# Integration of vector fields on cell complexes and Morse theory

Takeo Nishinou  Department of Mathematics, Rikkyo University, Toshima, Tokyo,
Japan<EMAIL_ADDRESS>
###### Abstract.

In this paper, we investigate vector fields on polyhedral complexes and their
associated trajectories. We study vector fields which are analogue of the
gradient vector field of a function in the smooth case. Our goal is to define
a nice theory of trajectories of such vector fields, so that the set of them
captures the topology of the polyhedral complex, as in classical Morse theory.
Since we do not assume the polyhedral complex to be a manifold, the definition
of vector fields on it is very different from the smooth case. Nevertheless,
we will show that there exist nice classes of functions and metrics which give
gradient vector fields with desired properties. Our construction relies on
Forman’s discrete Morse theory. In particular, the class of functions we use
is an improvement of Forman’s discrete Morse functions. A notable feature of
our theory is that our gradient vector fields are defined purely from
functions and metrics as in the smooth case, contrary to the case of discrete
Morse theory where we need the data of dimension of cells. This allows us to
implement several useful constructions which were not available in the
discrete case.

##  1\. Introduction

In this paper, we consider vector fields on cell complexes. Our goal is to
give a reasonable theory of integration of such vector fields. In other words,
we would like to give a definition of trajectories of these vector fields with
nice properties. As a measure of such ‘nice properties’, we refer to Morse
theory [4]. Namely, if we can calculate the homology of cell complexes in a
simple way from the data of trajectories of a vector field associated with a
suitable function, it may be reasonable to claim that the theory of
integrating vector fields is nice.

Since we deal with cell complexes which are not necessarily manifolds, a
definition of vector fields should be different from the usual one. In
particular, to capture the topological informations of a cell complex, it will
be necessary to allow multivaluedness at points where there is no regular
neighborhood. On the other hand, if we allow any object with such
multivaluedness, we will have to deal with extremely complicated objects which
exhibit pathological behavior. Therefore, it is desirable to assume that
vector fields are as simple as possible on each cell.

From this point of view, we consider gradient vector fields of piecewise
linear functions. However, it is easy to see that if one chooses the functions
arbitrarily, it would be impossible to deduce any meaningful result, even if
one chooses them generically. Different choices of functions will give
different results, and there would be no reasonable control. Therefore, for
the construction of a theory of integration of gradient vector fields, it is
essential to introduce a nice class of functions.

There are discrete, or piecewise linear, analogues of Morse theory, see for
example [1, 2, 6]. We adopt Forman’s _discrete Morse functions_ [2], see
Definition 2.4, to solve the problem. Forman used such functions to develop
_discrete Morse theory_ , a nice combinatorial analogue of usual Morse theory
on smooth manifolds. However, if we allow arbitrary discrete Morse functions,
we will still have difficulties. We need to choose a nice class of discrete
Morse functions, which we call _tame_ , to have a good control of gradient
trajectories, see Definition 3.1.

On the other hand, to consider the gradient vector field of a function, we
also need a metric. It turns out that the choice of a suitable class of
metrics is also essential. We choose _sharp piecewise affine metrics_ on the
base space, see Definition 4.3. This is also necessary to guarantee nice
behavior of trajectories of vector fields. On the other hand, on any finite
cell complex, plenty of tame discrete Morse functions and sharp piecewise
affine metrics exist, so these choices do not give a restriction to the
theory.

Using a tame discrete Morse function and a sharp piecewise affine metric, we
can define the gradient vector field and its trajectories, see Definition 5.5
and 7.3. They are the analogues of the corresponding objects in smooth case.
However, the nature of gradient vector fields defined here is quite different
from those in the smooth case. For example, even when the base space is a
manifold, several flows can go into (or emanate from) a point, which is not
necessarily a critical point, see, for example, Figure 1. Nevertheless, the
total behavior of the flows is good enough and we can naturally define a
complex from these flows which computes the homology of the base space, see
Section 8. We will prove this by comparing it to discrete Morse theory, see
Theorem 9.1.

As we mentioned at the beginning, our study is an attempt to construct a
theory of vector fields on cell complexes in a natural and tractable way. This
is achieved by geometrically realizing discrete Morse theory. One of the nice
features of our construction is that the gradient vector field of a function
is defined purely from the information of the function and the metric, as in
the smooth case. This is different from the case of discrete Morse theory,
where the definition of gradient paths requires information other than the
function and the metric, namely the dimension of the cell.

This allows us to give another useful construction which was not available in
discrete Morse theory. Namely, when the base space is a closed manifold, we
can consider the dual theory. In discrete Morse theory, the dual theory is
defined on the dual cell complex, and one cannot directly compare the gradient
vector fields of the original theory with those of the dual theory. On the
other hand, in our case, the gradient vector field of the dual theory is
obtained simply by reversing the directions, as in the smooth case. This
allows us to construct stable and unstable complexes. In fact, we can
introduce stable and unstable complexes even when the base space is not a
manifold. In such a case, the reversed vector field is not necessarily related
to a discrete Morse theory, since the dual complex of a polyhedral complex may
not make sense when the base space is not a manifold. Nevertheless, the vector
field still has nice properties, see Section 10.

##  2\. Review of discrete Morse theory

Our purpose is to study the behavior of gradient vector fields associated with
piecewise linear functions. To make sense of such vector fields, the base
space needs to have a piecewise linear structure. Simplicial complexes or
triangulated manifolds naturally have such structures. We will work with
affine complexes which we now introduce.

######  Definition 2.1.

A _convex polytope_ $P$ is a convex hull of finite points in the vector space
$\mathbb{R}^{n}$ for some $n$. The boundary $\partial P$ is naturally a union
$\cup P_{i}$ of convex polytopes such that for any $i\neq j$, $P_{i}\cap
P_{j}=P_{k}$ for some $k$, if $P_{i}\cap P_{j}$ is not empty. Each $P_{i}$ is
called a _face_ of $P$. A face of codimension one in $P$ is called a _facet_.
Note that any polytope has a natural affine linear structure. That is, the
notion of affine linear functions on a polytope makes sense.

######  Definition 2.2.

An _affine complex_ $X$ is a finite CW complex such that each cell has a
structure of a convex polytope in the sense of Definition 2.1 compatible with
the gluing. Here, a cell always means a closed cell in this paper.
Compatibility means that if $X_{j}$ is a face of some cell $X_{i}$ of $X$, the
affine linear structure on $X_{j}$ induced by its structure as a convex
polytope is isomorphic to the affine linear structure induced by that of
$X_{i}$. Note that no self-intersection of a cell is allowed.

In particular, an affine complex need not be a topological manifold, and it
can have boundary components.

######  Definition 2.3.

The _dimension_ of an affine complex $X$ is the maximum of the dimension of
the cells in $X$.

Let $X$ be an affine complex. If $\alpha$ is an $i$-dimensional cell of $X$,
we write it as $\alpha^{(i)}$ when we want to emphasize its dimension. We
often call it an $i$-cell. If $\alpha^{(i)}$ is a face of $\beta^{(j)}$, we
write $\alpha^{(i)}<\beta^{(j)}$. In this case, we say that $\alpha^{(i)}$ is
adjacent to $\beta^{(j)}$, and also that $\beta^{(j)}$ is adjacent to
$\alpha^{(i)}$.

As we mentioned in the introduction, we study vector fields modeled on the
gradient vector fields of piecewise linear functions. To introduce a nice
class of functions which are compatible with the piecewise linearity, we use
discrete Morse functions in the sense of Forman [2], instead of smooth
functions in usual Morse theory. Let us recall the definition.

######  Definition 2.4.

([2, Definition 2.1]) A real valued function

$F\colon\\{\rm{cells\;of\;X}\\}\to\mathbb{R}$

is a _discrete Morse function_ if for every cell $\alpha^{(i)}$ of $X$, the
inequalities

(1) $\\#\\{\beta^{(i+1)}>\alpha^{(i)}\;|\;F(\beta^{(i+1)})\leq
F(\alpha^{(i)})\\}\leq 1$

and

(2) $\\#\\{\gamma^{(i-1)}<\alpha^{(i)}\;|\;F(\gamma^{(i-1)})\geq
F(\alpha^{(i)})\\}\leq 1$

hold.

Roughly speaking, this definition says a discrete Morse function tends to have
larger values on larger dimensional cells. For example, the function
$F_{triv}$ defined by

$F_{triv}(\alpha^{(i)})=i$

for any $i$-dimensional cell $\alpha^{(i)}$ is a discrete Morse function,
where the left hand side of (1) and (2) are both zero. The condition of the
definition claims that there is at most one exception for each cell. The cells
without such an exception is called critical as in the following definition.

######  Definition 2.5.

([2, Definition 2.2]) Consider an affine complex and a discrete Morse function
$F$ on it. A cell $\alpha^{(i)}$ is _critical_ if

(3) $\\#\\{\beta^{(i+1)}>\alpha^{(i)}\;|\;F(\beta^{(i+1)})\leq
F(\alpha^{(i)})\\}=0$

and

(4) $\\#\\{\gamma^{(i-1)}<\alpha^{(i)}\;|\;F(\gamma^{(i-1)})\geq
F(\alpha^{(i)})\\}=0$

hold.

For example, when we take the discrete Morse function $F_{triv}$ above, all
the cells are critical.

We also need the definition of gradient flows in discrete Morse theory, which
is used to define the discrete Morse complex. We need it because later we will
compare the discrete Morse complex with a complex defined from piecewise
linear gradient flows, see Section 9.

Recall that there is a chain complex naturally associated with a CW complex.
Let $C_{i}(X,\mathbb{Z})$ denote the free abelian group generated by
$i$-dimensional cells of $X$, where on each cell an orientation is chosen. In
particular, $-\sigma\in C_{i}(X,\mathbb{Z})$ represents the same cell $\sigma$
with the opposite orientation. Let $\partial$ denote the boundary operator

$\partial:C_{i+1}(X,\mathbb{Z})\to C_{i}(X,\mathbb{Z}).$

For example, in the simplicial case, it is given by

$\partial\langle a_{0},\dots,a_{i+1}\rangle=\sum_{l=0}^{i+1}(-1)^{l}\langle
a_{0},\dots,\check{a}_{l},\dots,a_{i+1}\rangle,$

here $\check{a}_{l}$ means it is skipped. In general, we have

$\partial\beta=\sum_{l}sgn(\beta,\alpha_{l})\alpha_{l},$

where the sum runs over the set of facets of $\beta$, and
$sgn(\beta,\alpha_{l})\in\\{\pm 1\\}$ is determined as follows. Namely, given
an oriented polytope $\beta^{(i+1)}$, the orientation is represented by a non-
zero element $\tau_{i+1,x}\in\wedge^{i+1}T_{x}\beta^{(i+1)}$ at any point $x$
in the interior of $\beta^{(i+1)}$. This determines a constant section
$\tau_{i+1}$ of $\wedge^{i+1}T\beta^{(i+1)}$. On a facet $\alpha^{(i)}$ of
$\beta^{(i+1)}$, there is a natural orientation represented by an element
$\eta_{i}\in\wedge^{i}T_{y}\alpha^{(i)}$ at any point $y$ in the interior of
$\alpha^{(i)}$, determined by the condition
$n_{y}\wedge\eta_{i}=\tau_{i+1,y}$, where $n_{y}$ is an outward normal vector
of $\beta^{(i+1)}$ at $y$. Then,

$sgn(\beta,\alpha_{l})=\begin{cases}1\;\;\text{if the orientation on
$\alpha_{l}$ induced from $\beta$ equals to the fixed orientation on
$\alpha_{l}$},\\\ -1\;\;{\rm otherwise}.\end{cases}$

Now, introduce an inner product $\langle\,,\,\rangle$ on $C_{*}(X,\mathbb{Z})$
by requiring the set of cells of $X$ to be an orthonormal basis.

######  Definition 2.6.

([2, Definition 6.1],) A _gradient vector field_ $V$ on an affine complex
equipped with a discrete Morse function $F$ and an orientation for each cell
is defined as follows. Let $\alpha^{(i)}$ be an oriented $i$-cell. If there is
an $(i+1)$-cell $\beta^{(i+1)}$ satisfying $\beta^{(i+1)}>\alpha^{(i)}$ and
$F(\beta^{(i+1)})\leq F(\alpha^{(i)})$, we set

$V(\alpha^{(i)})=-\langle\partial\beta^{(i+1)},\alpha^{(i)}\rangle\beta^{(i+1)}.$

If there is no such $\beta^{(i+1)}$, set

$V(\alpha^{(i)})=0.$

For each $i$, we extend $V$ linearly to a map

$V:C_{i}(X,\mathbb{Z})\to C_{i+1}(X,\mathbb{Z}).$

The map $V$ is a discrete analogue of the gradient vector field of a Morse
function on a smooth manifold. The next is an analogue of the integral of such
a vector field.

######  Definition 2.7.

([2, Definition 8.4]) Let $X$ be an affine complex equipped with a discrete
Morse function $F$. A $gradient\,\,path$ of dimension $i$ is a sequence
$\gamma$ of $i$-cells of $X$

$\gamma=\alpha_{0},\alpha_{1},\alpha_{2},\dots,\alpha_{r}$

such that for every $l=0,\dots,r-1$,

(i) if $V(\alpha_{l})=0$, $\alpha_{l+1}=\alpha_{l}$.

(ii) if $V(\alpha_{l})\neq 0$, $\alpha_{l+1}<V(\alpha_{l})$ and
$\alpha_{l+1}\neq\alpha_{l}$.

We call $\gamma$ a gradient path of $length$ $r$.

###  2.1. Sign of gradient paths and Morse differential

Let us introduce the complex of Forman

$(\ast)\;\;\;\;\mathcal{M}:0\to\mathcal{M}_{n}\xrightarrow{\tilde{\partial}}\mathcal{M}_{n-1}\xrightarrow{\tilde{\partial}}\cdots\xrightarrow{\tilde{\partial}}\mathcal{M}_{0}\to
0,$

in terms of gradient paths, as in [2, Section 8]. Here, $\mathcal{M}_{i}$ is
the free abelian group generated by critical $i$-dimensional cells.

For this purpose, we need a remark on the orientation of the cells contained
in a gradient path, as in [2, page 125]. Suppose $\alpha$ and $\tilde{\alpha}$
are distinct $i$-cells and $\beta$ is an $(i+1)$-cell with $\alpha<\beta$ and
$\tilde{\alpha}<\beta$. Then, an orientation on $\alpha$ induces an
orientation on $\beta$ by the condition
$\langle\partial\beta,\alpha\rangle=-1$. Given this orientation on $\beta$, we
choose the orientation on $\tilde{\alpha}$ so that
$\langle\partial\beta,\tilde{\alpha}\rangle=1$. Equivalently, fixing an
orientation on $\alpha$ and $\beta$, an orientation is induced on
$\tilde{\alpha}$ by the condition

$\langle\partial\beta,\alpha\rangle\langle\partial\beta,\tilde{\alpha}\rangle=-1.$

On the other hand, if $\alpha=\tilde{\alpha}$, we just take the same
orientation on $\tilde{\alpha}$ as $\alpha$. Thus, if
$\gamma=\alpha_{0},\alpha_{1},\dots,\alpha_{r}$ is a gradient path, an
orientation on $\alpha_{0}$ induces an orientation on each $\alpha_{i}$, and,
in particular, on $\alpha_{r}$. Recall that we have fixed an orientation for
each cell of $X$. Write $m(\gamma)=1$ if the fixed orientation on $\alpha_{0}$
induces the fixed orientation on $\alpha_{r}$, and $m(\gamma)=-1$ otherwise.
For $i$-cells $\alpha$ and $\tilde{\alpha}$, let
$\Gamma_{r}(\alpha,\tilde{\alpha})$ denote the set of all gradient paths from
$\alpha$ to $\tilde{\alpha}$ of length $r$. If $\beta^{(i+1)}$ and
$\alpha^{(i)}$ are critical, the differential $\tilde{\partial}$ of the
sequence ($\ast$) is given by

(5)
$\langle\tilde{\partial}\beta^{(i+1)},\alpha^{(i)}\rangle=\sum_{\tilde{\alpha}^{(i)}<\beta^{(i+1)}}\langle\partial\beta^{(i+1)},\tilde{\alpha}^{(i)}\rangle\sum_{\gamma\in\Gamma_{N}(\tilde{\alpha}^{(i)},\alpha^{(i)})}m(\gamma)$

for any $N$ large enough. In the paper [2], Forman proved the next theorem
(for any finite CW complex which is not necessarily an affine complex).

######  Theorem 2.8.

(Forman [2], Theorems 8.2 and 8.10) $\tilde{\partial}$ is a differential, i.e,
$\tilde{\partial}^{2}=0$. The homology of the complex $(\ast)$ is precisely
the singular homology of $X$. ∎

##  3\. Tame discrete Morse functions

Our purpose is to define and study integration of vector fields on cell
complexes using discrete Morse theory as a model. However, if one uses
arbitrary discrete Morse functions, one will soon realize that the
corresponding piecewise linear theory may be ill-behaved. To remedy this, we
need several ideas, as we explained in the introduction. The first of these
ideas is to choose a nice class of discrete Morse functions, which we call
tame.

######  Definition 3.1.

We call a discrete Morse function $F$ _generic_ if for any pair of cells
$\alpha,\beta$ satisfying $\alpha<\beta$, we have $F(\alpha)\neq F(\beta)$. We
call $F$ _tame_ if $\alpha^{(i)}<\beta^{(j)}$ and $j\geq i+2$, the inequality
$F(\beta^{(j)})>F(\alpha^{(i)})$ holds.

The tameness of discrete Morse functions does not impose a strong restriction.
In fact, given a generic discrete Morse function, we can modify it to a tame
one without changing discrete Morse theoretic properties.

######  Lemma 3.2.

Let $F$ be a generic discrete Morse function on an affine complex $X$. Then,
there is a tame generic discrete Morse function $\tilde{F}$ such that there is
a natural identification between the associated complexes $\mathcal{M}$. That
is, the set of critical cells and the differential on them are the same.

###### Proof.

We prove this by induction on the number of pairs of cells
$(\alpha^{(i)},\beta^{(j)})$ in $X$ which violate the condition for the
tameness. That is, we consider those pairs of cells which satisfy

$\alpha^{(i)}<\beta^{(j)},\;\;j\geq i+2,\;\;F(\alpha^{(i)})>F(\beta^{(j)}).$

If this number is zero, $F$ itself is tame. Let $k$ be a positive integer.
Assume that for any $F$ such that there are $k^{\prime}$-pairs of cells
$(\alpha,\beta)$, $k^{\prime}\in\\{1,2,\dots,k\\}$, which violate the tameness
condition, there is a tame $\tilde{F}$ which gives the same complex as the
complex associated with $F$. We will prove the assertion for $F$ for which
there are $(k+1)$-pairs of cells $(\alpha,\beta)$ which violate the tameness
condition.

Let $\mathcal{P}$ be the set of such pairs of cells. We introduce a partial
order to $\mathcal{P}$ by the rule that
$(\alpha^{(i)},\beta^{(j)})>((\alpha^{\prime})^{(i^{\prime})},(\beta^{\prime})^{(j^{\prime})})$
if and only if $j>j^{\prime}$. Let $(\alpha^{(i)},\beta^{(j)})$ be a maximal
element with respect to this order. Note that if $\alpha^{(i)}<\beta^{(j)}$
and $j\geq i+2$, there are at least two $(i+1)$-cells adjacent to both
$\alpha^{(i)}$ and $\beta^{(j)}$. Then, by definition of discrete Morse
functions, there is at least one $(i+1)$-cell $\gamma^{(i+1)}$ such that

$\alpha^{(i)}<\gamma^{(i+1)}<\beta^{(j)}$

and

$F(\gamma^{(i+1)})>F(\alpha^{(i)})$

hold. Similarly, if we have $j\geq i+3$, there is at least one $(i+2)$-cell
$\delta^{(i+2)}$ such that

$\gamma^{(i+1)}<\delta^{(i+2)}<\beta^{(j)}$

and

$F(\delta^{(i+2)})>F(\gamma^{(i+1)})$

hold. In particular, we have $\alpha^{(i)}<\delta^{(i+2)}$ and
$F(\delta^{(i+2)})>F(\alpha^{(i)})$. Repeating this, we see that there is at
least one $(j-1)$-cell $\varepsilon^{(j-1)}$ which satisfies

$\alpha^{(i)}<\varepsilon^{(j-1)}<\beta^{(j)}$

and

$F(\varepsilon^{(j-1)})>F(\alpha^{(i)})>F(\beta^{(j)}).$

In particular, all the $(j-1)$-cells other than $\varepsilon^{(j-1)}$ adjacent
to $\beta^{(j)}$ have the smaller value of $F$ than $\beta^{(j)}$ by
definition of discrete Morse functions.

On the other hand, if $\eta^{(k)}>\beta^{(j)}$ (so that we also have
$\eta^{(k)}>\varepsilon^{(j-1)}$), by the maximality of the pair
$(\alpha^{(i)},\beta^{(j)})$ with respect to the given order on $\mathcal{P}$,
we have

(6) $F(\eta^{(k)})>F(\varepsilon^{(j-1)}).$

Then, modify the value $F(\beta^{(j)})$ to $\bar{F}(\beta^{(j)})$, where
$\bar{F}(\beta^{(j)})$ is a generic real number in the open interval
$(F(\alpha^{(i)}),F(\varepsilon^{(j-1)})).$ The values of $\bar{F}$ at the
other cells are the same as those of $F$. Clearly, $\bar{F}$ is still a
generic discrete Morse function. By the relation (6), this modification does
not produce a new pair of cells which violates the tameness condition. Also,
it does not change the associated complex. Then, by applying the induction
hypothesis to $\bar{F}$, we obtain a required function $\tilde{F}$.∎

##  4\. Piecewise linear functions associated with discrete Morse functions
and metrics on affine complexes

Now, we construct a piecewise linear function $f$ from a given discrete Morse
function $F$.

######  Definition 4.1.

Let $X$ be an affine complex and $F$ be a discrete Morse function on it.

1\. Take the barycentric subdivision $X_{1}$ of $X$. By definition, the
vertices of it correspond to the cells of $X$.

2\. To a vertex of $X_{1}$, give the value of $F$ at the corresponding cell.

3\. Extend it affine linearly to each maximal cell of $X_{1}$. It is possible,
because $X_{1}$ is a simplicial complex.

We call a vertex of $X_{1}$ corresponding to an $i$-dimensional cell of $X$ an
$i$-dimensional vertex. Also, we call an edge of $X_{1}$ connecting an
$i$-dimensional vertex and a $j$-dimensional vertex an $(i,j)$-edge.

One of the features of discrete Morse theory is that although it studies an
analogue of gradient flows, it does not require a metric on the base space.
However, since we consider a geometric flow on $X$ (contrary to the
combinatorial one, see Definition 2.7), we need a metric as in usual Morse
theory.

######  Definition 4.2.

Let $Y$ be a simplicial complex. A _piecewise affine metric_ on $Y$ is defined
by an affine metric on each maximal cell (i.e, a cell which is not contained
in the boundary of another cell) which is compatible in the following sense.
Namely, on any lower dimensional cell, the metrics induced by those on the
adjacent maximal cells are the same. Here, an affine metric on a polytope in
$\mathbb{R}^{n}$ is the metric obtained by the restriction of a constant
valued positive definite symmetric 2-tensor field
$g\in\Gamma(\mathbb{R}^{n},Sym^{2}T^{*}\mathbb{R}^{n})$.

We cannot use every such a metric, since otherwise the gradient flow will
exhibit undesirable behavior. This problem can be resolved by imposing the
following simple condition to the metric.

######  Definition 4.3.

We call a piecewise affine metric on a simplicial complex _sharp_ if on each
simplex $\sigma$, the following condition is satisfied. Namely, take any
vertex $a$ of $\sigma$, and a two dimensional plane $L$ containing $a$. Assume
the intersection $L\cap\sigma$ is a triangle, which we write by $\triangle
abc$. Then, it is acute-angled, that is, the interior angle at each vertex is
less than $\frac{\pi}{2}$ with respect to the induced metric.

A sharp piecewise linear metric always exists on simplicial complexes as the
following claim shows.

######  Proposition 4.4.

Given a simplicial complex, a piecewise affine metric for which any of its
simplex is equilateral of a given edge length is sharp.

###### Proof.

It suffices to prove that the metric on the standard simplex with vertices at
$a_{0}=(1,0,\dots,0),a_{1}=(0,1,0,\dots,0),\dots,a_{n}=(0,\dots,0,1)$ in
$\mathbb{R}^{n+1}$ induced from the Euclidean metric on $\mathbb{R}^{n+1}$ is
sharp. By symmetry, we can assume that the vertex $a$ in Definition 4.3 is
$a_{0}$. Then, using the notation in Definition 4.3, we can assume that the
vertices $b$ and $c$ have the coordinates of the form

$(0,x_{1},\dots,x_{k},0,\dots,0),\;\;x_{1},\dots,x_{k}\geq 0,$

and

$(0,\dots,0,y_{k+1},\dots,y_{n}),\;\;y_{k+1},\dots,y_{n}\geq 0,$

respectively. Here, the coordinates satisfy $\sum_{i=1}^{k}x_{i}=1$ and
$\sum_{i=k+1}^{n}y_{i}=1$.

By the cosine formula, we have

$\cos\angle
acb=\frac{\overline{ac}^{2}+\overline{bc}^{2}-\overline{ab}^{2}}{2\overline{ac}\overline{bc}}.$

Since we have

$\overline{ac}^{2}+\overline{bc}^{2}-\overline{ab}^{2}=2\sum_{i=k+1}^{n}y_{i}^{2}>0,$

we have $\cos\angle acb>0$ and $\angle acb<\frac{\pi}{2}$. We can apply the
same argument to the other angles. This finishes the proof.∎

##  5\. Piecewise linear gradient vector fields

In Sections 5 to 8, we will define piecewise linear Morse theory on affine
complexes. First, we will define the piecewise linear gradient flow. Let
$X,X_{1},F$ and $f$ be as in the previous section. We write $n=\dim X$. We fix
a piecewise affine metric on $X_{1}$. From now on, we assume the following
unless otherwise noted.

######  Assumption 5.1.

The metric is sharp and the discrete Morse function $F$ is generic and tame in
the sense of Definition 3.1.

By Lemma 3.2 and Proposition 4.4, this assumption does not give a restriction
to the applicability of the theory.

Let $\sigma$ be a cell in $X_{1}$. If $\sigma$ is a polytope in
$\mathbb{R}^{n}$ and $L$ is the minimal affine subspace of $\mathbb{R}^{n}$
containing $\sigma$, we define

$T\sigma=TL|_{\sigma}.$

The restriction of the function $f$ and the metric to $\sigma$ induces a
gradient vector field on it. By the piecewise linearity, this is a constant
vector field. Under the above assumption, we have the following.

######  Lemma 5.2.

Under Assumption 5.1, the direction of the gradient vector on each $\sigma$ is
not contained in the hyperplanes parallel to the facets of it.∎

######  Definition 5.3.

Let $\sigma$ be a cell in $X_{1}$. The constant vector field above gives the
_gradient flow_ on $\sigma$, which is the family of maps

$\phi_{t}\colon\sigma\to\sigma,\;\;t\in\mathbb{R}_{\geq 0}.$

Note that since $\sigma$ has boundary, $\phi_{t}$ is not a diffeomorphism.
Namely, once a point reaches the boundary by the flow at some $t>0$, it stays
there afterwards.

######  Definition 5.4.

Let $\sigma$ be a cell in $X_{1}$ and $\tau$ be its facet. Let $x$ be a point
in the interior of $\tau$. By Lemma 5.2, for sufficiently small $t$, either

  1. (i)

$\phi_{t}(x)$ is in the interior of $\sigma$, or

  2. (ii)

$\phi_{t}(x)=x$,

holds. In the case (i), we say that the facet $\tau$ has an _out-flow_ , and
in the case (ii), we say that the facet $\tau$ has an _in-flow_.

Let $x$ be a point in $X_{1}$. Let $\sigma_{0}$ be the minimal cell of $X_{1}$
which contains $x$ in its interior. We write its dimension by
$d_{0}=\dim\sigma_{0}$. Let $\\{\sigma_{j,1},\dots,\sigma_{j,a_{j}}\\}$ be the
set of $(d_{0}+j)$-dimensional cells which contain $\sigma_{0}$, here $1\leq
j\leq n-d_{0}$. For each $\sigma_{j,l}$, there is a gradient vector field
determined by the restriction of $f$ and the metric. In this paper, we assume
that the gradient vector points the direction in which the function decreases.
We write the value of it at $x$ as $\tau_{x;j,l}\in T_{x}\sigma_{j,l}$. Here,
we write $\sigma_{0}=\sigma_{0,1}$. The inclusions of the cells induce natural
inclusions of tangent spaces. Note that under the above assumption, the images
of gradient vectors under these inclusions are all different.

Figure 1. Examples of gradient vector fields on cell complexes. Each point in
the interior of the edge $PQ$ has three directions.

######  Definition 5.5.

The set of _gradient vectors_ of $f$ at $x$ is the set of tangent vectors
$\\{\tau_{x;j,l}\\}$. The _gradient vector field_ of $f$ is the assignment of
the set of gradient vectors to each point of $X_{1}$.

This definition of gradient vector fields is very different from the usual
gradient vector field in the smooth case, see Figure 1. There will be
different definitions of vector fields in piecewise linear theory, see [3, 5]
for example. In these references, vector fields on manifolds with piecewise
linear structures are studied. The definition here is different from these
definitions as well, in that it allows multivaluedness. This makes it possible
to develop a meaningful theory of gradient flows on spaces which are not
necessarily manifolds. On the other hand, the definition here is very simple
compared to the other definitions in that it is constant valued on cells. Of
course, such a trivial choice will not give a meaningful theory in general. It
makes sense only if it is combined with the tameness of the function and the
sharpness of the metric.

A nice feature of this definition of gradient vector fields is that, unlike
the case of discrete gradient paths (Definition 2.7), it is purely determined
by the function and the metric, without referring to other information such as
the dimension of the vertices. From this, one can immediately see the
following duality. Assume that the affine complex $X$ is a polyhedral
decomposition of a manifold without boundary. Consider the dual of our
piecewise linear theory on the dual complex $\bar{X}$ of $X$. Namely, since
there is a natural inclusion-reversing one-to-one correspondence between the
set of cells of $X$ and $\bar{X}$, we have a natural function $\bar{F}$ on the
set of cells of $\bar{X}$ induced from the discrete Morse function $F$ on $X$.
Then, $-\bar{F}$ is a discrete Morse function on $\bar{X}$. Note that if $F$
is tame and generic, so is $-\bar{F}$. Also, there is a natural identification
between $X_{1}$ and the barycentric subdivision of $\bar{X}$. Thus, we have a
piecewise linear theory on $X_{1}$ using the function induced from $-\bar{F}$
and the given sharp piecewise affine metric.

######  Proposition 5.6.

The associated gradient vector for the dual theory at a point $x\in X_{1}$ is
given by $\\{-\tau_{x;j,l}\\}$.

###### Proof.

This follows from the definition. ∎

This point is one of the advantages of considering piecewise linear theory,
compared to the discrete theory. That is, in the discrete case, when one
considers Poincaré duality for a polyhedral decomposition of a manifold
without boundary, the dual theory is defined on the dual complex of the
original, so that one cannot directly relate the original discrete gradient
path to the dual. Here, we can consider Poincaré duality on the same complex
just by reversing the direction of the gradient vector field.

##  6\. Properties of piecewise linear flows

Proposition 6.2 below will be a basic ingredient in the argument in the
following sections. First, we prove the following.

######  Lemma 6.1.

Consider a triangle $P$ with vertices $a,b,c$ and put an affine metric on it
which is not necessarily sharp. Take an affine linear function $f$ on $P$
which is generic so that the values at the vertices are different, say,
$f(a)<f(b)<f(c)$, and the gradient flow with respect to the metric is not
parallel to any of the edges. Then, if the edge $\overline{bc}$ of $P$ has an
in-flow, the angle $\angle abc$ is larger than $\frac{\pi}{2}$. Further, the
edge $\overline{ab}$ also has an in-flow.

###### Proof.

There is a point $b^{\prime}$ on the edge $\overline{ac}$ with the property
$f(b)=f(b^{\prime})$. Since $f$ is affine linear, $f$ is constant on the line
connecting $b$ and $b^{\prime}$. This line divides $P$ into two triangles
$P_{1}=\triangle{cbb^{\prime}}$ and $P_{2}=\triangle{bab^{\prime}}$. The
gradient flow is perpendicular to $\overline{bb^{\prime}}$, and as an edge of
$P_{1}$, clearly $\overline{bb^{\prime}}$ has an in-flow. By the assumption,
the edge $\overline{bc}$ also has an in-flow. It follows that there is a flow
into the vertex $b$ whose trajectory is perpendicular to
$\overline{bb^{\prime}}$. Then, we have

$\angle abc=\angle abb^{\prime}+\angle cbb^{\prime}>\angle
abb^{\prime}+\frac{\pi}{2}>\frac{\pi}{2}.$

The final assertion is obvious.∎

######  Proposition 6.2.

Let $X,X_{1},F$ and $f$ be as above. Put a sharp piecewise affine metric on
$X_{1}$. Let $\tau$ be a cell in $X_{1}$. If $\tau$ has an in-flow from a cell
$\sigma$, the cell $\tau$ contains the lowest value vertex of the Morse
function on $\sigma$.

###### Proof.

Note that $\tau$ is a facet of $\sigma$ by definition, see Definition 5.4. If
$\dim\sigma=1$, the claim is obvious. If $\dim\sigma=2$, the claim follows
from Lemma 6.1. Namely, using the notation in Lemma 6.1, if the cell $\tau$
does not contain the lowest value vertex, we have $\tau=\overline{bc}$. Thus,
if $\tau$ has an in-flow, the angle $\angle abc$ is larger then
$\frac{\pi}{2}$. However, this contradicts to the sharpness of the metric.

Assume $\dim\sigma$ is larger than two. Suppose the lowest value vertex $p$ of
$\sigma$ is not contained in $\tau$. Take a generic point $q$ on $\tau$. Let
$L$ be a two dimensional plane containing the edge $\overline{pq}$ and the
flow line from the interior of $\sigma$ to $q$. Then, the intersection
$\tau\cap L$ is a segment, which we write as $\overline{rs}$. We can assume
the inequalities

$f(p)<f(r)<f(s)$

hold. Then, the triangle $\triangle prs$ satisfies the condition of Lemma 6.1,
since there is an in-flow on $\overline{rs}$ by the assumption. It follows
that the angle $\angle prs$ is larger than $\frac{\pi}{2}$. However, this
contradicts to the sharpness of the metric.∎

Dually, we have the following.

######  Corollary 6.3.

Using the same notation as in Proposition 6.2, if $\tau$ has an out-flow to a
cell $\sigma$, the cell $\tau$ contains the largest value vertex of the Morse
function on $\sigma$.∎

##  7\. Gradient trajectories between critical vertices

Let $X,X_{1},F$ and $f$ be as before. Fix a sharp piecewise affine metric on
$X_{1}$.

######  Definition 7.1.

Critical points of an affine complex $X$ equipped with a discrete Morse
function are the vertices of $X_{1}$ corresponding to the critical cells of
the discrete Morse function in the sense of Forman. We call a critical point
$i$-dimensional if it is a vertex corresponding to an $i$-dimensional cell of
the original complex.

As in discrete theory, let $\mathcal{M}_{i}^{PL}$ be the free abelian group
generated by critical $i$-dimensional vertices. It is canonically isomorphic
to $\mathcal{M}_{i}$. We want to construct a complex

$0\to\mathcal{M}_{n}^{PL}\to\cdots\to\mathcal{M}_{0}^{PL}\to 0$

by counting gradient trajectories of the piecewise linear flow. First, let us
define gradient trajectories.

######  Definition 7.2.

Let $\sigma$ be a cell of $X_{1}$. Let $x,y$ be points on $\sigma$. A
_gradient segment_ from $x$ to $y$ is an affine linear map

$\gamma\colon[t_{0},t_{1}]\to\sigma,$

where $t_{0},t_{1}\in\mathbb{R}$, such that $\gamma(t_{0})=x,\gamma(t_{1})=y$
and for each $s\in[t_{0},t_{1}]$, $\gamma^{\prime}(s)\in\\{\tau_{x;j,l}\\}$
(see Definition 5.5). A _gradient trajectory_ from $x$ to $y$ is a piecewise
affine linear map

$\gamma\colon[t_{0},t_{k}]\to\sigma,$

where $k$ is a positive integer, such that there is a refinement

$t_{0}<t_{1}<\cdots<t_{k}$

of $[t_{0},t_{k}]$ where $\gamma|_{[t_{l-1},t_{l}]}$ is a gradient segment for
each $l=1,\dots,k$.

######  Definition 7.3.

A gradient trajectory of the gradient vector field on $X_{1}$ is an ordered
sequence

$\gamma_{1},\gamma_{2},\dots,\gamma_{k},$

of gradient trajectories $\gamma_{l}:[t_{l-1},t_{l}]\to\sigma_{l}$ on some
cell $\sigma_{l}$ such that

$\gamma_{l}(t_{l})=\gamma_{l+1}(t_{l}),\;\;l=1,\dots,k-1.$

We identify two gradient trajectories if their images give the same subsets of
$X$. Note that for a point $x\in X_{1}$, there can be more than one gradient
trajectories through $x$, contrary to the case of smooth manifolds.

##  8\. Piecewise linear Morse complex

######  Definition 8.1.

Let $i$ be a non-negative integer. Let $X^{(i)}$ be the $i$-th skeleton of
$X$. Let $X_{1}^{(i)}$ be the barycentric subdivision of $X^{(i)}$. We call
$X_{1}^{(i)}$ the _$i$ -th rib_ of $X_{1}$.

Note that $X_{1}^{(i)}$ is not the $i$-th skeleton of $X_{1}$, which consists
of all the simplices of $X_{1}$ of dimension at most $i$.

######  Proposition 8.2.

Assume $F$ is generic and tame. Let $p$ and $q$ be $i$\- and
$(i-1)$-dimensional critical vertices, respectively. Then, a gradient
trajectory $\gamma:[0,t]\to X_{1}$ connecting $p$ and $q$ is contained in a
subcomplex composed of edges whose ends are $i$\- and $(i-1)$-dimensional
vertices.

###### Proof.

First, we observe the following.

######  Claim 8.3.

Let $x\in X_{1}^{(i)}$ be a point. Let $\delta$ be a gradient trajectory
starting from $x$. Then, the inclusion $\delta\subset X_{1}^{(i+1)}$ holds.

###### Proof.

This is an immediate consequence of Corollary 6.3 and the tameness of $F$.∎

Since $p$ is an $i$-dimensional critical vertex, any of its adjacent
$(i+1)$-dimensional vertices has larger value of $f$ than $f(p)$. Then, again
by Corollary 6.3, for sufficiently small $\varepsilon$,
$\gamma([0,\varepsilon))$ is contained in $X^{(i)}$. Let $\sigma$ be the cell
of $X_{1}$ with the property that $\gamma((0,\varepsilon))\subset int(\sigma)$
for sufficiently small $\varepsilon$, where $int(\sigma)$ is the interior of
$\sigma$. Note that $\sigma$ is uniquely determined by this property. Let
$0<t_{1}$ be the minimal real number with the property
$\gamma(t_{1})\in\partial\sigma$. Let $\tau\subset\partial\sigma$ be the
unique cell with the property that $\tau$ contains $\gamma(t_{1})$ in its
interior. Then, since $p$ is a critical vertex, the inclusion $\tau\subset
X_{1}^{(i-1)}$ holds.

Thus, by Claim 8.3, the image of the restriction $\gamma_{[t_{1},t]}$ is
contained in $X_{1}^{(i)}$. It follows that we have $\gamma([0,t])\subset
X_{1}^{(i)}$. Since $q$ is a critical vertex, a positive dimensional cell in
$X_{1}^{(i)}$ which has $q$ as the lowest value vertex with respect to $f$
must be an edge connecting $q$ and an adjacent $i$-dimensional vertex. Thus,
by Proposition 6.2, the intersection of a small neighborhood of $q$ with
$\gamma$ must be a subset of an edge connecting $q$ and an $i$-dimensional
vertex $q_{1}$. Similarly, by the tameness of $F$, a positive dimensional cell
in $X_{1}^{(i)}$ which has $q_{1}$ as the lowest value vertex with respect to
$f$ must be an edge connecting $q_{1}$ and an $(i-1)$-dimensional vertex
$q_{2}$ (which is unique, if any). In this case, we have

$f(q_{1})<f(q_{2}).$

Note that any $(i-2)$-dimensional vertex adjacent to $q_{2}$ is also adjacent
to $q_{1}$. If $r$ is an $(i-2)$-dimensional vertex adjacent to $q_{1}$, we
have $f(r)<f(q_{1})$ by the tameness of $F$. In particular, for any
$(i-2)$-dimensional vertex $r$ adjacent to $q_{2}$, we have $f(r)<f(q_{2})$.

Thus, again, a positive dimensional cell in $X_{1}^{(i)}$ which has $q_{2}$ as
the lowest value vertex with respect to $f$ must be an edge connecting $i$\-
and $(i-1)$-dimensional vertices. Repeating this, we see that $\gamma$ is
contained in the union of edges connecting $i$\- and $(i-1)$-dimensional
vertices.∎

From this, the following is obvious.

######  Corollary 8.4.

In the notation of Proposition 8.2, the number of gradient trajectories
connecting $p$ and $q$ is finite.∎

###  8.1. Sign of gradient trajectory and Morse differential

We define the linear map

$d_{PL}:\mathcal{M}_{i}^{PL}\to\mathcal{M}_{i-1}^{PL}$

by linearly extending the map

$p\mapsto\sum_{q\in\mathcal{M}_{i-1}^{PL}}\sum_{\gamma\in\Gamma_{PL}(p,q)}\bar{m}(\gamma)q,$

where $\Gamma_{PL}(p,q)$ is the set of gradient trajectories connecting $p$
and $q$, and $\bar{m}(\gamma)$ is the sign of the gradient trajectory $\gamma$
determined as follows. Namely, to each $k$-dimensional critical vertex
$a^{(k)}$, $0\leq k\leq n$, we attach an orientation of the $k$-dimensional
cell corresponding to it. It is represented by a constant element of
$\Gamma(\sigma_{a^{(k)}},\wedge^{k}T\sigma_{a^{(k)}})$. As an element of
$\mathcal{M}_{k}^{PL}$, $-a^{(k)}$ represents the same vertex with the
opposite orientation of $\sigma_{a^{(k)}}$ attached.

By Proposition 8.2, the trajectory connecting $p^{(i)}\in\mathcal{M}_{i}^{PL}$
and $q^{(i-1)}\in\mathcal{M}_{i-1}^{PL}$ is contained in the union of edges
connecting $i$\- and $(i-1)$-dimensional vertices. It follows that if the
trajectory passes through the vertices as

$p^{(i)}\to s^{(i-1)}\to r^{(i)}\to\cdots\to q^{(i-1)},$

the orientation at $p^{(i)}$ naturally induces an orientation at each
$s^{(i-1)},r^{(i)},\dots,q^{(i-1)}$. Namely, if
$o_{p^{(i)}}\in\Gamma(\sigma_{p^{(i)}},\wedge^{i}T\sigma_{p^{(i)}})$ is the
orientation at $p^{(i)}$, it induces an orientation at $s^{(i-1)}$ by

$o_{s^{(i-1)}}=n_{p^{(i)}}\lfloor
o_{p^{(i)}}\in\Gamma(\sigma_{s^{(i-1)}},\wedge^{(i-1)}T\sigma_{s^{(i-1)}}),$

where $n_{p^{(i)}}$ is the unit outward normal vector of $\sigma_{p^{(i)}}$ on
the facet $\sigma_{s^{(i-1)}}$, and $\lfloor$ denotes the interior product
defined using the affine metric. Then, the orientation at $r^{(i)}$ is given
by

$o_{r^{(i)}}=-n_{r^{(i)}}\wedge o_{s^{(i-1)}},$

where $n_{r^{(i)}}$ is the unit outward normal vector of $\sigma_{r^{(i)}}$ on
the facet $\sigma_{s^{(i-1)}}$. Repeating this, an orientation is induced at
$q^{(i-1)}$. If we compare this with the given orientation, we set
$\bar{m}(\gamma)=1$ when they coincide, and $-1$ if not.

With this definition, we have a sequence of maps

(7)
$0\to\mathcal{M}_{n}^{PL}\xrightarrow{d_{PL}}\mathcal{M}_{n-1}^{PL}\xrightarrow{d_{PL}}\cdots\xrightarrow{d_{PL}}\mathcal{M}_{0}^{PL}\to
0.$

A priori, we do not know whether this gives a chain complex. In the next
section, we see this is indeed the case, and its homology is the same as the
singular homology of $X$.

##  9\. The main theorem

Let $X,X_{1},F$ and $f$ be as above. Namely, $X$ is an affine complex of
dimension $n$, $X_{1}$ is the barycentric subdivision of $X$, $F$ is a tame
generic discrete Morse function on $X$, and $f$ is the affine linear function
on $X_{1}$ constructed from $F$. Put a sharp piecewise affine metric on
$X_{1}$. Our purpose is to show the following result.

######  Theorem 9.1.

The sequence $({\mathcal{M}}_{\bullet}^{PL},d_{PL})$ in the previous section
is a complex, and its homology is isomorphic to the singular homology of $X$.

###### Proof.

We examine the relation between discrete and piecewise linear Morse theories.
Recall that a gradient path in discrete Morse theory (Definition 2.7) is an
appropriate sequence $\gamma$ of $i$-cells of $X$

$\gamma=\alpha_{0},\alpha_{1},\alpha_{2},\dots,\alpha_{r}.$

Let $\Gamma_{r}(\alpha_{0},\alpha_{r})$ be the set of such gradient paths.

Consecutive $i$-cells $\alpha_{l-1}$ and $\alpha_{l}$ are in the boundary of a
uniquely determined $(i+1)$-cell, if $\alpha_{l-1}\neq\alpha_{l}$. Let us
write this $(i+1)$-dimensional cell by $\beta_{l}$. If
$\alpha_{l-1}=\alpha_{l}$, take $\beta_{l}=\emptyset$. Let $\beta_{0}$ be an
$(i+1)$-dimensional cell adjacent to $\alpha_{0}$ with the property
$F(\beta_{0})>F(\alpha_{0})$, and consider the sequence

$\beta_{0},\alpha_{0},\beta_{1},\alpha_{1},\beta_{2},\cdots,\alpha_{r-1},\beta_{r},\alpha_{r}.$

Let $\Gamma_{r}(\beta_{0},\alpha_{r})$ denote the set of such sequences. An
element of $\Gamma_{r}(\beta_{0},\alpha_{r})$ is determined by an element of
$\Gamma_{r}(\alpha_{0},\alpha_{r})$ and the choice of $\beta_{0}$. Then, if
$\beta_{0}$ is an $(i+1)$-dimensional critical cell, the differential of
discrete Morse theory (5) can be written in the form

(8)
$\tilde{\partial}\beta_{0}=\sum_{\alpha\in\mathcal{M}_{i}}\sum_{\tilde{\gamma}\in\Gamma_{N}(\beta_{0},\alpha)}\tilde{m}(\tilde{\gamma})\alpha,$

where $N$ is a sufficiently large number and $\mathcal{M}_{i}$ is the set of
critical $i$-cells. The sign $\tilde{m}(\gamma)\in\\{\pm 1\\}$ is equal to
$m(\gamma)$, where $\gamma$ is the element of $\Gamma_{N}(\alpha_{0},\alpha)$
associated with $\tilde{\gamma}$. Here, $\alpha_{0}$ is a facet of $\beta_{0}$
with an induced orientation, that is,
$\langle\partial\beta_{0},\alpha_{0}\rangle=1$.

To adjust the terminology with those in piecewise linear theory, we redefine
the sequence

$\beta_{0},\alpha_{0},\beta_{1},\alpha_{1},\beta_{2},\cdots,\alpha_{r-1},\beta_{r},\alpha_{r}$

above as a gradient path in discrete Morse theory. We identify two sequences
$\\{\beta_{i},\alpha_{i}\\}$ and
$\\{\beta_{i}^{\prime},\alpha_{i}^{\prime}\\}$ which are the same up to
stabilization, that is, if the union
$\cup_{l}int({\beta}_{l})\bigcup\cup_{l}\alpha_{l}$ and
$\cup_{l}int({\beta}^{\prime}_{l})\bigcup\cup_{l}\alpha^{\prime}_{l}$ are the
same as a subset of $X$. Here, $int({\beta}_{l})$ and
$int({\beta}^{\prime}_{l})$ are the interior of $\beta_{l}$ and
$\beta_{l}^{\prime}$, respectively.

Clearly, an element in $\Gamma_{N}(\beta_{0},\alpha_{r})$ determines a
sequence

$p_{0}\to q_{0}\to p_{1}\to\cdots\to p_{r}\to q_{r},$

where $p_{i}$ is the $(i+1)$-dimensional vertex in $X_{1}$ corresponding to
$\beta_{i}$, $q_{i}$ is the $i$-dimensional vertex in $X_{1}$ corresponding to
$\alpha_{i}$, and $r$ is the smallest number at which the sequence
$\alpha_{0},\alpha_{1},\alpha_{2},\dots,\alpha_{r}$ stabilizes (that is, the
smallest number such that $\alpha_{r}=\alpha_{r+1}$ holds). This sequence has
the following properties:

  * •

Consecutive vertices in this sequence are connected by a unique edge in
$X_{1}$.

  * •

Following inequalities hold:

$f(p_{0})>f(q_{0})>f(p_{1})>\cdots>f(p_{r})>f(q_{r}).$

Conversely, a sequence of $i$\- and $(i+1)$-dimensional adjacent vertices with
the properties above determines a unique (up to stabilization) gradient path
in discrete Morse theory. By Proposition 8.2, we have the following.

######  Proposition 9.2.

Let $\beta$ be a critical $(i+1)$-dimensional cell and $\alpha$ be a critical
$i$-dimensional cell of discrete Morse theory. Let $p,q$ be corresponding
$(i+1)$\- and $i$-dimensional critical vertices of piecewise linear Morse
theory. There is a canonical one to one correspondence between the following
objects:

  * •

Gradient paths in discrete Morse theory connecting $\beta$ and $\alpha$ up to
stabilization.

  * •

Gradient trajectories in piecewise linear Morse theory connecting $p$ and $q$.

Recall that the differential
$d_{PL}\colon\mathcal{M}_{i+1}^{PL}\to\mathcal{M}_{i}^{PL}$ in piecewise Morse
theory is given by

$d_{PL}p=\sum_{q\in\mathcal{M}_{i}^{PL}}\sum_{\gamma\in\Gamma_{PL}(p,q)}\bar{m}(\gamma)q,$

and the differential
$\tilde{\partial}\colon\mathcal{M}_{i+1}\to\mathcal{M}_{i}$ in discrete Morse
theory is given by

$\tilde{\partial}\beta=\sum_{\alpha\in\mathcal{M}_{i}}\sum_{\tilde{\gamma}\in\Gamma_{N}(\beta,\alpha)}\tilde{m}(\tilde{\gamma})\alpha.$

Let $\tilde{\gamma}$ be the element of $\Gamma_{N}(\beta,\alpha)$
corresponding to $\gamma\in\Gamma_{PL}(p,q)$ by Proposition 9.2. Then, Theorem
9.1 is a consequence of the following claim and Theorem 2.8.

######  Proposition 9.3.

The equality $\bar{m}(\gamma)=\tilde{m}(\tilde{\gamma})$ holds.

###### Proof.

This follows from a straightforward comparison between the argument in
Subsections 2.1 and 8.1.∎

##  10\. Stable and unstable complexes

Let $X,X_{1},F$ and $f$ be as before. Let $i$ be a positive integer. In this
section, we will show that in piecewise linear Morse theory, we have nice
stable and unstable complexes associated with critical points. This is another
benefit from the fact that we can consider the original Morse theory and its
dual in the same complex, see Proposition 5.6. We use the same notation as in
the previous sections. First, we observe the following.

######  Lemma 10.1.

Let $C$ be a closed subset of $X_{1}$. Then, the union of gradient
trajectories starting from a point in $C$ is a closed subset of $X_{1}$.

###### Proof.

It suffices to prove the claim on a simplex $\sigma$. If $x_{1}$ is a point in
the interior of $\sigma$, there is a unique gradient vector at $x_{1}$ tangent
to $\sigma$. This does not depend on the choice of $x_{1}$, and gradient
vectors at the points on the boundary of $\sigma$ also contain this direction.
It follows that the subset of $\sigma$ swept by the flow along this direction
starting from points in $C$ is closed in $\sigma$. Namely, embed $\sigma$ in
some $\mathbb{R}^{d}$ isometrically and let $v\in\mathbb{R}^{d}$ be the
direction of the gradient flow on $\sigma$. If we define the set $C_{\geq}$ by

$C_{\geq}=\\{x+tv\;|\;x\in C,\;t\geq 0\\}\subset\mathbb{R}^{d},$

the subset of $\sigma$ swept by the flow starting from points in $C$ is
$C_{\geq}\cap\sigma$. Since $C_{\geq}$ is closed, the claim follows.

The same argument applies to the intersection of a face of $\sigma$ and $C$.
Since the union of gradient trajectories starting from a point in $C$ is the
union of these closed subsets on each face of $\sigma$, the claim follows.∎

######  Lemma 10.2.

Let $\sigma$ be a simplex in $X_{1}$. Let $p$ be the vertex of $\sigma$ at
which the function $f$ takes the highest value in $\sigma$. Then, $\sigma$ is
swept by the gradient trajectories starting from $p$.

###### Proof.

By the previous lemma, it suffices to prove that the interior of $\sigma$ is
swept by the gradient trajectories starting from $p$. We use an induction. If
$\dim\sigma=1$, the claim is obvious. Assume that the claim is proved for
$\sigma$ with $\dim\sigma\leq k$ for some positive integer $k$. Let us prove
the case where $\dim\sigma=k+1$.

Let $x$ be a point in the interior of $\sigma$. As in the proof of the
previous lemma, there is a unique gradient vector at $x$ tangent to $\sigma$.
Tracing back the gradient trajectory along this direction, we will hit a
boundary of $\sigma$. Let $q$ be the point in $\partial\sigma$ at which this
occurs for the first time. Let $\\{A_{1},\dots,A_{m}\\}$ be the set of facets
of $\sigma$ containing $q$. By Corollary 6.3, each $A_{l}$ contains $p$. Then,
by applying the induction hypothesis to any of the cells
$\\{A_{1},\dots,A_{m}\\}$, the claim follows.∎.

######  Lemma 10.3.

Let $q$ be an $(i-1)$-dimensional vertex and $p$ be an adjacent
$i$-dimensional vertex such that $f(q)>f(p)$ holds. Let $\sigma_{p}$ be the
cell of $X$ corresponding to $p$. Then, $\sigma_{p}$ is swept by the gradient
trajectories starting from $q$.

###### Proof.

By the tameness, if $\alpha$ is a maximal dimensional simplex in $\sigma_{p}$,
the vertex taking the highest value among the set of vertices of $\alpha$ is
$p$ or $q$. Since $p$ is contained in a gradient trajectory starting from $q$,
the claim follows from the previous lemma.∎

We also prove the following claim.

######  Lemma 10.4.

Let $p$ be an $i$-dimensional critical vertex. Let $\gamma$ be a gradient
trajectory starting from $p$. Then, $\gamma$ is contained in $X_{1}^{(i)}$,
the $i$-th rib of $X_{1}$ (see Definition 8.1).

###### Proof.

We assume $i>0$ since otherwise the claim is trivial. By Corollary 6.3, for
sufficiently small $\varepsilon$, the image $\gamma((0,\varepsilon))$ is
contained in $X_{1}^{(i)}$. Let $\tau$ be the unique cell of $X_{1}$
satisfying $\gamma((0,\varepsilon))\subset int(\tau)$. Let $t_{1}>0$ be the
minimal real number satisfying $\gamma(t_{1})\in\partial\tau$. Then, we have
$\gamma(t_{1})\in X_{1}^{(i-1)}$. The lemma follows from this and Claim 8.3.∎

Let $k$ be a positive integer.

######  Proposition 10.5.

Let $p$ be an $i$-dimensional critical vertex. Then, the subset of $X_{1}$
swept by the trajectories starting from $p$ is a subcomplex of $X^{(i)}$.

###### Proof.

By Corollary 6.3, for sufficiently small $\varepsilon$, the image
$\gamma((0,\varepsilon))$ is contained in $\sigma_{p}$, for any trajectory
$\gamma$ starting from $p$. By Lemma 10.2, the cell $\sigma_{p}$ is swept by
the trajectories starting from $p$. Thus, to prove that the union of the
trajectories is a subcomplex of $X$, it suffices to show that for any facet
$A$ of $\sigma_{p}$, the union of trajectories starting from points on $A$ is
a subcomplex of $X$. If this is proved, by Claim 8.3, it must be a subcomplex
of $X^{(i)}$.

We prove this from the following more general statement.

######  Claim 10.6.

Let $B$ be a cell in $X$. Then, the union of trajectories starting from points
on $B$ is a subcomplex of $X$.

###### Proof.

We prove this by induction on the dimension of $B$. First, let us consider the
case when $\dim B=0$. If $B$ is a critical cell, the claim is obvious since
there is no trajectory emanating from $B$. If $B$ is not critical, there is a
unique one dimensional cell $C$ adjacent to $B$ such that

$F(B)>F(C)$

holds. Higher dimensional cells adjacent to $B$ have larger value of $F$ than
$F(B)$ by the tameness.

By Corollary 6.3, any trajectory starting from $B$ is given by an out-flow
into $C$. On the other hand, again by the tameness, there is only one cell
$B^{\prime}$ adjacent to $C$ satisfying

$F(C)>F(B^{\prime}),$

and this $B^{\prime}$ is a zero dimensional cell. Repeating this argument, the
claim follows.

Assume that we have proved the claim when $\dim B\leq k-1$ for some positive
integer $k$. Let us prove the claim when $\dim B=k$.

We use further induction on the value of $F$. Let $B_{1}$ be a $k$-dimensional
cell of $X$ such that there is no other $k$-dimensional cell whose value of
$F$ is smaller than $F(B_{1})$. Then, there is no adjacent $(k+1)$-cell which
has a smaller value of $F$ than $F(B_{1})$, by definition of discrete Morse
functions. Then, $B_{1}$ is critical, or there is a facet $C$ of $B_{1}$
satisfying $F(C)>F(B_{1})$.

If $B_{1}$ is critical, as in the beginning part of the proof of Proposition
10.5, any trajectory starting from a point on $B_{1}$ touches $\partial
B_{1}$. Thus, the claim follows by the induction hypothesis on the dimension
of $B$. If there is a facet $C$ of $B_{1}$ satisfying $F(C)>F(B_{1})$, $B_{1}$
is swept by trajectories starting from $C$, by Lemma 10.3. Thus, it suffices
to show that the union of trajectories starting from $C$ is a subcomplex of
$X$, since it coincides with the union of trajectories starting from $B_{1}$.
However, this follows from the induction hypothesis on dimension again.

Now, let

$c_{1}<c_{2}<\cdots<c_{l}$

be the values of $F$ taken by $k$-dimensional cells of $X$. Assume that the
claim is proved for a $k$-dimensional cell whose value of $F$ is at most
$c_{i}$, $1\leq i<l$. We will prove the claim for a $k$-dimensional cell $B$
satisfying $F(B)=c_{i+1}$.

If $B$ is critical or there is a facet $C^{\prime}$ of $B$ satisfying
$F(C^{\prime})>F(B)$, the claim is proved as in the case of $B_{1}$ above.
Assume that there is a $(k+1)$-dimensional cell $D$ adjacent to $B$ satisfying
$F(B)>F(D)$. By Lemma 10.3, $D$ is swept by trajectories starting from $B$.
Moreover, if $x\in int(B)$, for sufficiently small $\varepsilon$, a trajectory
$\gamma$ starting from $x$ satisfies $\gamma((0,\varepsilon))\subset D$, by
definition of discrete Morse functions and Corollary 6.3. It follows that the
trajectory $\gamma$ hits some facet of $D$ other than $B$ itself. Therefore,
it suffices to prove that the union of trajectories starting from a facet $E$
of $D$ other than $B$ is a subcomplex of $X$. However, we have $F(E)<F(B)$ by
definition of discrete Morse functions. Therefore, we can apply the induction
hypothesis on the values of $F$ to $E$. This proves the claim and the
proposition.∎

Using this proposition, we can define stable and unstable complexes of
critical points in piecewise linear Morse theory.

######  Definition 10.7.

Assume $X$ is a manifold without boundary. We call the subcomplex of $X^{(i)}$
constructed in Proposition 10.5 the _unstable complex_ of the critical point
$p$. Consider the dual theory, see Section 5. Then, $p$ is an
$(n-i)$-dimensional critical point. We call its unstable complex the _stable
complex_ of $p$ of the original theory.

Note that the stable complex is not a subcomplex of $X$, but of its dual
complex. In any case, it is a subcomplex of $X_{1}$.

Since the gradient trajectory is defined purely in terms of a function and a
metric as we mentioned in Section 5, this definition can be extended to the
case when $X$ has boundary, or is not necessarily a manifold. Namely, if $p$
is a critical point, we define the unstable complex as the subcomplex of
$X^{(i)}$ as in Proposition 10.5, and define the stable complex as the
subcomplex with respect to the reversed gradient flow. Note that the reversed
gradient flow is not necessarily induced by a discrete Morse function, since
if $X$ is an affine complex which is not a manifold, a dual complex does not
make sense in general. In other words, it is not a gradient flow associated
with a discrete Morse function, although it is the gradient flow associated
with a piecewise linear function on $X_{1}$. Nevertheless, the set of gradient
trajectories gives a complex, which is isomorphic to the dual of the original
complex $\mathcal{M}$.

## Acknowledgement

The author was supported by JSPS KAKENHI Grant Number 18K03313.

## References

  * [1] M. Bestvina and N. Brady, Morse theory and finiteness properties of groups. Invent. Math. 129 (1997), 445–470.
  * [2] R. Forman, Morse theory for cell complexes. Adv. Math. 134 (1998), no. 1, 90–145.
  * [3] J. Llibre and E. Ponce, Bifurcation of a periodic orbit from infinity in planar piecewise linear vector fields. Nonlinear Anal. 36 (1999), 623–653.
  * [4] J. Milnor, Morse theory. Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J., 1963.
  * [5] R. Stern, On topological and piecewise linear vector fields. Topology 14 (1975), 257–269.
  * [6] M. Zaremsky, Bestvina-Brady discrete Morse theory and Vietoris-Rips complexes. To appear, Amer. J. Math.


FKN, first proof, rewritten

Ehud Friedgut and Gil Kalai and Assaf Naor

###### Abstract

About twenty years ago we wrote a paper, ”Boolean Functions whose Fourier
Transform is Concentrated on the First Two Levels”, [1]. In it we offered
several proofs of the statement that Boolean functions
$f(x_{1},x_{2},\dots,x_{n})$, whose Fourier coefficients are concentrated on
the lowest two levels are close to a constant function or to a function of the
form $f=x_{k}$ or $f=1-x_{k}$.

Returning to the paper lately, we noticed that the presentation of the first
proof is rather cumbersome, and includes several typos. In this note we
rewrite that proof, as a service to the public.

Here is the main theorem of FKN [1].

######  Theorem 0.1.

Let $f:\\{0,1\\}^{n}\rightarrow\\{0,1\\}$, $\|f\|_{2}^{2}=p$. If
$\sum_{|S|>1}\widehat{f}^{2}(S)=\delta$ then either $p<K^{\prime}\delta$ or
$p>1-K^{\prime}\delta$ or $\|f(x_{1},x_{2},\dots,x_{n})-x_{i}\|_{2}^{2}\leq
K\delta$ for some $i$ or $\|f(x_{1},x_{2},\dots,x_{n})-(1-x_{i})\|_{2}^{2}\leq
K\delta$ for some $i$. Here, $K^{\prime}$ and $K$ are absolute constants.

Proof: First observe that we may assume that $p=1/2$: replace $f$ by a
function $g:\\{0,1\\}^{n+1}\rightarrow\\{0,1\\}$ defined by

$g(x_{1},\ldots,x_{n},x_{n+1})=f(x_{1},\ldots,x_{n})\cdot\frac{1+\raise
2.2pt\hbox{$\chi$}_{n+1}}{2}+(1-f(1-x_{1},\ldots,1-x_{n}))\cdot\frac{1-\raise
2.2pt\hbox{$\chi$}_{n+1}}{2}$

and note that $|g|_{2}^{2}=1/2$, and
$\sum_{|S|>1}\widehat{f}^{2}(S)=\sum_{|S|>1}\widehat{g}^{2}(S)$. Then prove
that $g$ is close to a dictator (or anti-dictator). If the influential
coordinate for $g$ is $n+1$ then $f$ is necessarily close to a constant, and
if the influential coordinate for $g$ is some other $i$, then $f$ too is
necessarily close to a dictator or anti-dictator of the $i$th variable.

Let

$f^{=1}:=\sum\widehat{f}(i)\raise 2.2pt\hbox{$\chi$}_{i}.$

Note that $f^{=1}$ is an odd function, i.e. $f^{=1}(S)=-f^{=1}(S^{c})$, and
that

$|f^{=1}|_{2}^{2}=\sum\widehat{f}(i)^{2}=1/4-\delta.$

Note also that

$(1/4-\delta)=\langle f^{=1},f^{=1}\rangle=\sum\widehat{f}(i)^{2}=\langle
f,f^{=1}\rangle$
$=\sum_{S\subset[n]}2^{-n}f(S)f^{=1}(S)=\sum_{S\subset[n]}2^{-n}f(S)\frac{f^{=1}(S)-f^{=1}(S^{c})}{2}=$
$\sum_{S\subset[n]}2^{-n}f^{=1}(S)\frac{f(S)-f(S^{c})}{2}\leq\sum_{S\subset[n]}2^{-n}\frac{1}{2}\left|f^{=1}(S)\right|=\frac{1}{2}|f^{=1}|_{1}.$

So

$\frac{1}{2}-2\delta\leq|f^{=1}|_{1},|f^{=1}|_{2}=\sqrt{1/4-\delta}\leq\frac{1}{2}-\delta.$

Normalizing, let $h=\frac{f^{=1}}{|f^{=1}|_{2}}$, then

$|h|_{2}=1,|h|_{1}\geq\frac{1/2-2\delta}{1/2-\delta}\geq 1-4\delta$

So $h=\sum a_{i}\raise 2.2pt\hbox{$\chi$}_{i}$, with $\sum a_{i}^{2}=1$, and
we wish to prove that one of the $a_{i}$’s is close to 1 or -1. This follows
immediately from the following result of König, Schütt and Tomczak-Jaegermann,
[2].

######  Theorem 0.2.

Let $a_{1},a_{2},\dots,a_{n}$ be real numbers such that $\sum a_{i}^{2}=1$.
Let $E$ be the expected value of $|\sum a_{i}\raise 2.2pt\hbox{$\chi$}_{i}|$.
Then

$\left|E-\sqrt{\frac{2}{\pi}}\right|\leq\left(1-\sqrt{\frac{2}{\pi}}\right)\max_{1\leq
i\leq n}|a_{i}|.$ (0.1)

In our case this yields

$\max\\{|a_{i}|\\}\geq\frac{1-\sqrt{\frac{2}{\pi}}-4\delta}{1-\sqrt{\frac{2}{\pi}}}>1-20\delta.$

## References

  * [1] E. Friedgut, G. Kalai, A. Naor, Boolean functions whose Fourier transform is concentrated on the first two levels, Advances in Applied Mathematics 29 (2002), pp. 427–437.
  * [2] König, H., C. Schütt, N. Tomczak-Jaegermann , Projection constants of symmetric spaces and variants of Khintchine’s inequality. J. Reine Angew. Math. 511 (1999), 1–42.


# MilliKelvin microwave impedance microscopy in a dry dilution refrigerator

Leonard Weihao Cao  Department of Physics, University of California, San
Diego, CA 92093, USA  Chen Wu  Department of Physics, University of
California, San Diego, CA 92093, USA  Rajarshi Bhattacharyya  Department of
Physics, University of California, San Diego, CA 92093, USA  Ruolun Zhang
Department of Physics, University of California, San Diego, CA 92093, USA
Monica T. Allen<EMAIL_ADDRESS>Department of Physics, University
of California, San Diego, CA 92093, USA

###### Abstract

Microwave impedance microscopy (MIM) is a near-field imaging technique that
has been used to visualize the local conductivity of materials with nanoscale
resolution across the GHz regime. In recent years, MIM has shown great promise
for the investigation of topological states of matter, correlated electronic
states and emergent phenomena in quantum materials. To explore these low-
energy phenomena, many of which are only detectable in the milliKelvin regime,
we have developed a novel low-temperature MIM incorporated into a dilution
refrigerator. This setup, which consists of a tuning-fork-based atomic force
microscope with microwave reflectometry capabilities, is capable of reaching
temperatures down to $70\text{\,}\mathrm{m}\mathrm{K}$ during imaging and
magnetic fields up to $9\text{\,}\mathrm{T}$. To test the performance of this
microscope, we demonstrate microwave imaging of the conductivity contrast
between graphite and silicon dioxide at cryogenic temperatures and discuss the
resolution and noise observed in these results. We extend this methodology to
visualize edge conduction in Dirac semimetal cadmium arsenide in the quantum
Hall regime.

††preprint: AIP/123-QED

##  I Introduction

Figure 1:  Scanning probe microscopy system with combined AFM and MIM readout,
integrated into a dilution refrigerator. (a) Schematic of the scanning MIM
readout electronics and hardware integrated into a dilution refrigerator. The
shaded regions refer to the different thermal stages inside the fridge. (b)
Left panel: Photo of the microscope head, scanners and sample stage
(corresponding to the red box in the schematic). Right panels: Zoomed-in view
of the tip glued onto the tuning fork (blue box) and a scanning electron
microscope image of the etched tungsten tip used for combined AFM and MIM
imaging. (c) Plot of the reflect microwave power $S_{11}$ of the impedance
matching network (IMN) shows the fundamental resonance at the
$1.8\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. Inset: Circuit diagram of the
IMN with a $0.2\text{\,}\mathrm{p}\mathrm{F}$ capacitor with
$5\text{\,}\mathrm{c}\mathrm{m}$ coax connected in series. (d) Plots of the
oscillation amplitude of the tuning fork as a function of frequency, showing
the mechanical resonance used for height feedback. The upper and lower panels
show the resonance peak at room temperature and 70 mK, respectively.

Microwave impedance microscopy (MIM) has the unique capacity to probe the
local conductivity and permittivity of quantum materials with nanoscale
spatial resolutionLai  _et al._ (2007, 2011a, 2011b); Kundhikanjana  _et al._
(2013); Ma  _et al._ (2015a, b); Liu  _et al._ (2015); Seabron  _et al._
(2016); Chu  _et al._ (2020); Barber, Ma, and Shen (2022). This enables direct
visualization of the microscopic nature of electronic states, including the
real-space disorder landscape, multi-domain behavior, or the presence of
topological modes that propagate along the sample boundaries. By coupling
microwaves with a wavelength of 1-$100\text{\,}\mathrm{c}\mathrm{m}$ to a
sharp metallic probe and collecting the reflected signal, MIM characterizes
the complex admittance between the tip and the sample without the requirement
for the sample to be conductive, which is less restrictive than other
electronic imaging techniquesEriksson  _et al._ (1996); Döring, Eng, and Kehr
(2016); Lu  _et al._ (2017); McGilly  _et al._ (2019); Rosenberger  _et al._
(2020). As demonstrated in recent experiments, MIM can provide insight into
the real-space nature of correlated states and topological states in two-
dimensional heterostructuresLai  _et al._ (2008, 2010); Ma  _et al._ (2015c);
Wu  _et al._ (2018); Allen  _et al._ (2019); Chu  _et al._ (2020); Barber, Ma,
and Shen (2022). However, many of these states are characterized by low energy
scales and are therefore most robust at millikelvin temperatures, motivating
the development of cryogenic MIM instrumentation. Thus far, most state-of-the-
art MIM experiments have been performed in
1.5-$2\text{\,}\mathrm{K}$Kundhikanjana  _et al._ (2011) or He-3 cryostats,
which can reach of a minimum temperature of $450\text{\,}\mathrm{m}\mathrm{K}$
Allen  _et al._ (2019).

Here we report on the construction of a novel milliKelvin MIM, which will
support spatially-resolved detection of quantum electronic states at ultra-low
temperatures. This setup consists of scanning probe microscope with tuning-
fork-based height feedback integrated into a dry dilution refrigerator. A
sharp metallic probe driven by an AC signal at microwave frequency is coupled
to the tuning fork and scanned over the sample. Using reflectometry, MIM
detects the sample’s response to high frequency electromagnetic fields
emanating from the probe. To demonstrate the measurement capabilities of this
setup, we present MIM images of the conductivity contrast between graphite and
SiO2 down to temperatures of 70 mK. Finally, we also demonstrate microwave
imaging of edge states in Cd3As2 thin films in the quantum Hall regime at the
base temperature.

##  II Experimental Setup

This setup consists of an custom-designed tuning fork based atomic force
microscope (AFM) integrated into a Leiden Cryogenics CF-CS110 dilution
refrigerator. The microscope housing is in thermal equilibrium with the mixing
chamber plate on the cold-insertable probe, which is loaded into a dilution
refrigerator, as shown schematically in Figure 1(a). Figure 1(b) shows the
design of the microscope head, which houses an etched tungsten wire mounted
onto to the end of one prong of a tuning fork (TF) mechanical resonator (blue
box) Khan  _et al._ (2012). The oscillation amplitude of the TF is monitored
for continuous height feedback, which enables tapping-mode topographic
imagingEdwards  _et al._ (1997). Below the tip holder, the sample stage is
mounted on a stack of CuBe piezoelectric scanners (Attocube AN-Sxyz100) and
the positioners (ANPx(z)100), which control fine xyz scanning (up to
$$40\text{\,}\mu\mathrm{m}$\times$40\text{\,}\mu\mathrm{m}$$ below
$4\text{\,}\mathrm{K}$) and coarse positioning
($$5\text{\,}\mathrm{m}\mathrm{m}$\times$5\text{\,}\mathrm{m}\mathrm{m}$$
below $4\text{\,}\mathrm{K}$), respectively.

On the MIM circuitry side, GHz signals are generated by an analog signal
generator, and one split branch of the signal is coupled to the tip via an
impedance matching network (IMN) Pozar (2011), which is responsible for
minimizing the reflected signal [inset in Figure 1(c)] Cui, Ma, and Shen
(2016). A plot of the reflected microwave power $S_{11}$ of an example IMN is
shown in Figure 1(c), showing the first resonance at
$1.8\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. The reflected signal from the
tip passes through two directional couplers mounted on the probe-still plate
($1\text{\,}\mathrm{K}$) to cancel out the residual reflected power. The
signal from the sample is then amplified by a cryogenic amplifier (Cosmic
Microwave Technology CITCRYO1-12) mounted on the $3\text{\,}\mathrm{K}$ stage,
after which the signal propagates out of the probe and gets further amplified
and demodulated at room temperature, as shown in Figure 1(a).

During the tip approach procedure, active height feedback can be performed by
monitoring either the TF oscillation amplitude or the MIM signal. Here we use
a Nanonis SC5 controller to excite and track the TF oscillation and control
the fine scanners during imaging Cui, Ma, and Shen (2016). Figure 1(d)
displays a measurement of the oscillation amplitude of the tuning fork as a
function of excitation frequency, showing the resonance peak near 32.768 kHz.
The Q-factor of the resonance is around 500 - 2000 at room temperature (upper
panel), while at base temperature it can easily reach 10,000-100,000 (lower
panel).

The main technical challenge of microwave imaging in a dry dilution fridge is
the emergence of new noise sources, which impact both spatial resolution and
the signal-to-noise of the microwave reflectometry measurements. There are two
main sources of increased noise: (1) mechanical pulse tube vibrations, which
are associated with the cooling mechanism of the dilution fridge, place limits
on the lateral spatial resolution and add noise to the measured MIM signal,
and (2) the high Q factor of the tuning fork at mK temperatures leads to
fluctuations in the tip-sample distance, which also couples with the pulse
tube vibration. Our fridge is equipped with a pulse tube cryocooler operating
at $\sim$1.4\text{\,}\mathrm{H}\mathrm{z}$$ Wang and Gifford (2002); Chijioke
and Lawall (2010) generating vibrations that amplitude modulate the tuning
fork oscillation, and consequently also modulate the GHz MIM signal. To
mitigate these vibrations, we physically decoupled the pulse tube motion from
the microscope by unscrewing the rotary valve from the fridge and putting
isolation foam in between Li  _et al._ (2005), while the high-pressure helium
lines connected to the compressor are wrapped with acoustic pipe lagging.

We found that performing AC-mode MIM imaging, described below, largely
eliminates background oscillations in the images that arise from pulse tube
vibrations. In AC height-modulated imaging mode, a low frequency lock-in
amplifier (SR830) is added to the output of the GHz frequency mixer to
demodulate the reflected MIM signal at the tuning fork resonance frequency (32
kHz), after which low-pass filters can be used to attenuate noise Cui, Ma, and
Shen (2016). We note that because the GHz MIM signal (from the tip) is
amplitude-modulated by both the tuning fork oscillation at 32 kHz and the
pulse tube vibration, there are multiple side-bands around the measurement
frequency. Therefore, band-pass filters between 20-30 kHz are added to the
output of the GHz mixer to reduce noise, after which the MIM signal is fed
into the SR830 lock-in amplifier for demodulation. During this step, the lock-
in amplifier multiplies the MIM input signal with a TF reference signal
(provided by the measured piezo current from the tuning fork, after
amplification by a commercial low-noise charge amplifier) to extract the in-
phase components. Both the filters inside SR830 and the additional low-pass
filter added to the output of the lock-in are chosen to eliminate noise at the
pulse tube vibration frequency.

Figure 2:  Topographic imaging of a micropatterned dielectric film at mK
temperatures using tuning-fork-based atomic force microscopy. (a) Optical
image of an etched array of holes $\mathrm{SiO_{2}}$. The diameter and the
spacing of the holes are 1 $um$. The hole depth is
$20\text{\,}\mathrm{n}\mathrm{m}$. (b) AFM spatial scan at 70mK. The scan
covers 4$\times$ $4\text{\,}\mathrm{u}\mathrm{m}$ and the scan speed is 400
$nm/s$. (c) Cross-sectional line cut corresponding to the black line in (b).
Figure 3:  Microwave impedance microscopy of graphite at milliKelvin
temperatures. (a) Optical image of a graphite flake exfoliated onto a SiO2/Si
substrate. The dark purple region has a thickness of
3-$5\text{\,}\mathrm{n}\mathrm{m}$, and the light yellow region has a
thickness of $\sim$20\text{\,}\mathrm{n}\mathrm{m}$$. The blue box marks the
imaging window for (c) and (d). (b) Theoretical MIM response curves simulated
at $1.8\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, illustrating the evolution of
the MIM contrast with the sample conductivity. Inset: vertical cut of the
potential distribution for the tip-sample interaction, calculated using
finite-element analysis. (c) AFM and MIM imaging of the graphite flake at
$4\text{\,}\mathrm{K}$, with the scan window covering the
$20\text{\,}\mathrm{n}\mathrm{m}$ region (lower left),
3-$5\text{\,}\mathrm{n}\mathrm{m}$ region (middle), and the SiO2 region (upper
right). The scan speed is 0.5 $\mu m/s$ . (d) AFM and MIM images of the same
location at $70\text{\,}\mathrm{m}\mathrm{K}$. The scan speed is 0.2 $\mu
m/s$.

##  III Results and Discussion

We characterized the low temperature performance of the AFM on a sample
consisting of an array of etched $\mathrm{SiO_{2}}$ holes patterned on a Si
wafer, as depicted in the optical image in Figure 2(a). Cryogenic AFM
measurements are used to visualize the topographic profile of a 5 $\mu$m × 5
$\mu$m scan region at $70\text{\,}\mathrm{m}\mathrm{K}$, as depicted in Figure
2(b). Figure 2(c) shows a cross-sectional cut of this AFM image, whose
position is marked by the black line, revealing a noise level of roughly
$3\text{\,}\mathrm{n}\mathrm{m}$. To more carefully assess the magnitude of
the z-noise during AFM scanning, we performed 96 x 96-pixel noise scans over a
1 nm x 1 nm area, such that the spatial profile is irrelevant. Root mean
square (RMS) roughness was calculated using Gwyddion after line fitting, which
gives z-noise levels in the range of 1.8 - 2.2 nm. Furthermore, upon careful
inspection of Figure 2(b), we noticed that a tilted stripe pattern appears as
a background modulation in the AFM image. By taking a Fourier transform of
this data, we found that the stripe pattern has a frequency of
$1.4\text{\,}\mathrm{H}\mathrm{z}$, which coincides with the frequency of the
pulse tube.

Next, to demonstrate our combined AFM and MIM imaging capabilities at low
temperatures, we measured the spatial contrast of the MIM response across the
boundary between graphite and SiO2 at 70 mK. Figure 3(a) shows an optical
image of the graphite sample, which has terraces of varying thicknesses: the
purple region is $\sim$3\text{\,}\mathrm{n}\mathrm{m}$$ and the bright yellow
region is 15-$20\text{\,}\mathrm{n}\mathrm{m}$. In Figure 3, panels (b) and
(c) display AFM and MIM images of the graphite/SiO2 interface measured at 4K
and $70\text{\,}\mathrm{m}\mathrm{K}$, respectively. In both sets of AFM
images, the 3/$20\text{\,}\mathrm{n}\mathrm{m}$ step height in graphite is
clearly visible, while the graphite/SiO2 boundary only shows a faint contour,
as the z-movement of the scanner to compensate for fluctuations in the tip-
sample distance dominates over the $3\text{\,}\mathrm{n}\mathrm{m}$ boundary.
Meanwhile, we observe a sharp contrast in the MIM signal across the
graphite/SiO2 boundary due to the different conductivities of the two
materials, as predicted by the response curves in Figure 3(b). To explain the
experimental observations, one can model the tip-sample interaction for this
system using finite element analysis, which can be used to calculate the MIM
response curves as a function of sample conductivityBarber, Ma, and Shen
(2022). At a measurement frequency of 1.8 GHz, the imaginary part of the MIM
response should monotonically increase with the sample conductivity,
saturating when the resistivity is higher than $10^{-2}$ $\Omega\cdot m$
(insulating limit) or lower than $10^{-5}$ $\Omega\cdot m$ (conductive limit),
as shown in Figure 3(b). A cross-sectional profile of the penetration of the
tip potential into the sample is provided in the inset. We estimate the MIM
spatial resolution to be around $200\text{\,}\mathrm{n}\mathrm{m}$,
constrained by the apex geometry of the etched tungsten tip and mechanical
noise from pulse tube vibrations.

Figure 4:  Microwave imaging of edge modes in a cadmium arsenide film in the
quantum Hall regime. (a) Cross-sectional schematic of an epitaxially-grown
Cd3As2 heterostructure. (b) Transport measurement of the longitudinal
resistance $R_{xx}$ as a function of magnetic field at
$90\text{\,}\mathrm{m}\mathrm{K}$. The minima correspond to the emergence of
quantum Hall plateaus. (c) MIM image at $6.5\text{\,}\mathrm{T}$, revealing a
sharp enhancement of the reflected signal at the boundaries of a quantum Hall
insulator state. (d) MIM image at $5.4\text{\,}\mathrm{T}$, showing spatially
uniform conductivity at the transition between quantum Hall plateaus (e-f)
Cross-sectional line cuts of the MIM response across the sample extracted from
(c) and (d) respectively.

We also apply this methodology to visualize edge states at the boundaries of
thin film cadmium arsenide (Cd3As2), a novel three-dimensional Dirac semi-
metal, in the quantum Hall regimeSankar  _et al._ (2015); Schumann  _et al._
(2016). A cross-sectional schematic of the epitaxially-grown hetero-structure
is shown in Figure4 (a), where the film thickness is
$20\text{\,}\mathrm{n}\mathrm{m}$Goyal  _et al._ (2018); Lygo  _et al._
(2023). The Cd3As2 device is lithographically patterned and etched into strips
of width 10-15 $\mu$m, which are electrically grounded. Transport measurements
were performed to characterize the magnetic field behavior of the sample,
which reveal dips in the longitudinal resistance at around
$4.7\text{\,}\mathrm{T}$ and $6.5\text{\,}\mathrm{T}$, as shown in Figure
4(b). These minima should correspond to the emergence of quantum Hall
plateausGuo  _et al._ (2022).

To shed light on the real-space conductivity profile of Cd3As2 in the quantum
Hall regime and monitor its evolution across the topological phase transition
between plateaus, MIM measurements were performed at a series of magnetic
fields at a base temperature of 90 mK. Microwave imaging reveals a sharp
enhancement of the reflected MIM signal at the boundaries of the sample in the
quantum Hall insulator state, which rapidly decays into the bulk of the
sample, as shown in Figure 4(c). Meanwhile, we observed a spatially-uniform
conductivity at the transition between quantum Hall plateaus when the
longitudinal resistance deviates from zero at B = 5.4 T (Figure 4(d)). The
variation of the MIM signal between different lines comes both from the noise
in the MIM signal and spatial inhomogeneities in the sample.

To more clearly compare the spatial dependence of the MIM signal in these two
regimes, in Figure 4(e-f) we plot the cross-sectional profiles of the MIM
response across the sample extracted from panels (c) and (d), respectively.
These low temperature microwave images reveal sharply enhanced edge conduction
that encloses an insulating interior in the quantum Hall regime, which is
consistent with the results of transport measurements performed on this system
in prior experimental studies.

We note that one way to improve signal quality is to use “floating” AC-mode
MIM, where imaging is performed with the tip retracted a fixed distance
(60-$100\text{\,}\mathrm{n}\mathrm{m}$) above the sample surface. At this
distance, the AFM channel will not be modulated due to the topography
feedback, but the MIM tip can still interact with the sample via the
electromagnetic fields in the vicinity of the tip (when operated in the near-
field regime). Because periodic oscillations in the tip-sample distance at the
tuning fork resonance are decoupled from the surface roughness of the sample,
noise in the MIM response can be dramatically reduced in floating mode. Figure
5 shows the results of a floating mode MIM scan performed at
$3\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ and T=
$70\text{\,}\mathrm{m}\mathrm{K}$, with the tip lifted
$100\text{\,}\mathrm{n}\mathrm{m}$ above an hBN-covered graphite layer. The
tip apex is around $0.8\text{\,}\mathrm{u}\mathrm{m}$, which is reflected in
the spatial profile of the MIM signal change across the boundary between the
graphite flake and hBN. In this case, the signal-to-noise ratio is even better
than that observed in tapping mode MIM images (Figure 3(c-d), which is
especially useful for fixed-location MIM measurements. However, this advantage
comes at the expense of signal size, as the tip is further away from the
sample than for tapping mode.

The choice of tip-sample distance for floating-mode measurements is a
compromise between maximizing signal sensitivity and minimizing the risk of a
tip crash due to vertical fluctuations in the tip-sample distance, which arise
from pulse tube vibrations and are aggravated by the large Q factor of the
tuning fork at mK temperatures. For larger scan windows or rougher sample
surfaces, the tip may need to be retracted further. We expect the sensitivity
of floating mode to be around
$0.01-$0.1\text{\,}\mathrm{a}\mathrm{F}\mathrm{/}\sqrt{Hz}$$ at
$0.1\text{\,}\mathrm{u}\mathrm{W}$ input power, and in our case the noise is
mostly due to vertical modulations of the tip-sample distanceLai  _et al._
(2008).

Figure 5:  Demonstration of low-noise height-modulated MIM in a dry dilution
fridge.  (a) Optical image of a van der Waals heterostucture with a graphite
gate ( $5\text{\,}\mathrm{n}\mathrm{m}$ thick) covered by hBN (
$50\text{\,}\mathrm{n}\mathrm{m}$ thick). (b) A floating mode MIM image of the
region enclosed by the red square in panel (a), acquired at
$70\text{\,}\mathrm{m}\mathrm{K}$. The measurement frequency is
$3\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ and the tip is retracted
$100\text{\,}\mathrm{n}\mathrm{m}$ from the highest feature inside the scan
window.

##  IV Conclusion and outlook

In summary, we report on the development of a microwave impedance microscope
that operates at temperatures down to $70\text{\,}\mathrm{m}\mathrm{K}$. This
is achieved by incorporating a TF-based AFM with near-field GHz imaging
capabilities into a dry dilution refrigerator. Pushing the limits of MIM into
new low temperature regimes should enable local sensing of quantum phenomena
that only exist at low energy scales, including certain topological states of
matter, domain wall physics at phase transitions, quantum states arising from
geometric confinement in mesoscopic devices, and correlated states in two-
dimensional materials and van der Waals heterostructures. Because this
instrumentation is equipped with combined transport and imaging capabilities,
it can also illuminate the correspondence between macroscopic transport
behavior and the underlying microscopic nature of electronic states, including
the real-space disorder landscape or presence of edge modes.

During the preparation of this manuscript, we became aware of a pre-print on a
related topicJiang  _et al._ (2023).

###### Acknowledgements.

We thank Alex Lygo and Susanne Stemmer for providing cadmium arsenide devices
for these experiments, Yongtao Cui for inspiring discussions, and Evan Cobb
for helping develop some of the MIM circuitry. We gratefully acknowledge
funding support from the UC Office of the President, specifically the UC
Laboratory Fees Research Program (award LFR-20-653926), the AFOSR Young
Investigator Program (award FA9550-20-1-0035) and the AFOSR/ARO MURI Program
(award FA9550-22-1-0270). This work was performed, in part, at the San Diego
Nanotechnology Infrastructure (SDNI) of UCSD, a member of the National
Nanotechnology Coordinated Infrastructure, which is supported by the National
Science Foundation (Grant ECCS-2025752).

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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