# Shrinking Without Doing Much At All

Michael Freedman  Michael Freedman
Microsoft Research, Station Q, and Department of Mathematics
University of California, Santa Barbara
Santa Barbara, CA 93106  and  Michael Starbird  Michael Starbird
Department of Mathematics
The University of Texas at Austin
Austin, TX 78712

###### Abstract.

In 1952 Bing astonished the mathematical community with his wild involution on
$S^{3}$. It has been among the most seminal examples in topology. The example
depends on finding shrinking homeomorphisms of Bing’s decomposition of $S^{3}$
into points and arcs. If Bing’s original homeomorphisms are varied, Bing’s
original wild involution changes by conjugation, which preserve some analytic
properties [fs22] while altering others. In 1988, Bing published a second
paper “Shrinking Without Lengthening,” answering a question that one of the
present authors posed to him in an effort to understand the geometry of the
entire conjugacy class. In this paper we produce a counterintuitive
construction, namely, a method to shrink the Bing decomposition doing almost
nothing at all–neither lengthening much nor rotating much.

##  1\. Introduction

Bing’s 1952 wild involution $I$ on the three sphere $S^{3}$ is a landmark
example in topology and is essential to understanding the topological
category, not only in dimension 3, but also in dimension four [freedman82].
$I$ was constructed from the standard reflection on $S^{3}$ by quotienting out
an invariant lower semicontinuous decomposition $\mathcal{D}$, the Bing
decomposition. Decomposition space theory and shrinking will be reviewed
shortly. While the Bing decomposition $\mathcal{D}$ is reflection invariant,
and shrinkable, the shrinking homeomorphisms cannot be made equivariant, that
is, the necessity to break reflection symmetry during the shrink is the source
of the wildness. Recently, the authors were able to quantify this wildness
[fs22] analytically by studying features that must be shared by any shrink of
$\mathcal{D}$. That is, we were able to understand analytical properties
shared by all topological conjugates $I^{h}$ of $I$, $h$ a self-homeomorphism
of $S^{3}$. That investigation led to novel shrinking strategies for
$\mathcal{D}$. We present one such here as an homage to Bing’s final paper,
“Shrinking Without Lengthening” [bing88], which Bing wrote to answer a
question one of us asked him at the time. Bing’s decomposition is made by
intersecting finite stages called “Bing rings,” nested solid tori (see Figure
2). The shrink amounts to figuring out a strategy for stretching, twisting,
bending, and/or rotating each finer pair of _daughter_ solid tori within the
previous _mother_ stage. Bing found that the rings need be lengthened only
infinitesimally during the shrink, hence his title. We find that not only can
the length of the solid tori be nearly preserved, but the “rotations” of the
daughters within the mother can also be made arbitrarily small and can be made
to decay towards zero. Because of the unresolved interface between smooth and
topological manifolds in dimension 4, the analytic requirements of a Bing
shrink are still germane.

_Bing’s decomposition_. The Alexander Horned Sphere is the first known wild
embedding of $S^{2}$ in $S^{3}$. One method of creating that wild embedding of
$S^{2}$ is as follows: Start with a standard $S^{2}$ in $S^{3}$. For ease of
visualization, think of $S^{2}$ as the $y\\-z$ plane in $R^{3}$ with a point
at infinity that makes $R^{3}$ into $S^{3}$ and makes the $y\\-z$ plane a
standard embedding of $S^{2}$ in $S^{3}$. On the positive $x$ side of this
$S^{2}$, construct a specific Cantor set’s worth of arcs–one end of each arc
will be on $S^{2}$, while the other ends of those arcs will be rather
entangled among each other. Specifically, the Cantor set’s worth of arcs are
created by taking the components of an infinite intersection of families
$\\{C_{i}\\}_{i=0}^{\infty}$ of U-shaped solid cylinders. Each family $C_{i}$
has $2^{i}$ components. $C_{0}$ consists of a single U-shaped cylinder as
pictured in Figure 1. Every component of $C_{i}$ contains two U-shaped
cylinders of $C_{i+1}$ embedded as shown in Figure 1. Notice that the feet of
each cylinder in $C_{i}$ get increasingly close to one another as $i$
increases. So in the limit, each component of $\cap_{i=0}^{\infty}C_{i}$ is an
arc meeting $S^{2}$ in a single point, and the totality of those arc-endpoints
is a Cantor set on $S^{2}$. Shrinking that Cantor set’s worth of arcs is key
to creating the Alexander Horned Sphere embedding of $S^{2}$. The set
consisting of those arcs together with the remaining points of $S^{3}$ is an
upper semi-continuous decomposition $\mathcal{D_{A}}$ of $S^{3}$.

$S^{2}$$C_{0}$$C_{1}$$C_{2}$ Figure 1. $C_{2}$ has 4 components, continuing
the pattern as shown.

The Alexander Horned Sphere can be created as the image of $S^{2}$ under a
surjective map $g^{\prime}:S^{3}\rightarrow S^{3}$ whose point pre-images are
exactly the elements of $\mathcal{D_{A}}$. The function $g^{\prime}$ can
easily be constructed as the limit of homeomorphisms
$\\{g_{i}^{\prime}:S^{3}\rightarrow S^{3}\\}_{i=0}^{\infty}$ that increasingly
compress the arcs of $\mathcal{D_{A}}$ toward their endpoints off $S^{2}$.
Describing this process in terms of quotient spaces, the function $g^{\prime}$
shows that $S^{3}/\mathcal{D}_{A}\cong S^{3}$, so the 3-sphere has not been
changed, but the 2-sphere $g^{\prime}(S^{2})$ is the _wild_ Alexander Horned
Sphere. Its wildness is reflected in the fact that the right side component of
its complement is no longer simply connected but instead has an infinitely
generated fundamental group. The simple closed curve around the center of the
cylinder $C_{0}$ is an example of a non-trivial element of the fundamental
group of the wild component of $S^{3}-g^{\prime}(S^{2})$. The image under
$g^{\prime}$ of the arcs in $\mathcal{D_{A}}$ is a wild Cantor set that lies
on the Alexander Horned Sphere.

Consider the two _closed_ complementary regions of $S^{3}-g^{\prime}(S^{2})$.
The closed left side region is clearly homeomorphic to the ball $B^{3}$,
whereas the closed right side region, the Alexander Horned Ball, is clearly
not, having a non-simply connected interior.

Now we create a new decomposition $\mathcal{D}$ of $S^{3}$ by creating non-
trivial elements that are symmetric across $S^{2}$. Let $I_{0}$ be the
standard involution of $S^{3}$ with fixed point set $S^{2}$, that is,
$I_{0}(x,y,z)=(-x,y,z)$. The non-degenerate elements of $\mathcal{D}$ are the
components of the infinite intersection
$\bigcap_{i=0}^{\infty}\widetilde{\mathcal{T}}_{i}$, where each
$\widetilde{\mathcal{T}}_{i}=C_{i}\cup I_{0}(C_{i})$. So each
$\widetilde{\mathcal{T}}_{i}$ is the union of $2^{i}$ tori, each torus of
which meets $S^{2}$ in two meridional disks. We will denote each torus
component of $\widetilde{\mathcal{T}}_{i}$ by $\widetilde{T}_{\sigma}$ where
$\sigma$ is a binary string of length $i$ and $\widetilde{T}_{\sigma 0}$ and
$\widetilde{T}_{\sigma 1}$ are the two component tori of
$\widetilde{\mathcal{T}}_{i+1}$ contained in $\widetilde{T}_{\sigma}$.
$\widetilde{\mathcal{T}}_{0}$, $\widetilde{\mathcal{T}}_{1}$, and
$\widetilde{\mathcal{T}}_{2}$ are drawn in Figure 2. Notice that
$\bigcap_{i=0}^{\infty}\widetilde{\mathcal{T}}_{i}$ is a Cantor set’s worth of
arcs, each one piercing $S^{2}$ at its center point. Those arcs comprise the
non-degenerate elements of $\mathcal{D}$.

$S^{2}$$\widetilde{\mathcal{T}}_{2}$$\widetilde{\mathcal{T}}_{0}$$\widetilde{\mathcal{T}}_{1}$
Figure 2.

In his 1952 paper, Bing showed that there is a surjective map
$g:S^{3}\rightarrow S^{3}$ whose point pre-images are precisely the sets in
$\mathcal{D}$, thereby producing an involution $I:S^{3}\rightarrow S^{3}$
defined by $I(x)=gI_{0}g^{-1}(x)$. The involution $I$ is wild since its fixed
point set is the wild 2-sphere $g(S^{2})$, and the closure of each component
of the complement of $g(S^{2})$ is an Alexander Horned Ball. The involution
$I$ swaps sides across a wild sphere.

Bing’s map $g:S^{3}\rightarrow S^{3}$ is produced as the limit of
homeomorphisms $\\{g_{i}:S^{3}\rightarrow S^{3}\\}_{i=0}^{\infty}$. The
$g_{i}$ homeomorphisms eventually shrink the non-degenerate elements of
$\mathcal{D}$ to increasingly smaller diameters as $i$ increases in such a way
that the limit $g$ has the property that each non-degenerate element of
$\mathcal{D}$ shrinks to a point and $\\{g^{-1}(x)|x\in S^{3}\\}=\mathcal{D}$.
So the challenge is to produce the homeomorphisms $g_{i}$ that shrink the arcs
of $\mathcal{D}$.

_Bing’s shrinks_. The astonishing conclusion of [bing52] is that such
shrinking homeomorphisms exist. Let us review two shrinks of $\mathcal{D}$
that Bing published in [bing52] and [bing88] . In both shrinks, Bing reduces
the shrinks to an essentially 1D model where the only important measure of
diameter is displacement along the $x$-axis. Denoting the “Bing tori”
dyadically, he lays them out along the $x$-axis and measures their diameters
discretely by choosing a large integer $n$ and erecting parallel planes in
intervals of $\frac{1}{n}$. So, the original tori are positioned as in Figure
3.

$\widetilde{T}$$\widetilde{T}_{0}$$\widetilde{T}_{1}$$\widetilde{T}_{00}$$\widetilde{T}_{01}$$\widetilde{T}_{10}$$\widetilde{T}_{11}$$\longhookleftarrow$$\longhookleftarrow$$\hookleftarrow$$\hookleftarrow$$\hookleftarrow$$\hookleftarrow$$\dots$$\widetilde{T}$$\widetilde{T}_{0}$$\widetilde{T}_{1}$$\widetilde{T}_{00}$$\widetilde{T}_{01}$$\widetilde{T}_{11}$$\widetilde{T}_{10}$
Figure 3.

Bing’s original shrink in his 1952 paper was accomplished by describing a
sequence of homeomorphisms $\\{g_{i}\\}_{i=0}^{\infty}$. Each homeomorphism
$g_{i+1}$ agrees with $g_{i}$ on $S^{3}-\widetilde{\mathcal{T}_{i}}$. The
homeomorphisms $g_{i}$ are defined in sets, meaning we first define the first
$n_{1}$ $g_{i}$’s and then sort of pause while we celebrate a certain amount
of shrinking, for example, we could choose our first collection of
homeomorphisms such that for every component torus $\widetilde{T}_{\sigma}$ of
$\widetilde{\mathcal{T}}_{n_{1}}$, diam($g_{n_{1}}(\widetilde{T}_{\sigma})$)
meets some diameter goal, say $<\frac{2}{10^{1}}$. Then we start with a new
diameter goal, say, $<\frac{2}{10^{2}}$ and create the next set of $g_{i}$’s,
say $i=n_{1}+1,...,n_{2}$. It is only at the end of each set of homeomorphisms
that the diameters shrink. In other words, when we look at the diameters of
the components of
$\\{g_{j}(\widetilde{\mathcal{T}}_{j})\\}_{j=n_{1}+1}^{n_{2}-1}$, generally
those images of tori do not have increasingly smaller diameters; however,
finally at stage $n_{2}$, for each torus
$\widetilde{T}_{\sigma}\subset\widetilde{\mathcal{T}}_{n_{2}}$,
diam$(g_{n_{2}}(\widetilde{T}_{\sigma}))<\frac{2}{10^{2}}$.

The homeomorphisms $\\{g_{i}\\}_{i=1}^{n_{1}}$ are defined as follows: let’s
choose our first goal to be to shrink each torus at some stage to diameter
less than, say, $\frac{2}{10^{1}}$. Imagine the first stage torus as long,
thin, and flat. Then position sufficiently many parallel planes, say $n_{1}$
planes, such that the distance between consecutive parallel planes is less
than $\frac{1}{10^{1}}$ and such that the torus $\widetilde{T}_{0}$ intersects
each of the $n_{1}$ planes in a pair of meridional disks. The first
homeomorphism $g_{1}$ leaves $S^{3}-\widetilde{\mathcal{T}}_{0}$ fixed and
rotates $\widetilde{T}_{0}\cup\widetilde{T}_{1}$ in
$\widetilde{\mathcal{T}}_{0}$ such that each intersects one fewer plane (see
Figure 4). For simplicity, for each component torus
$\widetilde{T}_{\sigma}\subset\widetilde{\mathcal{T}}_{i}$, we will denote
$g_{i}(\widetilde{T}_{\sigma})$ by $T_{\sigma}$. So after the rotation and
using our new notation, $T_{0}$ and $T_{1}$ each intersects only $n_{1}-1$
planes. In general, each subsequent pair of daughter tori are rotated in their
already moved mother so that each of the $k$-stage daughters meet only
$n_{1}-k$ planes (Figure 4).

etc.$T_{1}$$T_{0}$$T_{00}$$T_{01}$$T_{00}$$T_{01}$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$$T_{01}$$T_{00}$$T_{000}$$T_{001}$$T_{011}$$T_{010}$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$
Figure 4.

After $n_{1}$ generations, no $T_{\sigma}$, $\sigma$ a binary word of length
$\left\lvert\sigma\right\rvert=n_{1}$, meets more than one of the planes, so
its $x$-axis extent is $<\frac{2}{10^{1}}$. Normal to the $x$-axis, we are
free to have chosen a strong compression, so this procedure produces a
homeomorphism $g_{n_{1}}$ that shrinks each $n_{1}$-stage torus to diameter
less than $\frac{2}{10}$. Shrinking the tori of course shrinks the
decomposition elements therein.

Next we choose a new diameter goal, say, $\frac{2}{10^{2}}$. Make many tick
marks (actually, meridional disks on parallel planes) along the partially
shrunk $T_{\sigma}$’s, $|\sigma|=n_{1}$ such that the distance between
consecutive (around $T_{\sigma}$) meridional disks chosen is less than
$\frac{1}{10^{2}}$. Now start our process over. That is, let $g_{n_{1}}+1$
rotate the two daughters in each $T_{\sigma}$ in such a way that those
daughters each intersect one fewer meridional disk. Continue defining the
$g_{i}$’s, each reducing the number of meridional disks intersected by each
stage torus, until we reach a number $n_{2}$ such that every $T_{\sigma}$
where $|\sigma|=n_{2}$ meets at most one meridional disk. Again by compressing
dimensions other than the $x$-extent means every such $T_{\sigma}$ has
diameter less than $\frac{2}{10^{2}}$, as desired. Notice that, because of the
folded nature of the $T_{\sigma}$’s where $|\sigma|=n_{1}$, the images under
the $g_{i}$’s starting with $i=n_{1}+1$ do not decrease the diameters of the
$T_{\sigma}$’s for a long time, but when we reach $n_{2}$, we can again pause
to celebrate successful shrinking.

So after sufficient celebration, we start again and repeat the process with an
even more ambitiously small diameter goal. Continue producing such $g_{i}$’s.
In the limit, the $g_{i}$’s converge to a surjective function
$g:S^{3}\rightarrow S^{3}$ whose non-degenerate point pre-images are precisely
the non-degenerate elements of $\mathcal{D}$. This then was Bing’s original
method of shrinking the decomposition $\mathcal{D}$.

Next we summarize Bing’s 1988 shrink, which he produced in answer to questions
we asked him at that time. In his 1988 shrink, every other rotation of tori is
_greedy_ , as it tries (usually in vain) to cut diameters in half. The
alternate rotations are _patient_. It turns out greed does not speed the
shrinking; it is only at steps indexed by $2^{n}-1$, that diameters _actually_
are halved. So, again, many steps are taken during which no diameter shrinking
is accomplished. In pictures here is the idea of Bing’s 1988 shrink [bing88]
(Figure 5).

$\downarrow$ Step 1. Diam halves$\swarrow$ Step 2. Diam
unchanged$\searrow$$\ddots$$\searrow$ Step 3. Diam
halves$\swarrow$$\ddots$$\downarrow$ Step $2^{n}-1$. Diam halves$\vdots$
Figure 5.

##  2\. A Small Displacement Bing Shrink

As described in the introduction, every known shrink of the Bing decomposition
consists of starting with a standard torus and describing at each stage how to
displace the clasp points of the two daughters relative to the two clasp
points of the mother. For known shrinks, the clasps can be imagined as
arbitrarily tight, and the reduction of diameter can be studied in a strictly
1D model where the starting torus is configured to tightly surround the unit
interval. In this way, the diameter of a torus at a future stage is measured
by the $x$-axis width that that folded solid torus covers of $[0,1]$.

Describing the positions of the folded tori at each stage can therefore be
captured as a binary tree of functions into $[0,1]$ as follows. We start with
an example.

$x$-axis10$\frac{1}{3}$$\frac{3}{4}$$T_{0}$$T_{\varnothing}$$T_{1}$ Figure 6.

The first, straight torus $T_{\varnothing}$ is modeled by the identity
function $f_{\varnothing}:[0,1]\rightarrow[0,1]$. In the example in Figure 6,
the clasp points of the daughters $T_{0}$ and $T_{1}$ are displaced by
distance $\frac{1}{3}$ at the left end and distance $\frac{1}{4}$ at the right
end.

The following two functions, $f_{0}$ and $f_{1}$, describe the configurations
of $T_{0}$ and $T_{1}$.
$f_{0}:[-\frac{1}{3},\frac{3}{4}]\rightarrow[0,\frac{3}{4}]$ takes the
interval $[-\frac{1}{3},0]$ backwards from $\frac{1}{3}$ to 0 and then
proceeds forwards from 0 to $\frac{3}{4}$, thus modeling the shape of $T_{0}$
in $T$. Likewise,
$f_{1}:[\frac{1}{3},\frac{1}{4}]\rightarrow[\frac{1}{3},\frac{5}{4}]$.

Next we describe how to inductively produce a binary tree of functions
$f_{\sigma}$ into $[0,1]$ whose images re-trace the patterns of the folded
tori created by displacing the clasping points of daughter pairs relative to
clasping points of their mothers. We define $f_{\varnothing}$ as the identity
map on $[0,1]$.

Suppose $\sigma$ is a finite, binary string and the function
$f_{\sigma}:[c_{\sigma},d_{\sigma}]\rightarrow[0,1]$ has been produced. The
string $\sigma$ has two daughters $\sigma 0$ and $\sigma 1$, and $f_{\sigma}$
will have two daughters $f_{\sigma 0}$ and $f_{\sigma 1}$. The passage from
$f_{\sigma}$ to its daughters depends on the choice of a pair of distinct
points $a_{\sigma},b_{\sigma}$ in $[c_{\sigma},d_{\sigma}]$, $c_{\sigma}\leq
a_{\sigma}\leq b_{\sigma}\leq d_{\sigma}$. We call $a_{\sigma}$ and
$b_{\sigma}$ the daughter clasp points. Technically, each is a pair of nearby
turning points, one for each daughter, but we abuse notation by denoting each
pair of turning points as a single clasp point. No harm will result. The rule
is that $f_{\sigma 0}$ re-traces $f_{\sigma}$ backwards from $a_{\sigma}$ to
$c_{\sigma}$, then forwards from $c_{\sigma}$ to $b_{\sigma}$, and $f_{\sigma
1}$ re-traces $f_{\sigma}$ forward from $b_{\sigma}$ to $d_{\sigma}$ and then
backwards from $d_{\sigma}$ to $a_{\sigma}$.

$2c-a$$c$$a$$b$$d$$2d-b$$f_{\sigma 0}$$f_{\sigma 1}$$f_{\sigma
0}:[2c-a,b]\rightarrow[0,1]$$f_{\sigma 0}(x)=f_{\sigma}(-x+2c),\ 2c-a\leq
x\leq c$$f_{\sigma 1}(x)=f_{\sigma}(x),\ c\leq x\leq b$, and$f_{\sigma
1}:[a,2d-b]\rightarrow[0,1]$$f_{\sigma 1}(x)=f_{\sigma}(x),\ a\leq x\leq
d$$f_{\sigma 1}(x)=f_{\sigma}(-x+2d),\ d\leq x\leq 2d-b$ Figure 7.

In Figure 7 all occurrences of $a$, $b$, $c$, and $d$ above implicitly carry a
$\sigma$ subscript, dropped for readability.

Abstracting the concept of shrinking a decomposition, specifically Bing’s
decomposition $\mathcal{D}$, there are many interesting choices for the tree
of _clasp points_ $\\{a_{\sigma},b_{\sigma}\\}$, and hence the tree of
functions $\\{f_{\sigma}\\}$.

######  Definition 1.

(1) A binary tree of functions $\\{f_{\sigma}\\}$ starting with
$f_{\varnothing}$ being the identity on $[0,1]$ and defined as above will be
called a _Bing tree of functions_. (2) A Bing tree of functions
$\\{f_{\sigma}\\}$ _shrinks_ iff
$\operatorname{diam}(\operatorname{im}(f_{\sigma}))\rightarrow 0$ whenever the
bit string length $\left\lvert\sigma\right\rvert\rightarrow\infty$.

Let’s refer to $\left\lvert a_{\sigma}-c_{\sigma}\right\rvert$ and
$\left\lvert d_{\sigma}-b_{\sigma}\right\rvert$ as displacements. If the two
displacements are equal, we can say the daughters are _rotated_ in the mother.
Earlier, we put rotation in quotes because, in our construction, the
displacements will not be exactly equal. The previously known shrinks of the
Bing decomposition all included some large displacements, that is, instances
where $\left\lvert a_{\sigma}-c_{\sigma}\right\rvert$ and $\left\lvert
d_{\sigma}-b_{\sigma}\right\rvert$ were relatively large compared to
$\left\lvert d_{\sigma}-c_{\sigma}\right\rvert$. However, we show in this
paper that it is possible to construct a Bing shrink, or equivalently, a Bing
tree of functions $\\{f_{\sigma}\\}$ that shrinks even though all the
displacements are small. The shrink we will produce has the additional
property that the length of each torus, or equivalently the domain of each
$f_{\sigma}$, grows by less than any desired quantity.

Before constructing our small displacement shrink, let’s make some
observations about the functions $f_{\sigma}$ in a Bing tree of functions.
When visualizing the following, it might be useful to imagine the
displacements, that is, the $\left\lvert a_{\sigma}-c_{\sigma}\right\rvert$’s
and $\left\lvert d_{\sigma}-b_{\sigma}\right\rvert$’s as very small compared
to the length of $\left\lvert d_{\sigma}-c_{\sigma}\right\rvert$. Note that
each function $f_{\sigma}:[c_{\sigma},d_{\sigma}]\rightarrow[0,1]$ is
piecewise linear with each piece having slope $\pm 1$.

For specificity, we will discuss the daughter $f_{\sigma 0}$ of $f_{\sigma}$,
the case for $f_{\sigma 1}$ being similar. The domain of $f_{\sigma}$, namely,
$[c_{\sigma},d_{\sigma}]$ shifts downward to create the domain of $f_{\sigma
0}$, namely,
$[c_{\sigma}-(a_{\sigma}-c_{\sigma})=2c_{\sigma}-a_{\sigma},b_{\sigma}]$.
Let’s think about the relationship between the images of $f_{\sigma}$ and
$f_{\sigma 0}$.

First notice that the domains of $f_{\sigma}$ and $f_{\sigma 0}$ share the
interval $[c_{\sigma},b_{\sigma}]$, and, therefore, agree there. The function
$f_{\sigma 0}$ is defined on the additional interval
$[2c_{\sigma}-a_{\sigma},c_{\sigma}]$, but $f_{\sigma
0}([2c_{\sigma}-a_{\sigma},c_{\sigma}])$
$=f_{\sigma}([c_{\sigma},a_{\sigma}])$ as sets, so no new points are added to
the image of $f_{\sigma 0}$ compared to $f_{\sigma}$. So
$\operatorname{im}(f_{\sigma 0})\subset\operatorname{im}(f_{\sigma})$. These
observations prove the following lemma that records how an $f_{\sigma}$
relates to its ancestors.

######  Lemma 1.

Let the binary string $\tau$ be an ancestor of $\tau^{\prime}$ in a Bing tree
of functions. Let
$M[\tau,\tau^{\prime}]=\max\\{a_{\mu}\mid\tau\leq\mu\leq\tau^{\prime}\\}$ and
$m_{[\tau,\tau^{\prime}]}=\min\\{b_{\mu}\mid\tau\leq\mu\leq\tau^{\prime}\\}$.
Then

  1. (1)

$f_{\tau^{\prime}}$ restricted to
$[M_{[\tau,\tau^{\prime}]},m_{[\tau,\tau^{\prime}]}]=f_{\tau}$ restricted to
$[M_{[\tau,\tau^{\prime}]},m_{[\tau,\tau^{\prime}]}]$

  2. (2)

$\operatorname{length}(f_{\tau^{\prime}}([c_{\tau^{\prime}},M_{[\tau,\tau^{\prime}]}]))\leq\max\\{\left\lvert
a_{\mu}-c_{\mu}\right\rvert\mid\tau\leq\mu\leq\tau^{\prime}\\}$

  3. (3)

$\operatorname{length}(f_{\tau^{\prime}}[m_{[\tau,\sigma]},d_{\tau^{\prime}}])\leq\max\\{\left\lvert
d_{\mu}-b_{\mu}\right\rvert\mid\tau\leq\mu\leq\tau^{\prime}\\}$,

where $\operatorname{length}()$ denotes the length of a subset of $[0,1]$.

During the coming construction of the $f_{\sigma}$’s, from time to time we
will pause at an $f_{\tau}$ and then construct descendants in a prescribed
manner until we pause again at an $f_{\tau^{\prime}}$. The pause positions
will have the property that either
$c_{\tau^{\prime}}=M_{[\tau,\tau^{\prime}]}$ or
$d_{\tau^{\prime}}=m_{[\tau,\tau^{\prime}]}$. In such a case,
$f_{\tau^{\prime}}$ consists of two parts—one part _re-traces_ $f_{\tau}$,
namely (1) $f_{\tau^{\prime}}$ restricted to
$[M_{[\tau,\tau^{\prime}]},m_{[\tau,\tau^{\prime}]}]=f_{\tau}$ restricted to
$[M_{[\tau,\tau^{\prime}]},m_{[\tau,\tau^{\prime}]}]$, but if
$c_{\tau^{\prime}}=M_{[\tau,\tau^{\prime}]}$ or
$d_{\tau^{\prime}}=m_{[\tau,\tau^{\prime}]}$, then either the
$M_{[\tau,\tau^{\prime}]}=c_{\tau^{\prime}}$ or
$m_{[\tau,\tau^{\prime}]}=d_{\tau^{\prime}}$, and (2) a highly folded mapping
with image $f_{\tau^{\prime}}([c_{\tau^{\prime}},M_{[\tau,\tau^{\prime}]}])$
or $f_{\tau^{\prime}}([m_{[\tau,\tau^{\prime}]},d_{\tau^{\prime}}])$. By the
Lemma, the length of the images of the folds is less than
$\epsilon_{D_{[\tau,\tau^{\prime}]}}=$ the maximum displacement among $\mu$’s
between $\tau$ and $\tau^{\prime}$, which is either the right hand side of (2)
or (3) in Lemma 1. We will call such an $f_{\tau^{\prime}}$ a
$[A,\epsilon$-$W]$ function, where $A$ is the part of the domain on which
$f_{\tau^{\prime}}$ agrees with or re-traces $f_{\tau}$ and $\epsilon$ is a
bound on the length of the image of the remainder of the domain—the ’wiggles’,
hence the use of the letter ’$W$’. Notice that in the domain of an
$[A,\epsilon$-$W]$ function, the $A$ part could either be an interval at the
lower end of the domain, as suggested by the notation, or the higher end of
the domain interval.

$c_{\tau}$$d_{\tau}$$c_{\tau}$$c_{\tau^{\prime}}$$d_{\tau}$$d_{\tau^{\prime}}$$]$$<\epsilon_{D_{\tau,\tau^{\prime}}}$
Figure 8. A $[A,\epsilon$-$W]$ function. Note that $d_{\tau^{\prime}}$ may be
less than $d_{\tau}$, and that the part with $\epsilon$-bounded image wiggles
could be at the lower end.

So now we are ready to construct a shrink that uses only small displacements.

######  Theorem 1.

For any $\epsilon_{L}>0$ and
$\\{\epsilon_{i}>0\mid\sum_{i=1}^{\infty}\epsilon_{i}^{2}=\infty\\}$, there
exists a Bing tree of functions $\\{f_{\sigma}\\}$ that shrinks such that for
every $\sigma$,

  1. (1)

$\left\lvert d_{\sigma}-c_{\sigma}\right\rvert<1+\epsilon_{L}$, and

  2. (2)

for $\left\lvert\sigma\right\rvert=i$, $\left\lvert
a_{\sigma}-c_{\sigma}\right\rvert<\epsilon_{i}$ and $\left\lvert
d_{\sigma}-b_{\sigma}\right\rvert<\epsilon_{i}$.

###### Proof.

We will describe the functions $f_{\sigma}$ by describing the domain intervals
$[c_{\sigma},d_{\sigma}$]. Notice that knowing the domain intervals of
$f_{\sigma}$ and of $f_{\sigma 0}$ automatically implies what $a_{\sigma}$ and
$b_{\sigma}$ are and what $f_{\sigma 1}$ is. Specifically, if the domain of
$f_{\sigma}$ is $[c_{\sigma},d_{\sigma}]$ and the domain of $f_{\sigma 0}$ is
$[c_{\sigma 0},d_{\sigma 0}]$, then
$a_{\sigma}=c_{\sigma}+(c_{\sigma}-c_{\sigma 0})=2c_{\sigma}-c_{\sigma 0}$ and
$b_{\sigma}=d_{\sigma 0}$. So the domain of $f_{\sigma 1}$ is also determined.

It will be convenient to associate each domain interval
$[c_{\sigma},d_{\sigma}]$ with the point in the plane
$(c_{\sigma},d_{\sigma})$. Every such point $(c_{\sigma},d_{\sigma})$ lies
above the main diagonal $\Delta$ and the horizontal (or vertical) distance
from $\Delta$ to $(c_{\sigma},d_{\sigma})$ equals the length of
$[c_{\sigma},d_{\sigma}]$. Notice that interval inclusion, that is,
$[u,v]\subset[c_{\sigma},d_{\sigma}]$, corresponds to the point
$(c_{\sigma},d_{\sigma})$ lying in the NW quadrant with respect to $(u,v)$.

Notice that the vector from the point in the plane $(c_{\sigma},d_{\sigma})$
to $(c_{\sigma 0},d_{\sigma 0})$,
$\overrightarrow{(c_{\sigma}-a_{\sigma},b_{\sigma}-d_{\sigma})}$, is the
negative of the vector from $(c_{\sigma},d_{\sigma})$ to $(c_{\sigma
1},d_{\sigma 1})$. These observations will allow us to construct the desired
$f_{\sigma}$’s by describing a binary tree of points in the plane.

Our goal is to create $(c_{\sigma},d_{\sigma})$’s with length
$<1+\epsilon_{L}$ and
$\operatorname{length}(f_{\sigma}([c_{\sigma},d_{\sigma}]))$ getting short.
Our iterative procedure always begins at a $[A,\epsilon$-$W]$ function.
Suppose $(c_{\tau},d_{\tau})$ is a $[A,\epsilon$-$W]$ function. Then
$\operatorname{length}(f_{\tau}(c_{\tau},d_{\tau}))$ is less than the length
of the domain of the re-trace part plus $\epsilon$, the maximum length of the
image of the part with small wiggles.

_The strategy in brief_. (1) Sometimes we work on shortening the domain of the
re-trace part while keeping the same $\epsilon$ of the wiggle part. Since all
functions are piecewise linear with slopes of each piece $\pm 1$, the image of
the re-trace part will always be less than or equal to the length of the
domain of the re-trace part. (2) After shortening the re-trace part, the
$\epsilon$ of the wiggle part limits how short the image is, hence the wiggle
part needs attention. So we then choose a smaller $\epsilon$, say
$\frac{\epsilon}{2}$, and begin again, that is, we think of the entire
interval $[c_{\tau},d_{\tau}]$ as the re-trace part and proceed to shorten the
re-trace part using $\frac{\epsilon}{2}$-wiggles.

This process of shortening the re-trace part domain—in a process that requires
many stages—with the same size $\epsilon$-wiggles, followed by starting again
with a smaller $\epsilon$ yields the result that the images of descendant
$f_{\sigma}$’s eventually have increasingly smaller images.

So now we need to describe the procedure for moving from point
$(c_{\sigma},d_{\sigma})$ to points $(c_{\sigma 0},d_{\sigma 0})$ and
$(c_{\sigma 1},d_{\sigma 1})$.

_Shrinking the re-trace part._ Suppose $f_{\tau}$ is a $[A,\epsilon$-$W]$
function. That is, $f_{\tau}:[c_{\tau},d_{\tau}]\rightarrow[0,1]$ is a
piecewise linear function with each piece having slope $\pm 1$, $\left\lvert
d_{\tau}-c_{\tau}\right\rvert<1+\epsilon_{L}$ and there is a
$t\in[c_{\tau},d_{\tau}]$ such that (i)
$\operatorname{length}(f_{\tau}([t,d_{\tau}]))<\epsilon$ or (ii)
$\operatorname{length}(f([c_{\tau},t]))<\epsilon$. Say (i) is the case (the
(ii) case being similar), so $[c_{\tau},t]$ is the re-trace part. Then there
exists a finite Bing tree of functions starting with $f_{\tau}$ such that for
every final descendant $f_{\tau^{\prime}}$, $f_{\tau^{\prime}}$ is a
$[A,\epsilon$-$W]$ function such that $\left\lvert
d_{\tau^{\prime}}-c_{\tau^{\prime}}\right\rvert<1+\epsilon_{L}$ and there is a
$t^{\prime}\in[c_{\tau^{\prime}},d_{\tau^{\prime}}]$ such that either
(i)$\left\lvert
t^{\prime}-c_{\tau^{\prime}}\right\rvert<\frac{2}{3}\left\lvert
t-c_{\tau}\right\rvert$ and
$\operatorname{length}(f_{\tau^{\prime}}([t^{\prime},d_{\tau^{\prime}}]))<\epsilon$
or (ii) $\left\lvert
t^{\prime}-d_{\tau^{\prime}}\right\rvert<\frac{2}{3}\left\lvert
t-c_{\tau}\right\rvert$ and
$\operatorname{length}(f_{\tau^{\prime}}([c_{\tau^{\prime}},t^{\prime}]))<\epsilon$.
In other words, the length of the domain of the re-trace part of
$f_{\tau^{\prime}}$ is less than $\frac{2}{3}$ the length of the domain of the
re-trace part of $f_{\tau}$, while the image of the remainder of the domain
has length less than $\epsilon$.

Figure 9 shows the NW quadrant over the point corresponding to the middle
third of the re-trace part domain of $f_{\tau}$. The re-trace part domain of
$f_{\tau}$ is $[c_{\tau},t]$. So, the middle third is
$[c_{\tau}+\frac{1}{3}(t-c_{\tau}),c_{\tau}+\frac{2}{3}(t-c_{\tau})]$.

$(c_{\tau},d_{\tau})$$(c_{\tau 0},d_{\tau 0})$$(c_{\tau 1},d_{\tau
1})$$(c_{\tau}+\frac{1}{3}(t-c_{\tau}),$$c_{\tau}+\frac{2}{3}(t-c_{\tau})$Blow-
up view$(c_{\tau 0},d_{\tau 0})$$(c_{\tau 00},d_{\tau 00})$$(c_{\tau
01},d_{\tau 01})$$(c_{\tau 001},d_{\tau 001})$$(c_{\tau 010},d_{\tau 010})$
Figure 9.

The figure also suggests a large number of concentric circles centered at some
distant point in that NW quadrant on the slope $-1$ ray from
$(c_{\tau}+\frac{1}{3}(t-c_{\tau}),c_{\tau}+\frac{2}{3}(t-c_{\tau}))$ heading
up and left. Recall that $\left\lvert
d_{\tau}-c_{\tau}\right\rvert<1+\epsilon_{L}$. The center point of the circles
is so distant that every point $(x,y)$ on the arc of the circle containing
$(c_{\tau},d_{\tau})$ in the pictured NW quadrant has $y-x<1+\epsilon_{L}$.
This choice of circle center will guarantee that no point
$(c_{\sigma},d_{\sigma})$ that is created during our process has length
greater than $1+\epsilon_{L}$.

The bullseye pattern of circles is chosen such that for any point on one of
those circles, the distance along the tangent to the next circle is
$<\min\\{\epsilon,\epsilon_{i}\\}$ where $\epsilon$ is the wiggle width
$\epsilon$ and $i=\left\lvert\sigma\right\rvert$ at each stage.

Now we are ready to construct our binary tree of points
$(c_{\sigma},d_{\sigma})$. We begin at $(c_{\tau},d_{\tau})$ and proceed along
the tangent of its circle in both directions until we hit the next circle.
Those two points will be $(c_{\tau 0},d_{\tau 0})$ and $(c_{\tau 1},d_{\tau
1})$. From each of those points we do the same thing—that is, from $(c_{\tau
0},d_{\tau 0})$ we move along the tangent of its circle to find the points
$(c_{\tau 00},d_{\tau 00})$ and $(c_{\tau 01},d_{\tau 01})$ on the next
circle.

We continue creating this binary tree of points until we arrive at a point
$(c_{\tau_{1}},d_{\tau_{1}})$ where
$c_{\tau_{1}}>c_{\tau}+\frac{1}{3}(t-c_{\tau})$ or
$d_{\tau_{1}}<c_{\tau}+\frac{2}{3}(t-c_{\tau})$, that is, when
$(c_{\tau_{1}},d_{\tau_{1}})$ does not lie in the NW quadrant with respect to
$(c_{\tau}+\frac{1}{3}(t-c_{\tau}),c_{\tau}+\frac{2}{3}(t-c_{\tau}))$. At such
a point we pause to state a simple lemma.

######  Lemma 2.

Let $\\{\epsilon_{i}>0\mid\sum_{i=0}^{\infty}\epsilon_{i}^{2}=\infty\\}$ and
let $\\{c_{i}\\}_{i=1}^{\infty}$ be a nested sequence of concentric circles in
$\mathbb{R}^{2}$ centered at point $c$ such that for every $i$, the radius of
$C_{i}$ is $r_{i}$ and the distance from a point on $C_{i}$ along the tangent
to $C_{i}$ to a point on $C_{i+1}$ is $\epsilon_{i}$. Then the sequence of
radii $(r_{i})_{i=1}^{\infty}$ is unbounded.

###### Proof.

Let $M\in\mathbb{R}^{+}$. Given the hypotheses we will show that there is a
$k$ such that $r_{k}\geq M$. If not, then for every $i$, $r_{i}<M$.

$C_{i+1}$$C_{i}$$c$$r_{i}$$\epsilon_{i}$$r_{i+1}$

By the Pythagorean Theorem,

$r_{i+1}^{2}-r_{i}^{2}=\epsilon_{i}^{2}\text{, so
}r_{i+1}-r_{i}=\frac{\epsilon_{i}^{2}}{r_{i+1}+r_{i}}>\frac{\epsilon_{i}^{2}}{2M}$

But $\sum\frac{\epsilon_{i}^{2}}{2M}$ diverges, meaning some $r_{k}>M$. ∎

Let us now notice two things: (1) Every path branch of the binary tree of
points stops by a predictable stage since our hypothesis about the divergence
of the sum of the squares of the $\epsilon_{i}$’s guarantees that there are
only a finite number of circles before the point
$(c_{\tau}+\frac{1}{3}(t-c_{\tau}),c_{\tau}+\frac{2}{3}(t-c_{\tau}))$ is
itself inside the circle. Any point constructed with our procedure on that
circle must be outside the NW quadrant. (2) Since we stop at the first time
that $c_{\tau_{1}}>c_{\tau}+\frac{1}{3}(t-c_{\tau})$ or
$d_{\tau_{1}}<c_{\tau_{1}}+\frac{2}{3}(t-c_{\tau})$, say
$c_{\tau}>c_{\tau}+\frac{1}{3}(t-c_{\tau})$, then $c_{\tau_{1}}$ must equal
$M_{[\tau,\tau_{L}]}$, so $f_{\tau_{1}}$ is a $[A,\epsilon$-$W]$ function.
That is, $f_{\tau_{1}}$ agrees with $f_{\tau}$ on
$[M_{[\tau,\tau_{1}]},m_{[\tau,\tau_{1}]}]=[c_{\tau_{1}},m_{[\tau,\tau_{1}]}]$
and $f_{\tau_{1}}([m_{[\tau,\tau_{1}]}],d_{\tau_{1}}])$ must have image length
less than $\epsilon$. Therefore, the re-trace part of each such $f_{\tau_{1}}$
must have domain length less than $\frac{2}{3}$ the domain length of the re-
trace part of $f_{\tau}$, while the image of the wiggly part has length less
than $\epsilon$; therefore,
$\operatorname{length}(f_{\tau_{1}}([c_{\tau_{1}},d_{\tau_{1}}])$ is less than
$\frac{2}{3}$ the domain length of the re-trace part of $f_{\tau}$; plus
$\epsilon$. And the domain of $f_{\tau_{1}}$, that is,
$[c_{\tau_{1}},d_{\tau_{1}}]$, has length less than $1+\epsilon_{L}$.

We can now repeat the process of shrinking the re-trace part using the same
$\epsilon$ size wiggles until the re-trace part itself has domain length less
than $\epsilon$. At that point, the total length of the image of that function
must be less than $2\epsilon$ and its domain length is less than
$1+\epsilon_{L}$. We now proceed to produce functions with images less than
$\epsilon$ by thinking of the entire domain as the re-trace part, but now
using $\frac{\epsilon}{2}$-wiggles. Then we repeat the entire process
infinitely often producing functions with image lengths less than
$\frac{\epsilon}{2}$, $\frac{\epsilon}{4}$, $\frac{\epsilon}{8},\dots$,
thereby creating the desired shrink.

∎

##  3\. Conclusion and Questions

Bing’s 1952 construction of a wild involution of $S^{3}$ opened the door to
many further insights—and many further questions. In [fs22], we proved that
all conjugates of Bing’s decomposition involution _must_ share certain
analytic features with the involution derived from Bing’s original shrinks.
That result confirmed what we guessed must be true. This paper contains an
example that was unexpected—previous shrinks of Bing’s decomposition suggested
that no example like the one in this paper could exist. So let us state some
questions without presuming to guess their answers.

######  Question 1.

The first question involves the rate at which small displacements can shrink.
Bing’s original method of shrinking and his shrinking without lengthening
method shrink stage tori to size about $\frac{1}{n}$ their original diameter
at stage order $n$. The analysis of [as89] shows that there is no faster way
to shrink the Bing decomposition $\mathcal{D}$. It appears if one imposes
constraints on displacement and lengthening, as we have here, shrinking must
be even slower. For example, if lengthening and displacement are restricted to
0.1%, our algorithm takes about $10^{12}$ stages to get from diameter $=1$ to
diameter $=0.001$. What is the actual functional form for shrinking using our
algorithm? And are there more efficient algorithms respecting the same
constraints?

######  Question 2.

Suppose we use the 1D model for shrinking in this paper. We showed that it is
possible to shrink the Bing decomposition with small displacement shrinks.
Bing’s shrinks and the shrink in this paper seem to require some insight or
even cleverness, but might that apparent cleverness be an illusion? Suppose
the $a_{\sigma}$’s and $b_{\sigma}$’s were simply chosen randomly in the
intervals $[c_{\sigma},d_{\sigma}]$. Would such a random selection lead to a
shrink of the Bing decomposition with probability $1$? If so, wouldn’t we feel
silly. Shrinking (or not) is a _tail event_ , meaning independent of any
initial segment of choices, so Kolmogorov’s 0-1 law tells us that within a
probabilistic model shrinking will occur with either probability 0 or 1.
Which, depends on the model. If the model is artificially concentrated near
our explicit shrink, the probability will be 1, but if
$\\{(a_{\sigma},b_{\sigma})\\}$ are independent and uniformly distributed, we
do not know.

It was surprising to us that Bing’s decomposition could be shrunk using only
tiny jiggles. Perhaps if we had not had an intuitive (and erroneous) sense of
what the truth should be, we might have discovered this construction sooner.
It is good to remember that the unknown is actually unknown.

Bing understood this. He told us that he would work for two hours trying to
prove that a conjecture is true and then he would spend two hours trying to
prove it was not true. He would learn things no matter which direction he was
exploring. He said it is important to not care which way it comes out.

## References


## Comment on ‘Open is not forever: a study of vanished open access journals’

Gerta Rücker
Institute of Medical Biometry and Statistics, Medical Center – University of
Freiburg
Stefan-Meier-Strasse 26
D-79115 Freiburg, Germany
e-mail<EMAIL_ADDRESS>
This is a comment to an article by Laakso, Matthias and Jahn
(arXiv:2008.11933).

With interest I read an article published by Laakso et al. [1] that was
already commented on by Shelomi [2]. Laakso et al. reported a study in which
they found 176 Open Access (OA) journals that had vanished from the web
between 2000 and 2019. As Shelomi noted, it was unfortunate that Laakso et al.
did not distinguish between predatory and reputable journals, and I agree. In
my present comment, however, I want to point to another weakness of the
article by Laakso et al., which is the statistical analysis. They consider
their sample of 176 vanished journals and present some descriptive statistics
on their characteristics, such as the time lag between the last publication
and vanishing (their Figure 3 and Table 3), the distribution of academic
disciplines (their Figure 4), their affiliation (whether academic or not), and
the geographic distribution (Figure 5 and Table 4). Of course, this is
legitimate as a description of a selected sample. However, the authors,
ignoring that their sample was selected conditionally on a journal having
vanished from the web, misinterpret their findings. Their statement that ‘our
study provides valuable insight into the types of OA journals that are
especially at risk of vanishing’ (page 22) is not justified by evidence,
simply because a risk can only be quantified on a population basis. Otherwise,
any analysis can only be descriptive.

Laakso et al. found that ‘North America and South Asia represent a
disproportionately larger share of vanished than active OA journals’ and
interpreted this as ‘journals published in North America …belong to the high
risk group’. For the geographic distribution this is justified, as they refer
to a list of active OA journals as a control group (Table 4). However, they
use the term ‘risk’ also for other factors, writing for example: ‘Moreover,
our study provides valuable insight into the types of OA journals that are
especially at risk of vanishing’, then referring to journals affiliated with
academic institutions or scholarly societies, without providing the numbers of
active OA journals for comparison. This might provoke a well known and common
error: the inversion fallacy, that is, confusing the denominators. To find out
whether scholarly-based journals are at higher risk of vanishing than other
journals, it is not correct to compare the proportions of journals by
affiliation _within vanished journals_. It would be necessary to compare
vanished and active journals and to adjust for covariates in a multivariable
analysis. The limitation of this study as a case series should have been
acknowledged by the authors.

## References

  * [1] Laakso M, Matthias L, Jahn N  (2020) Open is not forever: a study of vanished open access journals. (arXiv:2008.11933).
  * [2] Shelomi M  (2020) Comment on ”Open is not forever: a study of vanished open access journals”. (arXiv:2009.07561).


Laser-cooled gases of alkaline-earth atoms have led to revolutionary advances in atomic clocks [1] and precision measurement [2]. The magneto-optical trap (MOT), which consists of three pairs of intersecting laser beams and a magnetic quadrupole field, has been the dominant method of preparing ultracold atomic vapors since its invention decades ago [3]. The appropriate configuration of optical powers and state of circular polarization results in balanced conservative and dissipative forces that spatially confine and reduce the temperature of an atomic sample.  In a research laboratory, MOTs are typically built from an array of optical components that must be carefully aligned, taking up a volume of $\sim$1 m$^3$.  This level of complexity is a significant barrier to the development of portable cold atom systems [4, 5] that can address applications in geodesy [6] and inertial sensing [7].  In particular, integrated MOTs would greatly benefit the development of compact optical lattice clocks based on the ultra-narrow, spin-forbidden $^1$S$_0$ $\rightarrow$ $^3$P$_0$ transition in $^{87}$Sr and other alkaline-earth systems [1]. Recently, there has been a push towards using micro- or nanofabricated integrated photonic systems [8] that are compatible with foundry-scale manufacturing to reduce the MOT size, complexity and cost.

Metasurface (MS) functionality and their integration for two-stage Sr MOTs.
a) Each MS controls the deflection angle $\theta$, divergence angle $\alpha$ and the circular polarization (black arrow) of the 1$^{\text{st}}$-order beam.
b) SEM image of the MS.
c) Beam geometry and polarization in a conventional cubic MOT. LCP (RCP) stands for left (right) circular polarization.
d) Multi-wavelength MS integration on fiber-coupled, 3-inch wafers to produce all 12 trapping beams for 461 nm and 689 nm MOTs; wafer separation,  $\Delta z$.

Compared to alkali atoms like rubidium, strontium poses unique challenges to any photonic integration for MOTs. The broad linewidth of the $^1$S$_0$ $\rightarrow$ $^1$P$_1$ cooling transition at 461 nm and the lack of sub-Doppler cooling mechanisms requires a second stage MOT on the narrow $^1$S$_0$ $\rightarrow$ $^3$P$_1$ transition at 689 nm to reach $\mu$K temperatures. Previous work on integrated Sr MOTs with pyramid reflectors has suffered from limited optical access to the atom cloud [9]. The planarized geometry of grating MOTs [10] addresses this problem, but such systems are unable to achieve the level of precise beam control found in traditional MOTs built from bulk optics.  This limitation is problematic when cooling atoms with complex level structures such as $^{87}$Sr [11]. An alternative integration approach is to use metasurface (MS) optics which consist of periodic arrays of dielectric nanopillars [12].  These devices are capable of flexible control of the optical phase and offer the potential to perform the function of multiple traditional optics simultaneously in a single wafer-thickness optic. To date, atom trapping experiments with MSs have underutilized their multifunctional capabilities and have required auxiliary optical components to achieve the necessary level of control. Furthermore, the demonstrated MS beam delivery methods represent significant challenges to creating scalable systems.  Free space coupled MS systems [13, 14, 15] still require optical alignment and assembly  while guided mode approaches using photonic integrated circuits (PICs) [16, 17, 18] exhibit high propagation loss of distances of $\approx 10$ mm, especially at the visible wavelengths necessary to trap Sr [19].

Here, we demonstrate the integration of metasurface optics on a common substrate to generate the laser-beam configuration for Sr laser cooling at 461 nm and 689 nm. Our metasurface optics platform enables the generation of the complete beam configuration of a MOT directly from light emitted by optical fibers without any bulk free-space optics.   Additionally, our foundry-compatible integration strategy avoids the high-propagation loss found in PIC approaches and eliminates the need for assembly and optical alignment required for traditional MOTs. Instead, these features are automatically incorporated into our design at the nanophotonic-level by integrating multiple functionalities into a single MS optic, and at the system level by integrating MS designs for different beams and wavelengths on semiconductor wafers. Each MS simultaneously performs the beam formation functions of conventional steering mirrors and lenses, as well as the polarization transformation of quarter waveplates (QWPs). We present a comprehensive characterization of the design and performance of our nanofabricated structures, underscoring their role as fundamental building blocks in integrated, nanophotonic optical systems for diverse applications in cold-atom quantum computing and quantum sensing.

Figure <ref> introduces our metasurface optics devices and describes how we use them as compact replacements for traditional MOT optics.  The basis for beam control with MSs is the spatially dependent phase shift that results from the interaction of the optical beam with nanopillars of specific sizes placed across the transverse $x$-$y$ plane [12]. We set the size and distribution of the nanopillars to design a polarization-dependent MS phase function that transforms the linearly polarized input optical mode from an optical fiber into a $1^{\text{st}}$-order beam deflected by angle $\theta$ with efficiency $\eta$, focused to full divergence angle $\alpha$, and circularly polarized to ellipticity angle $\chi$; see Fig. <ref>(a). We use the standard definition for ellipticity angle $\chi = \arctan\left(b/a\right)$ where $a$ and $b$ are the major and minor axes of the polarization ellipse, $a \geq b$. The $1^{\text{st}}$-order output beam is used for trapping, but we note a residual, undeflected $0^{\text{th}}$-order beam and a weak negative $1^{\text{st}}$-order beam deflected by $-\theta$.

The MS device transforms the phase of the input beam according to the equation

\begin{eqnarray}
\phi(x,y) = k_0 x \sin\left(\theta\right) + k_0   \left(\lvert f_1 \rvert  - \sqrt{x^2 + y^2 +f_1^2}  \right)  \nonumber
\\
+ k_0   \left(\sqrt{x^2 + y^2 +f_2^2} -\lvert f_2 \rvert \right)
\label{eq:MSphase}
\end{eqnarray}

where $k_0 =\frac{2\pi}{\lambda}$ and $\lambda$ is the design wavelength.  We implement the phase map given by Eq. <ref> by wrapping the phase between 0 and $2\pi$. The first term in Eq. <ref> represents grating-like deflection at our designed angle of $\theta = \pi/4$ and the last two terms represent a superposition of focusing lenses with focal length $f_i$ that controls the beam shape.  $f_1$ is a convex lens used to collimate the input mode from the fiber and $f_2$ is a concave lens that controls the beam divergence angle $\alpha$.  The birefringent properties of our MS originate in the designed asymmetry in the nanopillars' transverse dimensions, $L_x$ and $L_y$; refer to the scanning electron microscope (SEM) image in Fig. <ref>(b).
To obtain beams with circular polarization, we choose a subset of pillars that results in a fixed quarter-wave phase lag between the two orthogonal components of the electric field, analogous to the fast and slow axes in a QWP.  This amounts to computing the phase shifts for light linearly polarized along $\hat{x}$ ($\phi_x$) and $\hat{y}$ ($\phi_y$) and determining which pillars satisfy $\phi_x =\phi_y + \pi/2$. Our MSs produce left circular polarization (LCP) when the input field is polarized along an azimuthal angle between $L_x$ and $L_y$ of $\beta_{\text{in}} = \pi/4$ and right circular polarization (RCP) for $\beta_{\text{in}} = -\pi/4$ where $\beta_{\text{in}}$ is measured from $L_x$.

Our MSs are composed of TiO$_2$ nanopillars arranged on a grid of periodicity $p$ = 220 nm (330 nm) for wavelength 461 nm (689 nm) with ($L_x,L_y$) varying from $0.3p$ to $0.8p$; an SEM image of a sample MS is shown in Fig. <ref>(b). We chose a pillar height of 900 nm to allow for efficient MS designs at each cooling wavelength to be integrated on single, 0.5-mm-thick fused silica wafer.  The nanostructures are fabricated by use of a Damascene process [20], which allows for the definition of high-aspect-ratio features.  We use electron beam lithography (EBL) to form 900-nm deep holes in a resist layer which is subsequently filled with atomic layer deposition of TiO$_2$.  The film is planarized with an etch step and the resist layer is removed with lift-off, leaving isolated TiO$_2$ pillars.

Our trapping beam geometry resembles the conventional MOT shown in Fig. <ref>(c) but is modified to enable the generation of three beams for each wavelength from a single wafer and all 12 beams from two wafers; see Fig. <ref>(d). We create counter-propagating beams that are circularly polarized with the same handedness (LCP or RCP) relative to fixed lab coordinates to appropriately drive the $\sigma^{\pm}$ transitions see loop arrows) in the atoms and direct the atoms toward the center of the trap. The diameter, $d$, and power $P_{\text{MOT}}$, of the beams are chosen to give the desired trapping volume and to efficiently cool atoms from our thermal oven source based on the saturation intensity for each transition as well as the available fiber-coupled laser power, $P_{\text{fiber}}$. Our system is designed to achieve a total beam intensity of 33 mW/cm$^2$ and 42 mW/cm$^2$ for the 461 nm and 689 nm transitions respectively.

Characterization of MS beams at 461 nm and 689 nm.  (a) and (b) show the degree of circular polarization, $DOCP$, (top panel) and deflection efficiency, $\eta$, (bottom panel) as a function of input linear polarization angle, $\beta_{\text{in}}$, for 461 nm and 689 nm, respectively. (c) Images of the 461 nm (blue) and 689 nm (red) beams at various heights $z$ showing a common intersection at $z=0$. (d) Measured beam widths vs $z$ and comparison to the expected trend of our design values for $\alpha = 10.3^{\circ}$ (461 nm) and $\alpha=2.6^{\circ}$ (689 nm).  The error bars in (a-d) are instrumental uncertainty at $95\%$ confidence.

Figure <ref> presents our characterization of the 461-nm and 689-nm MSs for the three functionalities outlined in Fig. <ref>(a). We first analyze the polarization transformation of the $1^{\text{st}}$-order beam through measurements of the degree of circular polarization, $DOCP$, as a function of the input azimuthal polarization angle, $\beta_{\text{in}}$; see the top panels of Figs. <ref>(a) and <ref>(b) respectively.  $DOCP$ represents the fraction of the optical power that is circularly polarized and is defined using the third Stokes parameter ($s_3$) so that $DOCP = \sin 2\chi$.  $DOCP = -1$ corresponds to LCP and $DOCP = +1$ to RCP.  Measurements at 461 nm and 689 nm are shown in the top panels of Fig. <ref>(a) and <ref>(b) respectively; the error bars are instrumental uncertainty at 95$\%$ confidence.  The equivalent QWP for our system should produce an output beam with an ellipticity angle $\chi_{\text{out}} = \beta_{\text{in}}$ from which we calculate the theoretical behavior $DOCP = \sin 2\beta_{\text{in}}$.  Our MSs reproduce this relation (blue and red lines) across the complete range of input angles, and reaches $|DOCP| \geq 0.95$ for both LCP and RCP at each wavelength.

Balanced optical powers in the counterpropagating beams are important for achieving balanced scattering forces and stable MOT operation.  We investigate the sensitivity of the beam power to misaligmnents in $\beta_{\text{in}}$ through measurements of the effective deflection efficiency defined as $\eta=P_{\text{MOT}}/P_{\text{fiber}}$.  We measure $\eta \approx 30 \%$ ($28.5 \%$) at 461 nm (689 nm), respectively, for both LCP and RCP and observe a weak sinusoidal dependence on input polarization angle; see bottom panels of Fig. <ref>(a) and <ref>(b) for data and fits.  We attribute the modulation to slightly different transmission for light polarized along the two transverse dimensions of each nanopillar.  We note that the overall efficiency is affected by a non-ideal overlap between the shape of the mode from the fiber and the shape of the finite-sized MS; see below for details about MS wafer layout.  When these effective aperture effects are taken into account, we find a fundamental MS efficiency of $\approx 45\%$ which is in good agreement with results from our simulations.  These measurements indicate that the dominant effect of misaligned polarization is to alter $DOCP$ and minimally affect $\eta$.

To reliably define the trapping volume and beam intensity, different MSs must generate beams with consistent orientations ($\theta = 45^{\circ}$) and shapes ($\alpha$).  We characterize these parameters by imaging the beams on a ground glass substrate at different heights above the wafer and performing routine image analysis to determine the beam positions and widths.  Figure <ref>(c) shows images of the beams generated by a single wafer at various heights.  All six beams come to an intersection at the trap center located at the midpoint between the wafers; see far right panel for relative height $z=0$. Figure <ref>(d) shows single beam measurements of the beam diameter with instrumental error bars.  At 461 nm (689 nm) we see excellent agreement with the expected trend shown as a blue line (red line) for the designed divergence angles of $\alpha = 10.3^{\circ}$ ($\alpha = 2.6^{\circ}$) and diameter at the trap center, $d=3$ mm ($d = 10$ mm).

System-level integration.  (a) A CAD drawing showing two wafers mounted over our vacuum chamber and magnetic field coils.  (b)  Wafer-level integration. (b,i) The 461 nm (blue ovals) and 689 nm MSs (red ovals) are laid out on the wafer with a minimal separation $\delta$, ensuring similar geometry in relation to the magnetic field gradient $\nabla |\mathbf{B}|$.  (b,ii) Fiber fixtures that mechanically set the mode diameter at the MS, $d_{\text{MS}}$, by controlling the distance $\Delta z_{\text{f}}$. c)  $DOCP$ (top) and $\eta$ (bottom) measurements for all twelve metasurfaces used in our system.

Figure <ref> presents the system-level integration of our photonics with our compact vacuum chamber and magnetic field coils for a dual-color Sr MOT; see Fig. <ref>(a) for a three-dimensional mechanical drawing of our system and the generated optical beams.  The 461-nm and 689-nm MSs are laid out on the wafer in a circle of radius $R$ with an azimuthal separation of $2\pi/3$; see Fig. <ref> (b,i) for a diagram of the top wafer as viewed from trap center.  We use elliptically shaped MSs with semi-major ($A$) and -minor ($B$) radii related by $A=\sqrt{2} B$ to achieve a circular beam cross-section after being deflected by $\theta = \pi/4$.  We choose major diameters of $2A=2.8$ mm (1.4 mm) at 461 nm (689 nm), which is compatible with EBL write-times $< 8$ hours.  We label the MS devices on the top (bottom) wafer by $T_i$ ($B_i$) where $i=1,2,3$ is an index that describes the MS location counting clockwise starting with the device opposite the wafer flat. The $i=1$ devices generate the `axial beams' that pass through both magnetic field coils with a direction of propagation (blue and red arrows) that is approximately parallel to the direction of the strongest magnetic field gradient, $\nabla |\textbf{B}|$ (black arrow). The top and bottom wafers differ only in the choice of $\delta$ (see description below) and a mirror flip along the wafer midline parallel to the wafer flat.  This results in the desired counter-propagating beam geometry when both wafers are mounted with their flats oriented opposite each other.  The distance between the wafers is set with precision mounts to be $\Delta z = 2R + \epsilon$ where $\epsilon$ is a small correction that accounts for angular beam deviation in the vacuum chamber windows (2.54 mm thick, refractive index $n\approx 1.52$).

Our MS fiber coupling strategy robustly generates beams with the desired diameter at the trap center in a compact, plug-and-play package.  We offset the 689 nm and 461 nm devices by an angle of $\delta$ to allow for the inclusion of compact, LC fiber connectors.  Our choice of $\delta =  -10^\circ$ ($\delta =  +10^\circ$) for the top (bottom) wafer is sufficiently small to ensure that the 689-nm beams approximately maintain the geometry of the 461-nm beams with respect to $\nabla |\mathbf{B}|$. We note that the lateral offset between our magnetic field coils and non-cubic propagation angle $\theta = \pi/4$ means that neither wavelength experiences the traditional projections along $\nabla |\textbf{B}| $ for a cubic MOT.  We manually rotate the fiber ferrule into the keyless fixture to set the input polarization to generate LCP or RCP as desired.  This step represents the only manual alignment in our system but we note that a small alignment key could be included in future systems.  We achieve the desired trapping beam diameter through control of $f_i$ in the second term in Eq. <ref> and the $1/e^2$ diameter of the optical mode incident on the MS, $d_{\text{MS}}$.  The fibers that deliver 461-nm (PM-S405-XP) and 689-nm (PM630-HP) light have similar NAs and divergence angles so that $d_{\text{MS}}$ is primarily controlled by the distance between the fiber and MS, $\Delta z_{\text{f}}$. To minimize aperture losses due to finite MS size, we set $d_{\text{MS}} = 2A$ by using precision mechanical mounts to define $\Delta z_{\text{f}}$ to be 16 mm at 461 nm and 7.3 mm at 689 nm; see Fig. <ref>(b,ii).  We also tightly control the lateral alignment between the optical fiber and the MS using high tolerance mechanical components to prevent errors in the beam-pointing direction that arise from off-axis coupling to the MSs.  These effects are most consequential for small-diameter beams where a small shift in position can lead to a large reduction in fractional overlap at the intersection volume.  For the 689-nm beams with a diameter of 3 mm at the trap center, a 140-$\mu$m error in the position of the fiber core can lead to a 50$\%$ reduction in the overlap at the trap center for two counterpropagating beams.

Finally, we evaluate the system-level performance of our MSs when used with COTS fiber components.  Figure <ref>(c) shows measurements of the average $(DOCP,\eta)$ that can be obtained over reasonable misalignments of the input polarization and the error bars represent instrumental uncertainty at $95\%$ confidence.  MS devices at 461 nm (689 nm) are grouped according to pairs of counterpropagating beams (e.g., $B_2$ and $T_3$) and displayed as blue (red) circles.  The $i=1$ axial beams are set to LCP ($DOCP = -1$) and the $i=2,3$ beams to RCP ($DOCP = +1$).  Some of the devices shown have lower polarization purity than the $|DOCP| \geq 0.95$ level observed in Fig. <ref>.   This reduction in $|DOCP|$ is caused by non-ideal performance of our visible PM fibers which are especially susceptible to misalignments of PM stress rods when using FC/APC connections. This misalignment results in a small degree of elliptical polarization at the output of the fiber (input to MS) that cannot be suppressed by azimuthal rotations of the ferrule to set $\beta_{\text{in}}$. Nevertheless, we find $|DOCP| \geq 0.85$ for devices in our system, which is sufficient for trapping [21].  As expected, $\eta$ is minimally affected by improper input polarization and we obtain an average efficiency across all beams of $\eta_{\text{avg}} = 30 \pm 3\%$ at 461 nm and $33 \pm 4\%$ at 689 nm (bottom panel); here the uncertainty range represents the statistical uncertainty at 65 $\%$ confidence.

In conclusion, our work demonstrates the successful multi-wavelength integration of six mm-scale metasurface optics on a single substrate to create a complex configuration of laser beams for optical cooling of Sr at 461 nm and 689 nm.  Our multifunction metasurface optics achieve precise control over beam pointing, divergence and polarization state without the need for bulk free-space optics. Our fiber-coupled approach streamlines the complexities of assembly and alignment through a foundry-compatible integration strategy and represents a significant step towards portable cold atom systems with applications in geodesy, inertial sensing and optical atomic clocks.

Defense Advanced Research Projects Agency(DARPA) (A-PhI);  National Institute of Standards
and Technology (NIST)

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