces a homotopic path which ``fails to be a section"
entirely on one side of a critical point. But there, $f$ is a fibrations, and
so the path can be straightened.

It will suffice to show that a subinterval $[a,b]\subset[s_{-},s_{+}]$ with
$p|_{p^{-1}((a,b))}$ having exactly 1 critical point is 0-sectional. We
consider only the type D case, as the type N case is symmetric. Then if we can
show that the conditions of Theorem 2.9 are satisfied, then we can
successfully compute $\pi_{0}\Gamma X$ by the formula of Theorem 1.1.

######  Proposition 3.6.

Suppose $X\xrightarrow{\pi}[a,b]$ is a tame function with a unique critical
point of type D. Then $\pi$ is 0-sectional, i.e. every path
$([a,b],a,b)\xrightarrow{\gamma}(X,\pi^{-1}(a),\pi^{-1}(b))$ is homotopic to a
section, and this section is unique up to homotopy of sections.

###### Proof.

First, choose a homeomorphism $([a,b],a,b)\xrightarrow{f}([a,b],a,b)$ such
that $\pi\circ\gamma\circ f(c)\geq c$ for every $c\in[a,b]$. Then define
$w(c)$ to be the point in $X$ that comes from flowing $\gamma\circ f(c)$ along
$(\pi\circ\gamma\circ f(c)-c)\xi$ for 1 unit of time, where $\xi$ is the
vector field from Proposition 3.5. Then $w:[a,b]\to X$ is a section, and by
construction it is homotopic to $\gamma$. Note that if $\gamma$ is already a
section, we can choose $f$ to be the identity map and get $w=\gamma$.

Now suppose $\gamma$ is also homotopic to a section $w^{\prime}:[a,b]\to X$.
We need to show that $w$ and $w^{\prime}$ are homotopic as sections: that is,
there is a homotopy $H:[a,b]\times I\to X$ such that $H_{t}$ is a section for
each $t\in I$. Let $G:[a,b]\times I\to X$ be a homotopy from $\gamma$ to
$w^{\prime}$. Then we can choose a continuous collection of homeomorphisms
$f_{t}:([a,b],a,b)\to([a,b],a,b)$ such that $\pi\circ G_{t}\circ f_{t}(c)\geq
c$ for $c\in[a,b]$, and $f_{1}$ is the identity map, and $f_{0}$ the same $f$
we used in constructing $w$. Applying the procedure of the preceding
paragraph, we obtain a homotopy of sections $w_{t}$ from $w_{0}=w$ to
$w_{1}=w^{\prime}$. ∎

We can now conclude Theorem 3.1 as a corollary of Proposition 3.6 and Theorem
1.1.

##  4 Application: sensor ball evasion

We return to the setting of the introduction now: if an understaffed
collection of guards is patrolling a space, when can we avoid them? Sensor
ball evasion is a model of this problem, in which our guards have a reasonably
controlled perception–for instance, they are motion detectors that observe
some ball centered on themselves.

Given a collection of continuous sensors
$\mathcal{S}=\\{\gamma_{j}:[0,1]\rightarrow\mathcal{D}\\}_{j\in J}$ moving in
a bounded domain $\mathcal{D}\subset\mathbb{R}^{d}$, an _evasion path_ is a
continuous map $\delta:[0,1]\rightarrow\mathcal{D}$ that avoids detection by
the sensors for the whole time interval $I=[0,1]$. Suppose each sensor
$\gamma_{j}$ observes a ball of fixed radius in $\mathbb{R}^{d}$ centered at
$\gamma(t)$ at every $t\in I$, and let $C_{t}$ be the time-varying union of
these sensor balls. Then the intruder $\delta$ avoids detection if $\delta(t)$
is not in $C_{t}$ for all $t\in I$.

The sensor ball evasion path problem asks for a criterion that determines
whether or not an evasion path exists. Ideally, we would do this with as
little information as possible: a common assumption is that the sensors only
detect other nearby sensors and intruders; in particular, they do not know
their coordinates in $\mathbb{R}^{d}$. The existence problem simply asks if
the space of evasion paths $\mathcal{E}$ is empty or not. We might also ask
after the homotopy type of $\mathcal{E}$, if it isn't empty.

This problem was first stated and studied in [SG06] by de Silva and Ghrist in
dimension $d=2$. Provided that the time-varying Čech complex changes only at
finitely many times, and by only simple changes at those times, the authors
establish a necessary condition for existence of an evasion path which states
that the connecting homomorphism on a relative homology group with respect to
the fence of the covered region must vanish.

In [AC15], Adams and Carlsson consider the same evasion problem in arbitrary
dimension $d\geq 2$, and they establish a necessary condition for the
existence of an evasion path based on zigzag persistent homology [CS10] of the
time-varying Čech complex. They also explain how this result is equivalent to
a generalization of de Silva and Ghrist's result to the case $d\geq 2$.

In both de Silva-Ghrist and Adams-Carlsson, the necessary condition for
existence of an evasion path is determined and easily computable from the
overlap information of sensor balls. Essentially, sensors only need to know
which other sensors are nearby; they don't need to know their coordinates in
$\mathbb{R}^{d}$.

Adams-Carlsson also study smarter sensors in the plane $\mathbb{R}^{2}$ that
know local distances to nearby sensors and the natural counterclockwise
ordering on nearby sensors. For these smarter sensors in $\mathbb{R}^{2}$,
they derive a necessary and sufficient condition for existence of an evasion
path as well as an algorithm to compute it, under the assumption that the
covered region $C_{t}$ is connected for all $t\in I$. They show that this
connectedness assumption is necessary for their methods.

In [GK17], Ghrist and Krishnan define positive homology and cohomology for
directed spaces over $\mathbb{R}^{q}$, and they compute it using sheaf-
theoretic techniques. They consider a general type of evasion problem where
the covered region is a pro-object in a category of smooth compact cobordisms.
They derive a criterion on positive cohomology that is necessary and
sufficient for existence of an evasion path, under the assumption that $C_{t}$
is connected.

In Section 4.1, we establish some notation and formally state the problem.
Section 4.2 then shows that the unpatrolled region is locally 0-sectional, so
that we can solve the existence problem by Theorem 1.1. However, this gives a
solution in terms of the unpatrolled space, which we don't always assume we
have access to: we would like to use less information, such as just the C̆ech
complex associated to the sensor balls. We can accomplish thisby using
Alexander Duality, which is the purpose of Section 4.3. Our results are then
summarized in Theorem 4.5.

###  4.1 Notation

Fix some sensing radii $r_{j}>0$ for each $j\in J$, and say that each sensor
$\gamma_{j}\in\mathcal{S}$ can detect objects within the closed ball

$B_{\gamma_{j}(t)}=\\{x\in\mathbb{R}^{d}\,\,|\,\,|\gamma(t)-x|\leq r_{j}\\}$

at all times $t\in I=[0,1]$. The time-$t$ covered region

$C_{t}=\bigcup_{\gamma\in\mathcal{S}}B_{\gamma(t)}\subset\mathbb{R}^{d}$ (2)

is the union of the sensor balls, and it is homotopy equivalent to the Čech
complex of the covering of $C_{t}$ by the sensor balls. The time-$t$ uncovered
region is the complement

$X_{t}=\mathcal{D}\setminus C_{t},$

where $\mathcal{D}\subset\mathbb{R}^{d}$ is a bounded domain that is
homeomorphic to a ball.

Assume that the boundary $\partial\mathcal{D}$ is covered by a collection of
immobile fence sensors $F\mathcal{S}\subset\mathcal{S}$, meaning
$\gamma(t)\in\mathbb{R}^{d}$ is constant for $\gamma\in F\mathcal{S}$, and
that the union of balls $B_{\gamma(t)}$ for $\gamma\in F\mathcal{S}$ covers
the boundary $\partial\mathcal{D}$ and is homotopy equivalent to
$\partial\mathcal{D}$. In particular, this ensures that an intruder $\delta$
can never escape from the domain $\mathcal{D}$.

We define the covered region

$C:=\bigcup_{t\in I}C_{t}\times\\{t\\}\subset\mathbb{R}^{d}\times I$ (3)

and the uncovered region

$X:=(\mathcal{D}\times I)\setminus C=\bigcup_{t\in I}X_{t}\times\\{t\\}.$

Then an evasion path is equivalent to a continuous section
$\delta:I\rightarrow X$ of the projection $\rho_{X}:X\rightarrow I$, i.e.,
$\rho_{X}\circ\delta=id_{I}$.

The sensor ball evasion path problem asks for a criterion for existence of an
evasion path.

###  4.2 Sensor ball covered and uncovered regions are locally 0-sectional

Both de Silva-Ghrist and Adams-Carlsson simplify the problem by looking at the
C̆ech complex associated to the union of sensors $C_{t}$, and the nerve
theorem tells us this simplicial complex computes the same homology. Moreover,
it is assumed that $C_{t}$ varies at only finitely many times
$\\{t_{1},...,t_{n}\\}$, and moreover, at any time $t_{i},0<i<n$, $C_{t}$
changes by either the addition of simplices or the removal of simplices, but
not both. In this section, we will show that under these hypotheses, both $X$
and $C$ are locally 0-sectional, subordinate to any open cover whose open sets
contain at most one of the $t_{i}$.

Let there be $\\{s_{i}\\}_{i=0,...,n}$ with
$0=s_{0}<t_{1}<s_{1}<...<t_{n}<s_{n}=1$. Up to homotopy equivalence, we may
replace $C$ by what Adams-Carlsson refer to as the _stacked C̆ech complex_ :
$SC\to I$ is a space equipped with a projection to the interval, with

$SC:=\coprod_{i=0,...,n}C_{s_{i}}\times[t_{i},t_{i+1}]/\sim,$

where $t_{0}=0,t_{n+1}=1$, and $\sim$ identifies either
$C_{s_{i}}\times\\{t_{i+1}\\}$ with a subset of
$C_{s_{i+1}}\times\\{t_{i+1}\\}$, or $C_{s_{i+1}}\times\\{t_{i+1}\\}$ with a
subset of $C_{s_{i}}\times\\{t_{i+1}\\}$ for each $i$. The map to $I$ is given
by projection onto the second factor of the products.

######  Definition 4.1.

We will say that $Y\to I$ is a _stacked simplicial complex_ if there are
distinct points $t_{1},...,t_{n}\in I$ so that

$Y\cong\coprod_{i=0,...,n}Y_{s_{i}}\times[t_{i},t_{i+1}]/\sim,$

where $s_{i}\in(t_{i},t_{i+1})$, $Y_{s_{i}}$ are simplicial complexes, and
$\sim$ identifies either $Y_{s_{i}}\times\\{t_{i+1}\\}$ with a subset of
$Y_{s_{i+1}}\times\\{t_{i+1}\\}$, or $Y_{s_{i+1}}\times\\{t_{i+1}\\}$ with a
subset of $Y_{s_{i}}\times\\{t_{i+1}\\}$ for each $i$.

We will sometimes say $Y$ is a _stacked C̆ech complex_ when $Y$ is a stacked
simplicial complex when we want to emphasize that the simplicial complex
structure $Y_{s_{i}}$ comes from a C̆ech nerve.

######  Proposition 4.2.

Let $Y\xrightarrow{p}I$ be a stacked C̆ech complex, with $t_{i}$ and $s_{i}$
as above. Then $p$ is locally 0-sectional, subordinate to any open cover whose
open sets contain at most one of the $t_{i}$.

###### Proof.

Suppose $[a,b]$ is a subinterval which contains $t_{i}$, $1\leq i\leq n$, and
does not contain any $t_{j}$ with $j\neq i$. Assume that a simpleces are added
at $t_{i}$; the case in which they are removed is symmetric.

Let us first produce an appropriate vector field on $p^{-1}([a,b])$. Since
$p^{-1}([a,b])=Y_{a}\times[a,t_{i}]\cup Y_{b}\times[t_{i},b]$, and
$Y_{a}\subset C_{b}$ at $t_{i}$, this is very straightforward: let $\xi$ be
the constant vector field on $[a,b]$ which points to the right with magnitude
1. Then $p^{*}\xi$ is a vector field on $p^{-1}([a,b])$ with the same formal
properties as the vector field of Proposition 3.5.

The remainder of the proof is entirely identical to the proof of Proposition
3.6. ∎

Our primary interest, however, is in the uncovered region $X$ of a sensor ball
network. To that end, we have

######  Lemma 4.3.

Let $\mathcal{D},C,X,p,SC$ be as above. Then $X$ is locally 0-sectional,
subordinate to the same collection of open covers as $SC$.

###### Proof.

$X_{(t_{i},t_{i+1})}$ is homotopic to a product, so it suffices to show that
$X$ restricted to neighborhoods of the $t_{i}$, $i=1,...,n$, is 0-sectional.
Suppose that at $t_{i}$ we are given an inclusion $SC_{s_{i+1}}\hookrightarrow
SC_{s_{i}}$; the case where we have the reverse inclusion is symmetric.

Since $\mathcal{D}$ is bounded, $\overline{X_{s_{i}}}$ and
$\overline{X_{s_{i+1}}}$ are compact. Take an open cover by convex sets of
$\overline{X_{s_{i}}}$, and reduce this to a finite subcover
$\\{U_{\alpha}\\}$. Let $SX_{s_{i},U_{\alpha}}$ denote the C̆ech nerve of this
complex. Then $X_{(t_{i},t_{i+1})}\cong
SX_{s_{i},U_{\alpha}}\times(t_{i},t_{i+1})$. We can build a C̆ech cover of
$\overline{X_{s_{i+1}}}$ by starting with (the translates of) the
$\\{U_{\alpha}\\}$, and extending it by adjoining to the $\\{U_{\alpha}\\}$ a
nice cover of the simplices of $SC_{s_{i}}$ which are removed at $t_{i+1}$,
and then translating along our chosen straightening.

In fact, we have built a restriction of a stacked simplicial complex
$SX|_{(t_{i},t_{i+2})}\to(t_{i},t_{i+2})$. By Proposition 4.2, $X$ is
therefore locally 0-sectional. ∎

######  Remark 4.4.

The deficit of this approach is that we don't get a detailed look at how
adding/removing simplices in the covered region affects the same in the
uncovered region. Morally, we see that removing a $k$-simplex in $SC$ should
be the same as adding a $(d-k)$-simplex in $SX$, as a form of parameterized
Alexander Duality. This will be expanded in a future work by the last author.

In particular, Theorem 1.1 applies, and we have a short exact sequence of sets
which computes $\pi_{0}\Gamma p$.

###  4.3 Sensor ball evasion with homology

By the previous sections, we know that the covered and uncovered regions are
both 0-sectional, and so there is a short exact sequence

$*\to R^{1}\lim\pi_{1}(ZX)\to\pi_{0}\Gamma p\to\lim\pi_{0}(ZX)\to*,$

as in Theorem 1.1, which computes the path components of the section space of
the uncovered region. But what if we don't have a good description of $X$? We
mentioned in the beginning of this section that we often assume only access to
the C̆ech nerve of the sensors. Under these more restricted assumptions, we
can still obtain a necessary and sufficient condition for $X$ to admit a
section.

######  Theorem 4.5.

Given a sensor ball evasion problem $X\xrightarrow{p}[a,b]$, we can compute
the number of solutions as the middle term in the short exact sequence

$*\to R^{1}\lim\pi_{1}ZX\to\pi_{0}\Gamma
p\to\lim\operatorname{Hom}_{k-alg}(H_{d-1}(ZC),k)\to*.$

In particular, there exists a section if and only if
$lim\operatorname{Hom}_{k-alg}(H_{d-1}(ZC),k)$ is nonempty.

There is a natural isomorphism
$\pi_{0}(A)\cong\operatorname{Hom}_{k-alg}(H^{0}(A),k)$. Thus we have a
surjection

$\pi_{0}\Gamma p\to\lim\operatorname{Hom}_{k-alg}(H^{0}(ZX),k).$

Alexander duality allows us to identify $H^{0}(X_{s_{i}})$ with
$H_{d-1}(C_{s_{i}})$. Unfortunately, we cannot directly identify
$H^{0}(X_{s_{i}}^{s_{i+1}})$ with $H_{d-1}(C_{s_{i}}^{s_{i+1}})$, which would
make the the theorem statement very nice.

Instead, we should note that for stacked simplicial complexes with a single
critical point, there is a deformation retract $X_{s_{i}}^{s_{i+1}}\to
X_{s_{i}}$ when simplices are removed at $t_{i}$, and a deformation retract
$X_{s_{i}}^{s_{i+1}}\to X_{s_{i+1}}$ when simplices are added at $t_{i}$.
Thus, in our diagram

${...}$${H^{0}X_{s_{i}}^{s_{i+1}}}$${...}$${H^{0}X_{s_{i}}}$${H^{0}X_{s_{i+1}}}$${H_{d-1}C_{s_{i}}}$${H_{d-1}C_{s_{i+1}}}$${...}$${H_{d-1}C_{s_{i}}^{s_{i+1}}}$${...}$$\scriptstyle{m}$$\scriptstyle{n}$$\scriptstyle{m^{\prime}}$$\scriptstyle{\sim}$$\scriptstyle{n^{\prime}}$$\scriptstyle{\sim}$

either $m$ and $n^{\prime}$ are both isomorphisms, or $m^{\prime}$ and $n$ are
both isomorphisms. The diagram commutes because Alexander duality is natural
with respect to inclusions.

Happily, we can remove isomorphic terms when taking inverse limits. For
instance, if in the above diagram $m$ was an isomorphism, we could replace

$H^{0}X_{s_{i}}\xleftarrow{m}H^{0}X_{s_{i}}^{s_{i+1}}\xrightarrow{n}H^{0}X_{s_{i+1}}$

with

$H^{0}X_{s_{i}}\xrightarrow{nm^{-1}}H^{0}X_{s_{i+1}}.$

Associated to this reduced diagram, we have a dual reduced diagram, with
arrows

$H_{d-1}C_{s_{i}}\xrightarrow{n^{\prime-1}m^{\prime}}H_{d-1}C_{s_{i+1}}.$

We therefore have an isomorphism of diagrams of $k$-algebras, and so
$\lim\operatorname{Hom}_{k-alg}(H_{d-1}(ZC),k)$ computes $\lim\pi_{0}(ZX)$.
This completes the proof of the theorem.

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