# Scaled Homology and Topological Entropy

Bingzhe Hou  Bingzhe Hou, School of Mathematics, Jilin university, Changchun,
P.R.China, 130012<EMAIL_ADDRESS>,  Kiyoshi Igusa  Kiyoshi Igusa,
Department of Mathematics, Brandeis university, Waltham, MA, 02453
<EMAIL_ADDRESS>and  Zihao Liu  Zihao Liu, Department of Mathematics,
Brandeis university, Waltham, MA, 02453<EMAIL_ADDRESS>
###### Abstract.

In this paper, we build up a scaled homology theory, $lc$-homology, for metric
spaces such that every metric space can be visually regarded as “locally
contractible” with this newly-built homology. We check that $lc$-homology
satisfies all Eilenberg-Steenrod axioms except the exactness axiom whereas its
corresponding $lc$-cohomology satisfies exactness axiom for cohomology. This
homology can relax the smooth manifold restrictions on the compact metric
space such that the entropy conjecture will hold for the first $lc$-homology
group.

###### 2020 Mathematics Subject Classification:

55N35; 37B40

##  1\. Introduction and preliminaries

The topological entropy of a map $f:M\rightarrow M$, denoted by $h(f)$,
measures the evolution of distinguishable orbits over time, thereby providing
an idea of how complex the orbit structure of a system is. Entropy
distinguishes a dynamical system where points that are close together remain
close from a dynamical system where sets of points move farther. On the other
hand, it measures how much $f$ mixes up the point set topology of $M$ whereas
its induced homomorphism $f_{*}:H_{*}(M;\mathbb{R})\rightarrow
H_{*}(M;\mathbb{R})$ measures how much $f$ mixes up the algebraic topology of
$M$.

In order to discover the connection between topological entropy and homology
theory, there is an entropy conjecture due to Shub [9] relating $h(f)$ to
$r(f_{*})$, the spectral radius of $f_{*}$. The idea is that $h(f)$ should be
bounded below by $\log r(f_{*})$. Unfortunately, Shub showed [9] that it is
not true in general for continuous maps $f$ of a manifold $M$ nor for $f$ a
homeomorphism of a finite complex.

One of the known results [5] is that the entropy conjecture holds for the
first homology group once $M$ is a compact smooth manifold. Moreover, it is
also shown in [5] that $M$ does not necessarily have to be a manifold. It
suffices that the space can be seen as “locally contractible”. Motivated by
this relaxation, we consider building up a new homology theory such that every
metric space can be regarded as locally contractible with this homology. Our
initial idea is to “rescale” the metric space. Naturally, when we observe
things around us, sometimes it is hard to guarantee that everything we see is
absolutely precise, especially for someone suffering from myopia. For example,
given a metric space $(X,d)$, for any $x\neq y$ with $d(x,y)=\epsilon$, they
can be seen as one point when we put a “scale” greater than $\epsilon$ onto
the space. So in the second section, we introduce the definitions of
comparatively “rough” continuous maps, $\epsilon$-continuous maps,
$\epsilon$-singular chain groups as well as $\epsilon$-singular homology
groups $H_{*}^{\epsilon}(X)$. If the diameter of a space is sufficiently
small, then the $\epsilon$-homology group of that space is trivial for $n>0$,
i.e., it is like a point or contractible!

Since we have an explicit chain complex for $\epsilon$-homology, we can easily
check that it satisfies the Eilenberg–Steenrod axioms except for homotopy
axiom. Indeed, the $\epsilon$-homology is not a functor from category of
metric spaces ${\mathcal{M}}et$ to category of abelian groups
${\mathcal{A}}b$. Concerning this drawback, we take the inverse limit of
$(H_{*}^{\epsilon}(X))_{\epsilon\in\mathbb{R}+}$ to obtain the infinitesimal-
scaled homology groups of the space and call them $lc$-homology groups where
$lc$ denotes the meaning of “local contractification”. It is then a homology
functor and satisfies most of the homology axioms, but it fails to satisfy
exactness axiom in general. In the following we discuss two counterexamples
and claim that it holds, however, for the corresponding $lc$-cohomology
theory.

Our work in the last section generalizes the existing results of entropy
conjecture, relaxing the restrictions on the compact metric space on which $f$
acts. The entropy conjecture will then hold for the first homology group with
$lc$-homology.

Through out this paper, all spaces are metric spaces. To begin with, let’s
recall the concept of topological entropy defined by Bowen [3].

Given a compact metric space $(X,d)$ and a continuous map $f:X\rightarrow X$,
we set $d_{n}:X\times X\rightarrow\mathbb{R}$ by

$d_{n}(x,y)=\max_{0\leq k<n}d(f^{k}(x),f^{k}(y)).$

The map $d_{n}$ is a metric on $X$ for each $n$ and we will consider the
metric $d_{n}$ to count distinguishable orbit segments at a fixed resolution.

######  Definition 1.1.

Fix $\epsilon>0$ and let $n\in\mathbb{N}$. A set $S\subset X$ is said to be an
$(n,\epsilon)$-spanning set for $f$ if for all $x\in X$, there exists $y\in S$
such that $d_{n}(x,y)<\epsilon$, and it is said to be an
$(n,\epsilon)$-separated set for $f$ if for all $x\neq y\in S$,
$d_{n}(x,y)\geq\epsilon$, i.e., there exists $k\in\\{0,1,\cdots,n-1\\}$ such
that $d(f^{k}(x),f^{k}(y))\geq\epsilon$.

Let $r_{n}(\epsilon,f)$ be the minimum cardinality of an
$(n,\epsilon)$-spanning set and $s_{n}(\epsilon,f)$ be the maximum cardinality
of an $(n,\epsilon)$-separated set. Then we have the following relation

$r_{n}(\epsilon,f)\leq s_{n}(\epsilon,f)\leq r_{n}(\frac{\epsilon}{2},f)$

and this implies the following equivalent definition of the topological
entropy:

######  Definition 1.2.

Let $f:X\rightarrow X$ be a continuous map. The topological entropy of $f$ is
defined as

$h(f)=\lim\limits_{\epsilon\to
0^{+}}\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\log
s_{n}(\epsilon,f)=\lim\limits_{\epsilon\to
0^{+}}\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\log r_{n}(\epsilon,f).$

In [9], Shub stated the following conjecture.

Entropy conjecture. Let $f$ be a continuous map on a compact finite-
dimensional manifold $M$ to itself. Then

$h(f)\geq\log\rho(f_{*}),$

where $f_{*}:H_{*}(M,\mathbb{R})\rightarrow H_{*}(M,\mathbb{R})$ denotes the
homomorphism induced by $f$ on the total homology of $M$, i.e.,

(1.1) $H_{*}(M,\mathbb{R})=\bigoplus\limits_{i=0}\limits^{\dim
M}H_{i}(M,\mathbb{R})$

and $\rho(f_{*})$ denotes the spectral radius of $f_{*}$, i.e.,
$\rho(f_{*})=\lim\limits_{n\rightarrow\infty}(\|f_{*}^{n}\|^{\frac{1}{n}})$,
which coincides with the maximum of the moduli of the eigenvalues of $f_{*}$.
The entropy conjectures means that $f_{*}$ must capture some but not
necessarily all of the mixing $f$ does.

Since the expansion (1.1) is clearly invariant under $f_{*}$, we have

$\rho(f_{*})=\max_{1\leq i\leq\dim M}\rho(f_{*i})$

where $f_{*i}$ denotes the restriction of $f_{*}$ to the space
$H_{i}(M;\mathbb{R})$. Hence the entropy conjecture is equivalent to the
system of inequalities

$h(f)\geq\log\rho(f_{*i}),\ i=1,\cdots,\dim M.$

In fact, the entropy conjecture only holds for some special cases. In [5],
Manning showed that

######  Theorem 1.3.

Let $M$ be a compact differentiable manifold without boundary and
$f:M\rightarrow M$ a continuous map. Then

$h(f)\geq\log\rho(f_{*1})$

where $f_{*1}:H_{1}(M;\mathbb{R})\rightarrow H_{1}(M;\mathbb{R})$ is the
induced map on the first homology group.

######  Corollary 1.4.

If $M$ has dimension $\leq 3$ and $f$ is a homeomorphism, then
$h(f)\geq\log\rho(f_{*})$ where $f_{*}$ is the map induced on the homology of
all dimensions.

In [7], the authors propose to imitate for homeomorphisms the “Markov
approximation” procedure for diffeomorphisms mentioned in [10] and construct a
dense set of homeomorphisms where the entropy conjecture holds.

######  Theorem 1.5.

If $\dim M\geq 5$, the entropy conjecture holds for all homeomorphisms
belonging to an open dense set of the space of all homeomorphisms of $M$ with
the $C^{0}$-topology.

Next, due to Nitecki in [6], this holds for diffeomorphisms with a hyperbolic
structure. Thin would imply that any sufficiently small perturbation in the
$C^{0}$-topology of such a homeomorphism could not decrease the topological
entropy.

For arbitrary homeomorphisms by [8], however, the entropy conjecture does not
hold.

It is also shown in [5] that it is not necessary to suppose that $M$ is a
manifold. It suffices that the following conditions be met:

  1. (1)

For any $\epsilon>0$, there exists a $\delta>0$ such that any two points $x$
and $y$ for which $d(x,y)<\delta$ can be joined by a path of diameter
$<\epsilon$.

  2. (2)

There exists an $\epsilon_{0}>0$ such that any loop of diameter
$<\epsilon_{0}$ is contractible in $M$.

Inspired by the relaxation of the restrictions Manning did on the space on
which $f$ acts, we attempt to generalize the existing results regarding smooth
manifolds to a metric scale.

##  2\. Scaled homology and cohomology

######  Definition 2.1.

Given $\epsilon>0$, a map $f:X\rightarrow Y$ is said to be
$\epsilon$-continuous if there exists $\delta=\delta(\epsilon)>0$ such that

$\sup\\{d_{Y}(f(x_{1}),f(x_{2})):d_{X}(x_{1},x_{2})<\delta\\}<\epsilon.$

Let $\Delta^{n}=\\{(t_{0},...,t_{n})\in\mathbb{R}^{n+1}|\sum_{i=0}^{n}t_{i}=1\
{\rm and}\ t_{i}\geq 0,\forall i\\}$, the standard $n$-simplex. An
$\epsilon$-continuous map $\sigma:\Delta^{n}\rightarrow X$ is said to be an
$\epsilon$-singular $n$-simplex. Moreover, we define the $n$-dimensional
$\epsilon$-singular chain group, denoted by $C_{n}^{\epsilon}(X)$, to be the
free abelian group generated by all $\epsilon$-singular $n$-simplices of $X$,
and $\partial_{n}:C_{n}^{\epsilon}(X)\rightarrow C_{n-1}^{\epsilon}(X)$ to be
the boundary operator defined as the one in classic singular homology theory,
i.e.,

$\partial_{n}\sigma=\sum_{i=0}^{n}(-1)^{i}\sigma|[v_{0},\cdots,\hat{v_{i}},\cdots,v_{n}].$

Then we call $(C_{*}^{\epsilon}(X),\partial)$ an $\epsilon$-singular chain
complex.

######  Definition 2.2.

Let $(C_{*}^{\epsilon}(X),\partial)$ be an $\epsilon$-singular chain complex.
Define $Z_{n}^{\epsilon}(X)=\operatorname{Ker}(\partial_{n})$ to be the
subgroup of cycles and $B_{n}^{\epsilon}(X)=\operatorname{Im}(\partial_{n+1})$
to be the subgroup of boundaries. Furthermore,
$H_{n}^{\epsilon}(X)=Z_{n}^{\epsilon}(X)/B_{n}^{\epsilon}(X)$ is said to be
the $n^{th}$ $\epsilon$-homology group of $X$.

######  Proposition 2.3.

If $(X,d)$ is a metric space with diameter $<\epsilon$, i.e., the maximum
distance between any two of its points is less than $\epsilon$, then
$H_{0}^{\epsilon}(X)=\mathbb{Z}$ and $H_{n}^{\epsilon}(X)=0$ for all $n>0$.

###### Proof.

As in the case of singular homology theory, it is easy to see that if $X$ is a
point $\\{{\rm pt}\\}$, then $H_{n}^{\epsilon}(X)=0$ for $n>0$ and
$H_{0}^{\epsilon}(X)\cong\mathbb{Z}$ since there is a unique
$\epsilon$-singular $n$-simplex $\sigma_{n}$ for all $n$.

Now let $f$ be a constant map sending all of $X$ to $x_{0}\in X$. Since ${\rm
diam}X<\epsilon$, $f$ is $\epsilon$-homotopic to the identity map
$\operatorname{id}:X\rightarrow X$, i.e., there exists $F:X\times I\rightarrow
Y$ given by $F(x,t)=f_{t}(x)$ that is $\epsilon$-continuous such that
$f_{0}=f$, $f_{1}=\operatorname{id}$.

Let $\Delta^{n}\times{0}=[v_{0},\cdots,v_{n}]$ and
$\Delta^{n}\times{1}=[w_{0},\cdots,w_{n}]$ and let
$P:C_{n}^{\epsilon}(X)\rightarrow C_{n+1}^{\epsilon}(X)$ be such that

$P(\sigma)=\sum_{i}(-1)^{i}F\cdot(\sigma\times\operatorname{id})|[v_{0},\cdots,v_{i},w_{i},\cdots,w_{n}]$

for $\sigma:\Delta^{n}\rightarrow X$, an $\epsilon$-continuous map. It is easy
to check that $P$ is a chain homotopy between the chain maps $f_{\sharp}$ and
$\operatorname{id}_{\sharp}$ that are induced by $f$ and $\operatorname{id}$
respectively, and $f_{*}:H_{n}^{\epsilon}(X)\rightarrow
H_{n}^{\epsilon}(\\{{\rm pt}\\})$ are isomorphisms for all $n$. Therefore,
$H_{0}^{\epsilon}(X)=\mathbb{Z}$ and $H_{n}^{\epsilon}(X)=0$ for all $n>0$. ∎

Thus, a metric space $(X,d)$ with $\epsilon$-scale can be informally seen as a
“locally contractible” space. Like in the singular homology theory, we can
define the reduced $\epsilon$-homology groups
$\widetilde{H}_{*}^{\epsilon}(X)$ to be the homology groups of the augmented
chain complex

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{2}^{\epsilon}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{2}}$$\textstyle{C_{1}^{\epsilon}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{1}}$$\textstyle{C_{0}^{\epsilon}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $\varepsilon(\sum_{i}k_{i}\sigma_{i})=\sum_{i}k_{i}$. It can be easily
checked that
$H_{0}^{\epsilon}(X)=\widetilde{H}_{0}^{\epsilon}(X)\oplus\mathbb{Z}$ and
$H_{n}^{\epsilon}(X)=\widetilde{H}_{n}^{\epsilon}(X)$ for all $n>0$.

For a subspace $A\subset X$, let
$C_{n}^{\epsilon}(X,A)=C_{n}^{\epsilon}(X)/C_{n}^{\epsilon}(A)$. Then
$\partial:C_{n}^{\epsilon}(A)\rightarrow C_{n-1}^{\epsilon}(A)$ induces a
quotient boundary map $\partial:C_{n}^{\epsilon}(X,A)\rightarrow
C_{n-1}^{\epsilon}(X,A)$ and we have $\partial^{2}=0$. So we can define the
relative $\epsilon$-homology group
$H_{n}^{\epsilon}(X,A)=\operatorname{Ker}\partial/\operatorname{Im}\partial$.

Thus, a metric space $(X,d)$ with $\epsilon$-scale can be informally seen as a
“locally contractible” space. Like in the singular homology theory, we can
define the reduced $\epsilon$-homology groups
$\widetilde{H}_{*}^{\epsilon}(X)$ to be the homology groups of the augmented
chain complex

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{2}^{\epsilon}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{2}}$$\textstyle{C_{1}^{\epsilon}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{1}}$$\textstyle{C_{0}^{\epsilon}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $\varepsilon(\sum_{i}k_{i}\sigma_{i})=\sum_{i}k_{i}$. It can be easily
check that
$H_{0}^{\epsilon}(X)=\widetilde{H}_{0}^{\epsilon}(X)\oplus\mathbb{Z}$ and
$H_{n}^{\epsilon}(X)=\widetilde{H}_{n}^{\epsilon}(X)$ for all $n>0$.

For a subspace $A\subset X$, let
$C_{n}^{\epsilon}(X,A)=C_{n}^{\epsilon}(X)/C_{n}^{\epsilon}(A)$. Then
$\partial:C_{n}^{\epsilon}(A)\rightarrow C_{n-1}^{\epsilon}(A)$ induces a
quotient boundary map $\partial:C_{n}^{\epsilon}(X,A)\rightarrow
C_{n-1}^{\epsilon}(X,A)$ and we have $\partial^{2}=0$. So we can define the
relative $\epsilon$-homology group
$H_{n}^{\epsilon}(X,A)=\operatorname{Ker}\partial/\operatorname{Im}\partial$.

An interesting object is the limit of $H_{n}^{\epsilon}(X)$ as
$\epsilon\rightarrow 0^{+}$. Note that an $n$-dimensional $\mu$-singular
simplex is naturally an $n$-dimensional $\epsilon$-singular simplex if
$0<\mu\leq\epsilon$. Then, let $\varphi_{\mu\epsilon*}$ be the homomorphism
induced by the natural inclusion chain map
$\varphi_{\mu\epsilon}:(C_{n}^{\mu}(X),\partial)\hookrightarrow(C_{n}^{\epsilon}(X),\partial)$.
In particular, $\varphi_{\epsilon\epsilon*}$ is the identity on
$H_{n}^{\epsilon}(X)$.

Since $(\mathbb{R}^{+},\leq)$ is a directed partially-ordered set, we obtain
an inverse system

$((H_{n}^{\epsilon}(X))_{\epsilon\in\mathbb{R}^{+}},(\varphi_{\mu\epsilon*})_{\mu\leq\epsilon\in\mathbb{R}^{+}}).$

Furthermore, we can take the inverse limit

$\begin{split}H_{n}^{lc}(X)&=\lim\limits_{\overleftarrow{\epsilon\in\mathbb{R}^{+}}}((H_{n}^{\epsilon}(X))_{\epsilon\in\mathbb{R}^{+}},(\varphi_{\mu\epsilon*})_{\mu\leq\epsilon\in\mathbb{R}^{+}})\\\
&=\\{\overrightarrow{[a]}\in\prod\limits_{\epsilon\in\mathbb{R}^{+}}H_{n}^{\epsilon}(X):[a_{\epsilon}]=\varphi_{\mu\epsilon*}([a_{\mu}]){\rm\
for\ all}\ \mu\leq\epsilon\ {\rm in}\ \mathbb{R}^{+}\\}.\end{split}$

Let $\\{\epsilon_{m}\\}$ be a non-increasing sequence of $\mathbb{R}^{+}$
converging to $0$. Then we have

$\lim\limits_{\overleftarrow{\epsilon\in\mathbb{R}^{+}}}((H_{n}^{\epsilon}(X))_{\epsilon\in\mathbb{R}^{+}},(\varphi_{\mu\epsilon*})_{\mu\leq\epsilon\in\mathbb{R}^{+}})\cong\varprojlim((H_{n}^{\epsilon_{m}}(X))_{m=1}^{\infty},(\varphi_{\epsilon_{k}\epsilon_{l}*})_{l\leq
k\in\mathbb{N}}),$

since $\\{\epsilon_{m}\\}$ is a cofinal subset of the directed index set
$\mathbb{R}^{+}$. We call the inverse limit above the $n^{th}$ infinitesimal-
scaled homology group of $X$ or $lc$-homology group of $X$ (the homology group
under infinitesimal scale or the homology group of the “local
contractification” of $X$).

The $lc$-homology group of $X$ is said to be stable if there exists
$\epsilon>0$ such that for all $0<\epsilon_{2}\leq\epsilon_{1}\leq\epsilon$,
$\varphi_{\epsilon_{2},\epsilon_{1}*}$ is an isomorphism, i.e., for any
$0<\mu\leq\epsilon$, $H_{n}^{\mu}(X)\cong H_{n}^{\epsilon}(X)$. Later we will
show that when $X$ is a Riemannian manifold, its $lc$-homology group will be
stable.

To begin with, we check that $lc$-homology is a functor from category of
topological spaces ${\mathcal{M}}et^{u}$ to category of abelian groups
${\mathcal{A}}b$, where ${\mathcal{M}}et^{u}$ denotes the category that has
metric spaces as its objects and uniformly continous maps between metric
spaces as its morphisms.

######  Proposition 2.4.

If $f:X\rightarrow Y$ is a uniformly continuous map, then $f$ induces a
homomorphism $f^{lc}_{*}:H_{n}^{lc}(X)\rightarrow H_{n}^{lc}(Y),\forall n\geq
0$.

###### Proof.

Let $\sigma:\Delta^{n}\rightarrow X$ be an $\epsilon$-continuous map.
Composing with $f$, we get a $\mu$-continuous map
$f_{\sharp}^{\epsilon\mu}(\sigma)=f\sigma:\Delta^{n}\rightarrow Y$, where
$\mu$ is determined by $f$ and $\sigma$. In fact, we can choose
$\mu=\inf\\{r>0:{\rm whenever}\
d_{X}(\sigma(v_{1}),\sigma(v_{2}))<\epsilon,d_{Y}(f(\sigma(v_{1})),f(\sigma(v_{2})))<r\\}$.
Extend $f_{\sharp}^{\epsilon\mu}:C_{n}^{\epsilon}(X)\rightarrow
C_{n}^{\mu}(Y)$ linearly by
$f_{\sharp}^{\epsilon\mu}(\sum_{i}n_{i}\sigma_{i})=\sum_{i}n_{i}f_{\sharp}^{\epsilon\mu}(\sigma_{i})$
and we have $f_{\sharp}^{\epsilon\mu}\partial=\partial
f_{\sharp}^{\epsilon\mu}$, which implies that $f_{\sharp}^{\epsilon\mu}$ takes
cycles to cycles and takes boundaries to boundaries. Hence $f_{\sharp}$
induces a homomorphism $f_{*}^{\epsilon\mu}:H_{n}^{\epsilon}(X)\rightarrow
H_{n}^{\mu}(Y)$.

Let $\\{\epsilon_{n}\\}$ be a monotonically non-increasing sequence such that
$\lim_{n\to\infty}\epsilon_{n}=0$. For each $\epsilon_{k}$-continuous map
$\varphi:\Delta^{n}\rightarrow X$, composing with $f$, we get a corresponding
$\mu_{k}$-continuous map
$f_{\sharp}^{\epsilon_{k}\mu_{k}}(\varphi)=f\varphi:\Delta^{n}\rightarrow Y$,
where $\mu_{k}=\inf\\{r_{k}>0:{\rm if}\
d_{X}(\sigma(v_{1}),\sigma(v_{2}))<\epsilon_{k},d_{Y}(f(\sigma(v_{1})),f(\sigma(v_{2})))<r_{k}\\}$.
Since $f$ is uniformly continuous, $\mu_{n}\rightarrow 0$ as
$\epsilon_{n}\rightarrow 0$. Hence, after taking the inverse limit, we obtain
the induced homomorphism $f^{lc}_{*}:H_{n}^{lc}(X)\rightarrow H_{n}^{lc}(Y)$.
∎

Naturally, we can define the corresponding $lc$-cohomology group of $X$ in the
following ways.

Let $G$ be an abelian group. For the $\epsilon$-singular chain complexes

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{n+1}^{\epsilon}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{C_{n}^{\epsilon}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{C_{n-1}^{\epsilon}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

we apply the $\operatorname{Hom}(-,G)$ functor and obtain the corresponding
$\epsilon$-singular cochain complex

$\textstyle{\cdots}$$\textstyle{C^{n+1}_{\epsilon}(X;G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{n}_{\epsilon}(X;G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{C^{n-1}_{\epsilon}(X;G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $C_{\epsilon}^{n}(X;G)=\operatorname{Hom}(C_{n}^{\epsilon}(X);G)$ and
$\delta=\partial^{*}:C_{\epsilon}^{n}(X;G)\rightarrow C_{\epsilon}^{n+1}(X;G)$
sending $\varphi\in C_{\epsilon}^{n}(X;G)$ to $\varphi\partial\in
C_{\epsilon}^{n+1}(X;G)$. Then we can define the $\epsilon$-singular
cohomology group

$H_{\epsilon}^{n}(X;G)=\operatorname{Ker}\delta/\operatorname{Im}\delta.$

Let $i_{\mu\epsilon}^{*}:H_{\epsilon}^{n}(X;G)\rightarrow H_{\mu}^{n}(X;G)$ be
the homomorphism induced by the natural inclusion
$i_{\mu\epsilon}:C_{n}^{\mu}(X)\hookrightarrow C_{n}^{\epsilon}(X)$ and we
obtain a direct system
$((H_{\epsilon}^{n}(X))_{\epsilon\in\mathbb{R}^{+}},(i_{\mu\epsilon}^{*})_{\mu\leq\epsilon\in\mathbb{R}^{+}})$.
Then we can take the direct limit

$H_{lc}^{n}(X)=\lim\limits_{\overrightarrow{\epsilon\in\mathbb{R}^{+}}}((H_{\epsilon}^{n}(X))_{\epsilon\in\mathbb{R}^{+}},(i_{\mu\epsilon}^{*})_{\mu\leq\epsilon\in\mathbb{R}^{+}})=\bigoplus\limits_{\epsilon\in\mathbb{R}^{+}}H_{\epsilon}^{n}(X)/S$

where $S$ is generated by

$\\{q_{\mu}i_{\mu\epsilon}^{*}([\varphi_{\epsilon}])-q_{\epsilon}[\varphi_{\epsilon}]:[\varphi_{\epsilon}]\in
H_{\epsilon}^{n}(X),q_{\epsilon}\ {\rm is\ the\
embedding}:H_{\epsilon}^{n}(X)\hookrightarrow\bigoplus\limits_{\epsilon\in\mathbb{R}^{+}}H_{\epsilon}^{n}(X)\\}.$

Next, we will check whether the $lc$-homology satisfies the Eilenberg-Steenrod
axioms of homology theory. Obviously, the $lc$-homology theory satisfies the
dimension axiom.

######  Theorem 2.5 (Dimension Axiom).

$H_{n}^{lc}(\\{pt\\})=0$ for all $n>0$.

######  Theorem 2.6 (Homotopy Axiom).

Let $f,g:X\rightarrow Y$ be two uniformly continuous maps. If $f$ and $g$ are
uniformly homotopic, i.e., there is a family of uniformly continuous maps
$f_{t}:X\rightarrow Y,t\in[0,1]$ such that $f_{0}=f$, $g_{0}=g$ and the
associated map $F:X\times[0,1]\rightarrow Y$ given by $F(x,t)=f_{t}(x)$ is
uniformly continuous, then $f^{lc}_{*}=g^{lc}_{*}:H_{n}^{lc}(X)\rightarrow
H_{n}^{lc}(Y),\forall n\geq 0$.

###### Proof.

Let $\\{\epsilon_{n}\\}$ be a non-increasing sequence of $\mathbb{R}^{+}$ with
$\lim_{n\to\infty}\epsilon_{n}=0.$ and we can take

$\begin{split}H_{n}^{lc}(X)&=\varprojlim((H_{n}^{\epsilon_{m}}(X))_{m=1}^{\infty},(\varphi_{\epsilon_{k},\epsilon_{l}*})_{l\leq
k\in\mathbb{N}})\\\
&=\\{\overrightarrow{[a]}\in\prod\limits_{m=1}^{\infty}H_{n}^{\epsilon_{m}}(X):[a_{\epsilon_{l}}]=\varphi_{\epsilon_{k},\epsilon_{l}*}([a_{\epsilon_{k}}]){\rm\
for\ all}\ l\leq k\in\mathbb{N}\\}.\end{split}$

Then, we have

$\begin{split}f_{*}^{lc}(\overrightarrow{[a]})=\\{[\overrightarrow{f(a)}]:[f(a_{\epsilon_{l}})]=\varphi_{\mu_{k},\mu_{l}*}([f(a_{\epsilon_{k}})]){\rm\
for\ all}\ l\leq k\in\mathbb{N}\\}\ {\rm and}\\\
g_{*}^{lc}(\overrightarrow{[a]})=\\{[\overrightarrow{g(a)}]:[g(a_{\epsilon_{l}})]=\varphi_{\omega_{k},\omega_{l}*}([g(a_{\epsilon_{k}})]){\rm\
for\ all}\ l\leq k\in\mathbb{N}\\},\end{split}$

where an $\epsilon_{*}$-continuous map composed with $f$ and $g$ will be a
$\mu_{*}$-continuous map and an $\omega_{*}$-continuous map respectively.
Since a $\mu_{k}$-continuous map is naturally an $\omega_{k}$-continuous map
if $\mu_{k}\leq\omega_{k}$, without loss of generality, we can assume
$\mu_{*}=\omega_{*}$.

Next, given a homotopy $F:X\times I\rightarrow Y$ from $f$ to $g$ with
associated maps uniformly continuous and an $\epsilon_{k}$-singular simplex
$\sigma:\Delta^{n}\rightarrow X$, we can form the composition
$F\cdot(\sigma\times\mathrm{id}):\Delta^{n}\times I\rightarrow X\times
I\rightarrow Y$ with associated maps $\theta_{k}$-continuous. As in the case
of singular homology theory, we can construct a _prism operator_ which cuts
$\Delta^{n}\times I$ into a sum of $(n+1)$-simplices in $\Delta^{n}\times I$
by

$P(\sigma)=\sum_{i=0}^{n}(-1)^{i}F\cdot(\sigma\times\mathrm{id})|_{[v_{0},\cdots,v_{i},w_{i},\cdots,w_{n}]}$

where $\Delta^{n}\times{0}=[v_{0},\cdots,v_{n}]$ and
$\Delta^{n}\times{1}=[w_{0},\cdots,w_{n}]$. Then $P(\sigma)\in
C_{n+1}^{\theta_{k}}(Y)$. Pick $\tau_{k}=\max\\{\mu_{k},\theta_{k}\\}$ and we
have $P(\sigma),g_{\sharp}-f_{\sharp}-P\partial\in C_{n+1}^{\tau_{k}}(Y)$. As
we did in singular homology theory,

$\partial P(\sigma)=g_{\sharp}(\sigma)-f_{\sharp}(\sigma)-P\partial(\sigma).$

Hence, $f$ and $g$ induce the same homomorphism
$f_{*}=g_{*}:H_{n}^{\epsilon_{k}}(X)\rightarrow
H_{n}^{\tau_{k}}(Y),k=1,2,\cdots$. By taking the inverse limit,
$f^{lc}_{*}=g^{lc}_{*}:H_{n}^{lc}(X)\rightarrow H_{n}^{lc}(Y),\forall n\geq
0$. ∎

######  Corollary 2.7.

The maps $f_{*}^{lc}:H_{n}^{lc}(X)\rightarrow H_{n}^{lc}(Y)$ induced by a
uniform homotopy equivalence $f:X\rightarrow Y$ are isomorphisms for all $n$.
In particular, if X is compact and contractible, then
$\widetilde{H}_{n}^{lc}(X)=0$ for any $n$.

######  Remark 2.8.

Actually, we cannot simply omit the restriction ”uniformly homotopic”. Indeed,
let $p\in S^{1}$ and consider a space $S^{1}\setminus\\{p\\}$. If $\epsilon$
is sufficiently small, then we cannot distinguish between $C_{n}(S^{1})$ and
$C_{n}^{\epsilon}(S^{1}\setminus\\{p\\})$. Naturally, we have
$H_{n}(S^{1})\cong H_{n}^{\epsilon}(S^{1}\setminus\\{p\\})$ (we will give a
rigorous proof later in this section) since
$B(p,\frac{1}{2}\epsilon)\setminus\\{p\\}$ is contractible under the
”$\epsilon$-scale” and can be regarded as a point. By taking inverse limit, we
have $H_{1}^{lc}(S^{1}\setminus\\{p\\})=\mathbb{Z}$, even though
$S^{1}\setminus\\{p\\}$ is obviously contractible. The reason for this is that
the homotopy map connecting the retraction of $S^{1}\setminus\\{p\\}$ to a
point and the identity map is not uniformly continuous.

######  Theorem 2.9 (Excision Axiom).

Let $Z$ and $A$ be two subspaces of $X$ such that the closure of $Z$ is
contained in the interior of $A$, i.e., $\bar{Z}\subseteq\mathring{A}$ and
${\rm dist}(\bar{Z},\partial A)>0$. Then the inclusion map
$i:(X-Z,A-Z)\rightarrow(X,A)$ induces an isomorphism
$i^{lc}_{*}:H_{n}^{lc}(X-Z,A-Z)\rightarrow H_{n}^{lc}(X,A)$ for all $n$.
Equivalently, for subspaces $A,B\subseteq X$ with
$X=\mathring{A}\cup\mathring{B}$, the inclusion $(B,A\cap
B)\hookrightarrow(X,A)$ induces isomorphisms $H_{n}^{lc}(B,A\cap B)\rightarrow
H_{n}^{lc}(X,A)$ for all $n$.

###### Proof.

For the cover $\\{A,B\\}$ with $X=A\cup B$ and $Z=X-B$, pick $0<\epsilon<{\rm
dist}(\bar{Z},\partial A)$ and denote $C_{n}^{\epsilon\\{A,B\\}}(X)$ by
$C_{n}^{\epsilon}(A+B)$, the formal sums of chains in $A$ and chains in $B$
(here $\epsilon$ is globally fixed for this space). As we do for singular
homology theory, by barycentric subdivision of chains, we can construct a
chain map $S:C_{n}^{\epsilon}(X)\rightarrow C_{n}^{\epsilon}(X)$ and a chain
homotopy $T:C_{n}^{\epsilon}(X)\rightarrow C_{n+1}^{\epsilon}(X)$, s.t.

$\partial T+T\partial={\rm id}-S.$

In addition, for the cover ${A,B}$ of $X$ and any $\epsilon$-simplex
$\sigma:\Delta^{n}\rightarrow X$, there exists $m\in\mathbb{N}$, s.t.
$S^{m}\sigma\in C_{n}^{\epsilon\\{A,B\\}}(X)$ since $\operatorname{dist}(A\cap
B)>\epsilon$. It will follow that $S$, $T$ restrict to maps on
$C_{n}^{\epsilon\\{A,B\\}}(X)$. Let $D_{m}=\sum_{0\leq i<m}TS^{i}$ and it
satisfies

$\partial D_{m}+D_{m}\partial={\rm id}-S^{m}.$

Hence the iterate $S^{m}$ is chain homotopic to ${\rm id}$.

Then the inclusion $C_{n}^{\epsilon}(A+B)/C_{n}^{\epsilon}(A)\hookrightarrow
C_{n}^{\epsilon}(X)/C_{n}^{\epsilon}(A)$ induces an isomorphism on homology.
In addition, we have a natural isomorphism
$C_{n}^{\epsilon}(B)/C_{n}^{\epsilon}(A\cap B)\rightarrow
C_{n}^{\epsilon}(A+B)/C_{n}^{\epsilon}(A)$ since both sides are free abelian
groups generated by the $\epsilon$-singular $n$-simplices in $B$ that are not
contained in $A$. Composing these two isomorphisms on homology, we get
$H_{n}^{\epsilon}(B,A\cap B)\cong H_{n}^{\epsilon}(X,A)$ for all $n$. By
taking the inverse limit, $H_{n}^{lc}(B,A\cap B)\cong H_{n}^{lc}(X,A)$. ∎

As a corollary of excision theorem, we naturally have:

######  Theorem 2.10 (Refined Additivity Axiom).

Let $X=\bigsqcup\limits_{\lambda\in\Lambda}X_{\lambda}$. If there exists
$\epsilon>0$ such that for any $k,\lambda\in\Lambda$,
$\operatorname{dist}(X_{k},X_{\lambda})>\epsilon$, then
$H_{n}^{lc}(X)\cong\bigoplus\limits_{\lambda\in\Lambda}H_{n}^{lc}(X_{\lambda})$
for all $n$.

######  Example 2.11.

$X=\\{\displaystyle\frac{1}{2^{k}};k=0,1,2,\cdots\\}\cup\\{0\\}$.

Take $\epsilon_{k}=1/2^{k+1}$. For a fixed $\epsilon_{k}$,
$H_{0}^{\epsilon_{k}}(X)$ is the free abelian group generated by elements of
$X$ that are $\geq 1/2^{k+1}$ and $\\{0\\}$, i.e., elements of
$H_{0}^{\epsilon_{k}}(X)$ are given by
$(a_{0},a_{1},\cdots,a_{k+2})\in\mathbb{Z}^{k+3}$. Hence the inverse limit of
$(H_{0}^{\epsilon_{k}}(X))_{n\in\mathbb{N}}$ is
$H_{0}^{lc}(X)=\mathbb{Z}^{|X|}$, the direct product of $\mathbb{Z}$, where
$|X|$ denotes the cardinality of $X$.

Next, for $\epsilon_{k}$, $X$ can be written as $X=X_{1}^{(k)}\bigsqcup
X_{2}^{(k)}$ with

$X_{1}^{(k)}=\\{0,1,\frac{1}{2},\cdots,\frac{1}{2^{k}}\\},X_{2}^{(k)}=\\{\frac{1}{2^{k+1}},\frac{1}{2^{k+2}},\cdots\\}$

and for $n\geq 1$, $H_{n}^{\epsilon_{k}}(X)\cong
H_{n}^{\epsilon_{k}}(X_{1}^{(k)})\oplus H_{n}^{\epsilon_{k}}(X_{2}^{(k)})$. By
dimension axiom,

$H_{n}^{\epsilon_{k}}(X_{n}^{(k)})\cong H_{n}^{\epsilon_{k}}(\\{0\\})\oplus
H_{n}^{\epsilon_{k}}(\\{1\\})\oplus\cdots\oplus
H_{n}^{\epsilon_{k}}(\\{\frac{1}{2^{k}}\\})=0.$

Since the diameter of $X_{2}^{(k)}$ is less than $\epsilon_{k}$, by
proposition 2.3, $H_{n}^{\epsilon_{k}}(X_{2}^{(k)})=0$. Therefore,
$H_{n}^{\epsilon_{k}}(X)=0$ holds for all $\epsilon_{k}$ and by taking the
inverse limit, $H_{n}^{lc}(X)=0$ for $n\geq 1$.

Consider the commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{n}^{\epsilon}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\epsilon}}$$\scriptstyle{\partial}$$\textstyle{C_{n}^{\epsilon}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\epsilon}}$$\scriptstyle{\partial}$$\textstyle{C_{n}^{\epsilon}(X,A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{n-1}^{\epsilon}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\epsilon}}$$\textstyle{C_{n-1}^{\epsilon}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\epsilon}}$$\textstyle{C_{n-1}^{\epsilon}(X,A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $i_{\epsilon}$ is inclusion and $j_{\epsilon}$ is the quotient map. By
the Snake Lemma, we obtain an induced long exact sequence in
$\epsilon$-homology:

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{n}^{\epsilon}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\epsilon*}}$$\textstyle{H_{n}^{\epsilon}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\epsilon*}}$$\textstyle{H_{n}^{\epsilon}(X,A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{\epsilon}}$$\textstyle{H_{n-1}^{\epsilon}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

Then we have the following proposition.

######  Proposition 2.12.

Let $A$ be a subspace of $X$. Then we have $H_{n}^{lc}(A)\cong
H_{n}^{lc}(\bar{A})$ for all $n$.

###### Proof.

First we claim that for any $\epsilon>0$, $H_{n}^{\epsilon}(A)\cong
H_{n}^{\epsilon}(\bar{A})$. Indeed, by the long exact sequence in
$\epsilon$-homology

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{n+1}^{\epsilon}(\bar{A},A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{\epsilon}}$$\textstyle{H_{n}^{\epsilon}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\epsilon*}}$$\textstyle{H_{n}^{\epsilon}(\bar{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\epsilon*}}$$\textstyle{H_{n}^{\epsilon}(\bar{A},A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

it suffices to show that $H_{n}^{\epsilon}(\bar{A},A)=0$ for all $n$.

Let $[\alpha]\in H_{n}^{\epsilon}(\bar{A},A)$ where $\alpha\in
C_{n}^{\epsilon}(\bar{A})$ and $\partial\alpha\in C_{n-1}^{\epsilon}(A)$.
Write $\alpha=\sum_{i=1}^{k}n_{i}\alpha_{i}$ where
$\alpha_{i}:\Delta^{n}\rightarrow\bar{A}$ is $\epsilon$-continuous. For each
$\alpha_{i}$, $\exists\delta_{i}>0$, s.t.
$\omega_{i}=\sup\\{d(\alpha_{i}(x_{1}),\alpha_{i}(x_{2})):d(x_{1},x_{2})<\delta_{i}\\}<\epsilon$.
By definition,

$\partial\alpha=\sum_{i=1}^{k}\sum_{j=0}^{n}n_{i}\alpha_{i}|[v_{0},\cdots,\hat{v_{j}},\cdots,v_{n}]$

which is a formal sum of $\epsilon$-continuous maps. Hence the restrictions of
$\alpha_{i}:\Delta^{n}\rightarrow\bar{A}$ to the faces of $\Delta^{n}$ consist
of $\epsilon$-continuous maps whose images are contained in either $\bar{A}$
or $A$, and in the formal sum of $\partial\alpha$, all of those
$\epsilon$-continuous maps whose images intersect $\partial A$ are cancelled.

Then, we construct $\beta_{i}:\Delta^{n}\times I\rightarrow\bar{A}$ by

$\beta_{i}(\sum_{\begin{subarray}{c}l=0\\\ l\neq
j\end{subarray}}^{n}\lambda_{l}v_{l},t)\equiv\alpha_{i}(\sum_{\begin{subarray}{c}l=0\\\
l\neq j\end{subarray}}^{n}\lambda_{l}v_{l})$

where $\sum_{\begin{subarray}{c}l=0\\\ l\neq
j\end{subarray}}^{n}\lambda_{l}=1$ for $j=0,1,\cdots,n$ and

$\beta_{i}(\sum_{l=0}^{n}\lambda_{l}v_{l},0)=\alpha_{i}(\sum_{l=0}^{n}\lambda_{l}v_{l})$

where $\sum_{l=0}^{n}\lambda_{l}=1$.

Moreover, for other points $(x,t)\in\Delta^{n}\times I$, if $\alpha_{i}(x)\in
A$, then we let $\beta_{i}(x,t)=\alpha_{i}(x)$. On the other hand, if
$\alpha_{i}(x)\notin A$, we do a perturbation for $\alpha_{i}(x)$ under the
scale of $(\epsilon-\omega_{i})/3$ and we obtain $\widetilde{\alpha_{i}}(x)$.
Then we let $\beta_{i}(x,t)=\widetilde{\alpha_{i}}(x)$. Clearly, $\beta_{i}$
is still $\epsilon$-continuous and since $\Delta^{n}\times I$ is a simplicial
complex, $\beta_{i}$ can be seen as a formal sum of $\epsilon$-singular
$n+1$-simplices. Let $\beta=\sum_{i=1}^{k}n_{i}\beta_{i}\in
C_{n+1}^{\epsilon}(\bar{A})$. Then $\partial\beta-\alpha\in
C_{n}^{\epsilon}(A)$. So $[\alpha]=0$ in $H_{n}^{\epsilon}(\bar{A},A)$ and
$H_{n}^{\epsilon}(A)\cong H_{n}^{\epsilon}(\bar{A})$ for $\forall\epsilon>0$.

Finally, by taking the inverse limit, $H_{n}^{lc}(A)\cong H_{n}^{lc}(\bar{A})$
for all $n$. ∎

But unfortunately, the inverse limit functor preserving left exactness is
still not exact in our case.

######  Example 2.13.

Let $A$ be the set of positive integers and let $X$ be the set of all
$n+\frac{i}{n}$ where $i,n$ are integers with $0\leq i<n$, i.e.,

$X=\\{1,\ 2,2\tfrac{1}{2},\ 3,3\tfrac{1}{3},3\tfrac{2}{3},\
4,4\tfrac{1}{4},4\tfrac{1}{2},4\tfrac{3}{4},\ 5,5\tfrac{1}{5},\cdots\\}$

Consider the short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}i_{\epsilon*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{0}^{\epsilon}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Im}i_{\epsilon*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $H_{0}^{\epsilon}(A)$ is the same for all $0<\epsilon<1$, which is the
direct sum of countably infinitely many copies of $\mathbb{Z}$ and
$H_{0}^{\epsilon}(X)$ is a finitely generated abelian group. Without loss of
generality, take $\epsilon_{n}=1/n$. Then elements of
$i_{\epsilon_{n}*}(H_{0}^{\epsilon_{n}}(A))\subset H_{0}^{\epsilon_{n}}(X)$
are given by a sequence of integers $(a_{0},a_{1},\cdots,a_{n})$, i.e.,
$i_{\epsilon_{n}*}(H_{0}^{\epsilon_{n}}(A))$ is generated by only $n$
generators.

Let $([a]_{\epsilon_{n}})\in\varprojlim\operatorname{Im}(i_{\epsilon_{n}})$,
which can be written as

$((a_{1},a_{2}),(a_{1},a_{2},a_{3}),\cdots,(a_{1},a_{2},\cdots,a_{m}),\cdots)$

and can have countably infinite length (the number of nonzero coefficients).
But the element of $\varprojlim H_{0}^{\epsilon_{n}}(A)$ can be characterized
by the element of the direct sum of countably infinitely many copies of
$\mathbb{Z}$, so its length can only be finite. Therefore, there is no element
of $H_{0}^{lc}(A)$ that can go to the elements with infinitely many nonzero
coordinates in $\varprojlim\operatorname{Im}i_{\epsilon_{n}*}$, i.e.,
$H_{0}^{lc}(A)\rightarrow\varprojlim\operatorname{Im}i_{\epsilon_{n}*}$ is not
surjective and we fail to show the exactness at $H_{0}^{\epsilon}(X)$.

Furthermore, we can give a counterexample showing that the $lc$-homology
doesn’t satisfy the exactness axiom.

######  Example 2.14.

Let $B$ be the set of all $n+\frac{i}{n}$ where $i,n$ are integers with $0\leq
i<n$, i.e.,

$B=\\{1,\ 2,2\tfrac{1}{2},\ 3,3\tfrac{1}{3},3\tfrac{2}{3},\
4,4\tfrac{1}{4},4\tfrac{1}{2},4\tfrac{3}{4},\ 5,5\tfrac{1}{5},\cdots\\}.$

Take $\epsilon_{n}$ to be such that $1/n<\epsilon_{n}\leq 1/(n-1)$. Clearly,
$H_{0}^{\epsilon_{n}}(B)$ is the free abelian group generated by the elements
of $B$ which are $\leq n$. For instance, for $n=3$,
$H_{0}^{\epsilon_{3}}(B)=\mathbb{Z}^{4}$, which is freely generated by
$\\{[1],[2],[2.5],[3]=[3\frac{1}{3}]=[3\frac{2}{3}]\\}$. Thus, elements of
$H_{0}^{\epsilon_{3}}(B)$ are given by the set of tuples
$\\{(a_{1},a_{2},a_{5/2},a_{3})\in\mathbb{Z}^{4}\\}$. Add a “dummy variable”
$a_{0}=-(a_{1}+a_{2}+a_{5/2}+a_{3})$ and we obtain a sequence of five integers
$(a_{0},a_{1},a_{2},a_{5/2},a_{3})$ whose sum is $0$.

Take $B_{+}=B\coprod\\{0\\}$. Then the inverse limit of
$(H_{0}^{\epsilon_{n}})_{n\in\mathbb{N}}$ is
$H_{0}^{lc}(B)=\mathbb{Z}^{|B_{+}|}$, the set of all infinite sequences of
integers $(a_{0},a_{1},a_{2},a_{5/2},\dots)$ indexed by the elements of
$B_{+}$.

Let $A=B\times\\{0,1\\}=B\coprod B$, the disjoint union of two copies of $B$
with distance $>\epsilon_{n}$. Then $H_{0}^{lc}(A)=H_{0}^{lc}(B)\oplus
H_{0}^{lc}(B)$ whose elements are given by pairs of infinite sequences of
integers

$\left((a_{0},a_{1},a_{2},a_{5/2},\dots),(b_{0},b_{1},b_{2},b_{5/2},\dots)\right).$

Take $X=A\cup\\{1,2,\cdots\\}\times I$. For any $\epsilon<1$,
$H_{1}^{\epsilon}(X,A)$ is the free abelian group generated by the set of
vertical line segments in $X$, which is the direct sum of countable copies of
$\mathbb{Z}$. Therefore, $H_{1}^{\epsilon}(X,A)\cong H_{1}^{lc}(X,A)$ and
elements in $H_{1}^{lc}(X,A)$ can only have finite nonzero coefficients.

Note that the homomorphism $H_{0}^{lc}(A)\to H_{0}^{lc}(X)$ adds together the
coefficients $a_{n},b_{n}$ for all integers $n$, so the kernel consists of all
pairs of infinite sequence $(a_{\ast},b_{\ast})$ such that $a_{k}+b_{k}=0$ for
all integers $k$ and $a_{x}=b_{x}=0$ for all $x\in B$ that are not integers.
So the kernel of $H_{0}^{lc}(A)\rightarrow H_{0}^{lc}(X)$ is a product of
infinitely many copies of $\mathbb{Z}$, i.e., it can have countably infinite
nonzero coefficients that cannot be the image of $H_{1}^{lc}(X,A)$.

Therefore, the sequence

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}^{lc}(X,A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{0}^{lc}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{0}^{lc}(X)}$

is not exact at $H_{0}^{lc}(A)$.

######  Remark 2.15.

However, $lc$-cohomology satisfies the exactness axiom. Indeed, the
$lc$-cohomology sequence

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{n}_{lc}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{n}_{lc}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{n+1}_{lc}(X,A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{n+1}_{lc}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

with any coefficients is exact since it is the direct limit of the exact
sequences

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{n}_{\epsilon}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{n}_{\epsilon}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{n+1}_{\epsilon}(X,A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{n+1}_{\epsilon}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

and the direct limit serves as an exact functor.

In particular, for our counterexample, the $lc$-cohomologies are all countably
generated since they are countable direct limits of countably generated
groups.

Therefore, the Universal Coefficient Theorem does not hold for $lc$-cohomology
theory in general.

To end this section, we will show the equivalence of $lc$-homology and
singular homology in one special case.

######  Theorem 2.16.

Let $(X,d)$ be a compact $n$-dimensional manifold endowed with a Riemannian
structure. Then the homomorphisms $H_{n}(X)\rightarrow H_{n}^{lc}(X)$ are
isomorphisms for all $n$.

###### Proof.

Since $X$ is a Riemannian manifold, by Theorem 5.1 in [2], every point has a
geodesically convex neighborhood. So for any $x\in X$, there exists
$\epsilon_{x}>0$ such that $B(x,\epsilon_{x})\cong\mathbb{R}^{n}$ and
$X=\bigcup_{x\in X}B(x,\epsilon_{x})$. By compactnesss, there exists $N>0$
such that $X$ can be expressed as
$\bigcup_{i=1}^{N}B(x_{i},\epsilon_{x_{i}})$. Let $\delta>0$ be a Lebesgue
number for this cover, i.e., for $\forall x\in X$, $B(x,\delta)\subset
B(x_{k},\epsilon_{x_{k}})$ for some $1\leq k\leq N$. Hence each point of $X$
has a $\delta$-ball homeomorphic to $\mathbb{R}^{n}$.

Now for $\mu\leq\epsilon<\delta$, consider the short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{n}^{\mu}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{n}^{\epsilon}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{n}^{\epsilon}(X)/C_{n}^{\mu}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

and by Snake Lemma, we have the following long exact sequence

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{n+1}(X^{\epsilon},X^{\mu})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{n}^{\mu}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{n}^{\epsilon}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{n}(X^{\epsilon},X^{\mu})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

where $H_{n}(X^{\epsilon},X^{\mu})$ denotes the homology group of

$\partial:C_{n}^{\epsilon}(X)/C_{n}^{\mu}(X)\rightarrow
C_{n-1}^{\epsilon}(X)/C_{n-1}^{\mu}(X).$

Then we claim that $H_{n}(X^{\epsilon},X^{\mu})=0$ for all $n$.

Let $[\alpha]\in H_{n}(X^{\epsilon},X^{\mu})$ where $\alpha\in
C_{n}^{\epsilon}(X)$ and $\partial\alpha\in C_{n-1}^{\mu}(X)$. Write
$\alpha=\sum_{i=1}^{k}n_{i}\sigma_{i}$ with $\sigma_{i}:\Delta^{n}\rightarrow
X$ an $\epsilon$-continuous map. By definition,

$\partial\alpha=\sum_{i=1}^{k}\sum_{j=0}^{n}n_{i}\sigma_{i}|[v_{0},\cdots,\hat{v_{j}},\cdots,v_{n}]$

which is a formal sum of $\mu$-continuous maps. Hence the restrictions of
$\sigma_{i}:\Delta^{n}\rightarrow X$ to the faces of $\Delta^{n}$ consist of
$\mu$-continuous maps and $\epsilon$-continuous maps and in the formal sum of
$\partial\alpha$, all of those $\epsilon$-continuous maps are cancelled.

For $\sigma_{i}$, for any $x\in\operatorname{Im}(\sigma_{i})\subset X$,
$\bigcup_{x\in\operatorname{Im}(\sigma_{i})}B(x,\delta)$ forms an open cover
of its image in $X$. Define $S:C_{n}^{\epsilon}(X)\rightarrow
C_{n}^{\epsilon}(X)$ by sending each $\epsilon$-singular $n$-simplex
$\sigma:\Delta^{n}\rightarrow X$ to $\sigma_{\sharp}S\Delta^{n}$ where
$S\Delta^{n}$ is the signed sum of the $n$-simplices in the barycentric
subdivision of $\Delta^{n}$ and $S\sigma$ is the corresponding signed sum of
the restrictions of $\sigma$ to the $n$-simplices of the barycentric
subdivision of $\Delta^{n}$. Since $\mu\leq\epsilon<\delta$, we can guarantee
that there exists $m>0$, s.t.
$S^{m}\sigma_{i}=\sum_{l=1}^{N}k_{l}\sigma_{i_{l}}$ with $k_{l}=\pm 1$,
$\operatorname{Im}(\sigma_{i_{l}})\subset B(x_{l},\delta)$ for some
$x_{l}\in\operatorname{Im}(\sigma_{i})$, where
$\sigma_{i_{l}}:\Delta_{l}^{n}\rightarrow X$ and
$\bigcup_{l=1}^{N}\Delta_{l}^{n}=\Delta^{n}$.

Next, for each $\sigma_{i_{l}}:[v_{0},v_{1},\cdots,v_{n}]\rightarrow X$, let
$\varphi_{l}:B(x_{l},\delta)\rightarrow\mathbb{R}^{n}$ be the coordinate map
and let
$f_{il}=\varphi_{l}\circ\sigma_{il}:\Delta_{l}^{n}\rightarrow\mathbb{R}^{n}$.
Then we do the following construction,

$\displaystyle F_{il}:\Delta_{l}^{n}\times I$
$\displaystyle\rightarrow\mathbb{R}^{n}$
$\displaystyle(\sum_{j=0}^{n}\lambda_{j}v_{j},t)$
$\displaystyle\mapsto(1-t)f_{l}(\sum_{j=0}^{n}\lambda_{j}v_{j})+t\sum_{j=0}^{n}\lambda_{j}f_{l}(v_{j}).$

$(v_{0},1)$$(v_{1},1)$$(v_{2},1)$$(v_{0},0)$$(v_{1},0)$$(v_{2},0)$$(\sum_{j=0}^{2}\lambda_{j}v_{j},0)$$\Delta_{l}^{2}\times
I$$X$$B(x_{l},\delta)$$\sigma_{il}$$\varphi_{l}$$f_{l}=\varphi_{l}\circ\sigma_{il}$

Let $\beta_{il}=\varphi_{l}^{-1}\circ F_{il}:\Delta_{l}^{n}\times I\rightarrow
X$ that can be seen as a homotopy equivalence between an $\epsilon$-continuous
map and a continuous map. Since $\Delta_{l}^{n}\times I$ is a simplicial
complex, $\beta_{il}$ can be regarded as a formal sum of $\epsilon$-singular
$n+1$-simplices.

Finally, given $[\alpha]\in H_{n}(X^{\epsilon},X^{\mu})$, we can take
$\beta=\sum_{i=1}^{k}n_{i}\sum_{l=1}^{N}\beta_{il}$ to be such that
$\partial\beta$ is a formal sum of $\alpha$ as well as some $\mu$-continuous
maps, i.e., $\alpha-\partial\beta\in C_{n}^{\mu}(X)$. Therefore,
$H_{n}(X^{\epsilon},X^{\mu})=0$ and $H_{n}^{\mu}(X)\cong H_{n}^{\epsilon}(X)$
for $0<\mu\leq\epsilon<\delta$. Actually, from the argument above, we can even
see that $H_{n}^{\epsilon}(X)\cong H_{n}(X)$ for $0<\epsilon<\delta$.
Therefore, by universal property of the inverse limit, $H_{n}^{lc}(X)\cong
H_{n}(X)$ for all $n$. ∎

######  Example 2.17.

Let $p$ be a point in $S^{n}$, then
$H_{n}^{lc}(S^{n}\setminus\\{p\\})\cong\mathbb{Z}$ and
$H_{i}^{lc}(S^{n}\setminus\\{p\\})=0$ for $i\neq n$. Indeed,
$\overline{S^{n}\setminus\\{p\\}}=S^{n}$ and $S^{n}$ is a compact Riemannian
manifold, so

$H_{i}^{lc}(S^{n}\setminus\\{p\\})\cong H_{i}^{lc}(S^{n})\cong
H_{i}^{n}(S^{n}).$

##  3\. Topological entropy and lc-homology

In this section, we give a generalization of the entropy conjecture and
consequently establish a connection between topological entropy and
$lc$-homology.

Let $(X,d)$ be a compact metric space with
$\operatorname{rank}(H_{i}^{lc}(X))<\infty$ for each $i\geq 0$ and
$f:X\rightarrow X$ be a continuous map. For the induced homomorphism
$f^{lc}_{*i}:H^{lc}_{i}(X)\rightarrow H^{lc}_{i}(X)$, denote the spectral
radius of $f^{lc}_{*i}$ by $\rho(f^{lc}_{*i})$, i.e.,

$\rho(f^{lc}_{*i})=\lim\limits_{n\rightarrow\infty}\|(f^{lc}_{*i})^{n}\|^{\frac{1}{n}}$

where $\|(f^{lc}_{*i})^{n}\|$ is the operator norm of the linear operator
$(f^{lc}_{*i})^{n}$.

######  Theorem 3.1.

$h(f)\geq\log\rho(f^{lc}_{*1})$.

###### Proof.

Let
$H_{1}^{lc}(X)=\varprojlim((H_{1}^{\epsilon_{n}}(X))_{n=0}^{\infty},(\varphi_{\epsilon_{k}\epsilon_{l}*})_{l\leq
k})$ and fix $\epsilon_{0}>0$. Then any loop of diameter $<\epsilon_{0}$ can
be seen as contractible under $\epsilon_{0}$-scale. Choose $\mu<\epsilon_{0}$
such that whenever $d(x,y)<\mu$, $d(f(x),f(y))<\epsilon_{0}$. Then take
$\delta<\frac{1}{3}\mu$ and clearly there exists an $\epsilon_{0}$-continuous
path from $x$ to $y$ whenever $d(x,y)<3\delta$. Let $P$ be a $\delta$-net in
$X$ and define $C_{1}^{\epsilon_{0}}(P,3\delta;\mathbb{R})$ to be the free
abelian group generated by $\\{[x,y]:x,y\in P,d(x,y)<3\delta\\}$, where
$[x,y]$ denotes some fixed $\epsilon_{0}$-path from $x$ to $y$ of diameter
$<\mu$, with real coefficients. Similarly, let
$C_{0}^{\epsilon_{0}}(P;\mathbb{R})$ be the free abelian group generated by
$P$ with real coefficients and we can define
$\partial_{\epsilon_{0}}:C_{1}^{\epsilon_{0}}(P,3\delta;\mathbb{R})\rightarrow
C_{0}^{\epsilon_{0}}(P;\mathbb{R})$ by $\partial_{\epsilon_{0}}[x,y]=y-x$.

For any $\epsilon_{1}<3\delta$, consider the $\epsilon_{0}$-coordinate of
$\varprojlim((H_{1}^{\epsilon_{n}}(X))_{n=0}^{\infty},(\varphi_{\epsilon_{k}\epsilon_{l}*})_{l\leq
k})$ that is contained in
$\varphi_{\epsilon_{1}\epsilon_{0}*}(H_{1}^{\epsilon_{1}}(X))$. Let
$H_{1}^{\epsilon_{1}\epsilon_{0}}(P,3\delta;\mathbb{R})$ be the subgroup of
$H_{1}^{\epsilon_{0}}(P,3\delta;\mathbb{R})$ whose representatives are in
$C_{1}^{\epsilon_{1}}(P,3\delta;\mathbb{R})$ and it is naturally isomorphic to
$\varphi_{\epsilon_{1}\epsilon_{0}*}(H_{1}^{\epsilon_{1}}(X;\mathbb{R}))$.
Indeed, every $\epsilon_{0}$-homology class in
$\varphi_{\epsilon_{1}\epsilon_{0}*}(H_{1}^{\epsilon_{1}}(X;\mathbb{R}))$ has
a representative in $C_{1}^{\epsilon_{1}}(P,3\delta;\mathbb{R})$ (defined
similarly as $C_{1}^{\epsilon_{0}}(P,3\delta;\mathbb{R})$) obtained by
breaking down $\epsilon_{1}$-paths in $C_{1}^{\epsilon_{1}}(X;\mathbb{R})$
into combinations of short $\epsilon_{1}$-paths joining points of $P$, since
$\epsilon_{1}<3\delta$. Then we can define a norm $\|\ \|$ on
$C_{1}^{\epsilon_{1}}(P,3\delta;\mathbb{R})$ by

$\|\sum_{i}a_{i}\sigma_{i}\|=\sum_{i}|a_{i}|$

and a norm $\|\ \|^{\prime}$ on
$H_{1}^{\epsilon_{1}\epsilon_{0}}(P,3\delta;\mathbb{R})$ by

$\|[u]\|^{\prime}=\inf_{[\sigma]=[u]}\|\sigma\|$

(since $H_{1}^{\epsilon_{1}\epsilon_{0}}(P,3\delta;\mathbb{R})$ is finite
dimensional, all norms on it are equivalent).

Take $\epsilon_{1}$ to be sufficiently small. For a non-zero class $[u]\in
H_{1}^{\epsilon_{1}\epsilon_{0}}(P,3\delta;\mathbb{R})$, take
$\sigma=\sum_{i=1}^{n}a_{i}\sigma_{i}\in
C_{1}^{\epsilon_{1}}(P,3\delta;\mathbb{R})$ to be its cycle with
$\|\sigma\|\leq 2\|[u]\|^{\prime}$.

Let $Q_{k}$ be a minimal $(k,\delta)$-spanning set for $f$. For each $i$, we
do the following construction with $\sigma_{i}:I\rightarrow X$. Define

$F_{k}=\operatorname{id}\times f\times f^{2}\times\cdots\times
f^{k-1}:X\mapsto X\times f(X)\times\cdots\times f^{k-1}(X)$

and then the points of $F_{k}(Q_{k})$ have $\delta$-neighbourhoods in $X^{k}$
(endowed with $d_{\infty}$ metric that takes the largest of the distances in
each of the $k$ factors) that cover $F_{k}(X)$. For the sequence of these
neighbourhoods through which $F_{k}(\sigma_{i}(I))$ passes, we pick a sequence

$x_{1},x_{2},\cdots,x_{b}\in Q_{k}$

of some length b such that each $F_{k}(x_{p})$ is in a $\delta$-neighbourhood
of $F_{k}(\sigma_{i}(I))\subset X^{k}$,
$d(f^{j}(x_{p-1}),f^{j}(x_{p}))<3\delta$ for $1<p\leq b$, and
$d(f^{j}(\sigma_{i}(0)),f^{j}(x_{1})),d(f^{j}(x_{b}),f^{j}(\sigma_{i}(1)))<\delta$
for $0\leq j<k$. Then $\sigma_{i}$ is homologous to

$\tau_{i}=[\sigma_{i}(0),x_{1}]\ast[x_{1},x_{2}]\ast[x_{2},x_{3}]\cdots\ast[x_{b-1},x_{b}]\ast[x_{b},\sigma_{i}(1)]$

where $\ast$ denotes the composition of paths.

It is possible that the length $b>|Q_{k}|$(the cardinality of $Q_{k}$). If
$x_{j}=x_{l}$ for some $j\neq l$, then $\tau_{i}$ contains a loop from $x_{j}$
to $x_{l}=x_{j}$ and this loop is contained in an $\epsilon_{1}+\delta$
neighbourhood of $\sigma_{i}I$ which itself has diameter $<\epsilon_{1}$. So
the diameter of the loop is $<\epsilon_{0}$. Hence we could suppress
$x_{j+1},\cdots,x_{l}$ and assume that $b\leq|Q_{k}|$.

Since $\sigma_{i}$ is $\epsilon_{0}$-continuous, without loss of generality,
we can assume that $f^{m}\sigma_{i}$ is $\epsilon_{0}^{(m)}$-continuous with
$\epsilon_{0}^{(k-1)}\geq\epsilon_{0}^{(k-2)}\geq\cdots\geq\epsilon_{0}^{(0)}=\epsilon_{0}$.
We claim that under $\epsilon_{0}^{(k-1)}$-scale, $f^{k-1}\sigma_{i}$ is
homologous ($\sim$) to $f^{k-1}\tau_{i}\sim v_{i}$, where

$v_{i}=[f^{k-1}(\sigma_{i}(0)),f^{k-1}(x_{1})]\ast\cdots\ast[f^{k-1}(x_{b-1}),f^{k-1}(x_{b})]\ast[f^{k-1}(x_{b}),f^{k-1}(\sigma_{i}(1))]$

Indeed, $f[x_{j-1},x_{j}]\sim[f(x_{j-1}),f(x_{j})]$ under
$\epsilon_{0}^{(1)}$-scale since $f$ does not extend the path
$[x_{j-1},x_{j}]$ so much. Similarly,
$f[f(x_{j-1}),f(x_{j})]\sim[f^{2}(x_{j-1}),f^{2}(x_{j})]$ under
$\epsilon_{0}^{(2)}$-scale etc. and so
$f^{k-1}[x_{j-1},x_{j}]\sim[f^{k-1}(x_{j-1}),f^{k-1}(x_{j})]$ under
$\epsilon_{0}^{(k-1)}$-scale and

$f^{k-1}\sigma\sim\sum a_{i}f^{k-1}\sigma_{i}\sim\sum
a_{i}f^{k-1}\tau_{i}\sim\sum a_{i}v_{i}=v.$

Let $p:X\rightarrow P$ be a map such that $d(x,px)<\delta$ for $\forall x\in
X$. Then we replace $\alpha$ by the $\epsilon_{0}^{(k-1)}$-singularly
homologous cycle $\alpha$ obtained by replacing each
$\epsilon_{0}^{(k-1)}$-singular simplex $[f^{k-1}(x),f^{k-1}(y)]$ in $v$ by
$[pf^{k-1}(x),pf^{k-1}(y)]$ which is a generator of
$C_{1}^{\epsilon_{0}^{(k-1)}}(P,3\delta;\mathbb{R})$ since
$d(pf^{k-1}(x),pf^{k-1}(y))<3\delta$.

Since $f_{\epsilon_{0}*1}^{k-1}$ is the induced homomorphism on the
$\epsilon_{0}$-coordinate of
$\varprojlim((H_{1}^{\epsilon_{n}}(X))_{n=0}^{\infty})$ to
$H_{1}^{\epsilon_{0}^{(k-1)}}(X;\mathbb{R})$, by the choice of $\|\
\|^{\prime}$, we have

$\|f_{\epsilon_{0}*1}^{k-1}([u])\|^{\prime}\leq\|\alpha\|\leq(1+|Q_{k}|)\sum|a_{i}|$

whereas on $H_{1}^{\epsilon_{1}\epsilon_{0}}(P,3\delta;\mathbb{R})$,

$\|[u]\|^{\prime}\geq\frac{1}{2}\|\sigma\|=\frac{1}{2}\sum|a_{i}|.$

Therefore

$\frac{\|f_{\epsilon_{0}*1}^{k-1}([u])\|^{\prime}}{\|[u]\|^{\prime}}<2(1+|Q_{k}|)$

and this holds for any non-zero $[u]\in H_{1}^{\epsilon_{0}}(X;\mathbb{R})$
and all $k$. Hence $\|f_{\epsilon_{0}*1}^{k-1}\|^{\prime}<2(1+|Q_{k}|)$. But
$\rho(f_{\epsilon_{0}*1})=\lim\limits_{k\to\infty}\|f_{\epsilon_{0}*1}^{k}\|^{\prime\frac{1}{k}}$,
so for the $\epsilon_{0}$-coordinate of
$\varprojlim((H_{1}^{\epsilon_{n}}(X))_{n=0}^{\infty})$,

$\begin{split}\log\rho(f_{\epsilon_{0}*1})&=\lim\frac{1}{k}\log\|f_{\epsilon_{0}*1}^{k}\|^{\prime}\leq\limsup\frac{1}{k}\log
2(1+|Q_{k+1}|)\\\ &=\limsup\frac{1}{k}\log|Q_{k}|=h(f,\delta)\leq
h(f).\end{split}$

Similarly, this will hold for each coordinate of the given inverse limit.
Hence,

$\log\rho(f_{*1}^{lc})\leq h(f).$

∎

## References

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  * [11] M. Shub and R. Williams, _Entropy and stability_ , Topology 14 (1975), 329–338.


# Inferior interfacial superconductivity in 1 UC FeSe/SrVO3/SrTiO3 with
screened interfacial electron-phonon coupling

Nan Guo  Xiaoyang Chen  Tianlun Yu  Yu Fan  [  Qinghua Zhang  Beijing National
Laboratory for Condensed Matter Physics and Institute of Physics, Chinese
Academy of Sciences, Beijing 100190, China  Minyinan Lei  [  Xiaofeng Xu
Xuetao Zhu  Jiandong Guo  Lin Gu  Beijing National Laboratory for Condensed
Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  Haichao Xu<EMAIL_ADDRESS>[  Rui Peng
<EMAIL_ADDRESS>[  Donglai Feng<EMAIL_ADDRESS>National Synchrotron
Radiation Laboratory and School of Nuclear Science and Technology, University
of Science and Technology of China, Hefei 230026, China

###### Abstract

Monolayer FeSe/TiOx and FeSe/FeOx interfaces exhibit significant
superconductivity enhancement compared to bulk FeSe, with interfacial
electron-phonon coupling (EPC) playing a crucial role. However, the reduced
dimensionality in monolayer FeSe, which may drive superconducting
fluctuations, complicates the understanding of the enhancement mechanisms.
Here we construct a new superconducting interface: monolayer
FeSe/SrVO3/SrTiO3, in which the itinerant electrons of highly metallic SrVO3
films can screen all the high-energy Fuchs-Kliewer phonons, including those of
SrTiO3, making it the first FeSe/oxide system with screened interfacial EPC
while maintaining the monolayer FeSe thickness. Despite comparable doping
levels, the heavily electron-doped monolayer FeSe/SrVO3 exhibits a lower
pairing temperature ($T_{\mathrm{g}}$ $\sim$ 48 K) than FeSe/SrTiO3 and
FeSe/LaFeO3. Our findings disentangle the contributions of interfacial EPC
from dimensionality on enhancing $T_{\mathrm{g}}$ in FeSe/oxide interfaces,
underscoring the importance of interfacial EPC in $T_{\mathrm{g}}$
enhancement. This FeSe/VOx interface also provides a platform for studying the
interfacial superconductivity.

###### keywords:

Iron selenide, angle-resolved photoemission spectroscopy, interfacial
electron-phonon coupling

Fudan University] Advanced Materials Laboratory, State Key Laboratory of
Surface Physics, and Department of Physics, Fudan University, Shanghai 200433,
China Fudan University] Advanced Materials Laboratory, State Key Laboratory of
Surface Physics, and Department of Physics, Fudan University, Shanghai 200433,
China Fudan University] Advanced Materials Laboratory, State Key Laboratory of
Surface Physics, and Department of Physics, Fudan University, Shanghai 200433,
China Shanghai Research Center for Quantum Sciences, Shanghai 201315, China
Fudan University] Advanced Materials Laboratory, State Key Laboratory of
Surface Physics, and Department of Physics, Fudan University, Shanghai 200433,
China Shanghai Research Center for Quantum Sciences, Shanghai 201315, China
Advanced Materials Laboratory, State Key Laboratory of Surface Physics, and
Department of Physics, Fudan University, Shanghai 200433, China New
Cornerstone Science Laboratory, University of Science and Technology of China,
Hefei 230026, China Collaborative Innovation Center of Advanced
Microstructures, Nanjing, 210093, China EPC

##  1 Introduction

Single unit-cell FeSe thin films grown on SrTiO3 (001) substrate (1 UC
FeSe/STO) have attracted intensive research interests over the past decades 1,
2, 3, 4, 5. 1 UC FeSe/STO hosts a superconducting pairing temperature
($T_{\mathrm{g}}$) as high as 65 K 3, 6 , which is close to the boiling point
of liquid nitrogen, and significantly enhanced compared with the transition
temperature ($T_{\mathrm{c}}$) of the bulk FeSe ( $\sim$ 8 K) 7. It is
commonly recognized that the interfacial charge transfer from the oxygen
vacancies in STO induces heavy electron doping in the top monolayer FeSe, and
contributes to the enhancement of $T_{\mathrm{g}}$ 2, 3, 8, 9, 10. However,
the superconducting $T_{\mathrm{c}}$ of various electron-doped FeSe 11, 12, 8
is lower than the $T_{\mathrm{g}}$ of 1 UC FeSe/STO and there should be
additional factors that further boost the superconducting pairing. In angle-
resolved photoemission spectroscopy (ARPES) studies 4, 13, 14, “replica bands”
were observed, suggesting that the electrons in FeSe couple with the Fuchs-
Kliewer (FK) phonons in STO at small momentum transfer 5, 8, 15. Such
interfacial electron-phonon coupling (EPC) is theoretically proposed to
effectively enhance the superconducting pairing in heavy electron-doped
FeSe/oxides 16, 17, 18, 19. On the other hand, it is recently proposed that
pairing strength is possibly enhanced by two-dimensionality 20. For example,
in (TBA)xFeSe, by intercalating organic ion layers between adjacent FeSe
layers to push the system closer to the ideal two-dimensional limit, the
superconducting gap is enhanced 20, and the $T_{\mathrm{g}}$ is observed at a
higher temperature than the $T_{\mathrm{c}}$ 20, similar to the behavior in 1
UC FeSe/STO 21. It remains perplexing whether both interfacial EPC and two-
dimensionality or one of the two factors contribute to the superconducting
pairing enhancement in 1 UC FeSe/STO.

Numerous studies have focused on engineering the interfacial EPC to gain
insights into its role in superconducting pairing. For instance, experiments
on increasing the thickness of FeSe grown on SrTiO3 from 1 unit cell to 20
unit cells and tuning the doping level with surface potassium dosing show that
the decay of the superconducting gap correlates with the decay of interfacial
EPC, emphasizing the importance of interfacial EPC in superconductivity 22,
17. However, the varied FeSe thickness between 1 unit cell and 20 unit cells
in FeSe/STO makes it challenging to isolate the effect of interfacial EPC from
that of dimensionality. More recently, the interfacial EPC strength has also
been successfully modulated in 1 UC FeSe/STO, revealing a linear relationship
between the gap size and the interfacial EPC constant ($\lambda$) 23. However,
1 UC FeSe/STO in the limit of zero interfacial EPC could not be reached, and
the role of dimensionality can not be distinguished. To discern the effect of
interfacial EPC and two-dimensionality, a promising approach is to completely
screen the interfacial EPC while maintaining the monolayer thickness of FeSe.

To screen the FK phonons, we here insert highly metallic films between FeSe
and STO, whose itinerant electrons provide strong screening effects. Note that
the 93 meV FK phonons generate a strong dipolar field perpendicular to the
interface, which could not be screened by two-dimensional electron gas 24.
Metallic film with three-dimensional bulk metallicity should be used. Here we
designed and obtained high-quality 1 UC FeSe/SrVO3/Nb:SrTiO3 (FeSe/SVO/STO)
heterostructures. By systematic studies on the interfacial atomic structure,
electronic structure, and phonon spectra, we demonstrate that 1 UC
FeSe/SVO/STO is a high-quality interfacial superconductor with superconducting
$T_{\mathrm{g}}$ of $\sim$ 48 K, in which the interaction between FeSe
electrons and the FK phonons in oxides is fully screened. Our experimental
findings reveal a superconducting pairing temperature of 48 K in the absence
of interfacial EPC, while the increase from 48 K to 65 K can be solely
attributed to the effect of interfacial EPC. These results disentangle the
contributions from dimensionality and interfacial EPC on the enhancement of
the pairing temperature in 1 UC FeSe/oxides.

##  2 Results and discussion

Figure 1: Electronic structure of SVO/STO and screening of the FK phonons. a
Low-energy electron diffraction (LEED) patterns of 42 UC SVO/STO. Blue and
orange filled circles represent the $\sqrt{2}\times\sqrt{2}$ R$45^{\circ}$ and
$\sqrt{5}\times\sqrt{5}$ R$26.6^{\circ}$ surface reconstructions,
respectively. b In-plane photoemission intensity map of 42 UC SVO/STO
integrated over [$E_{\mathrm{F}}-100$ meV, $E_{\mathrm{F}}+100$ meV] using
21.2 eV photons. Green dashed lines schematically illustrate the Fermi
surfaces of SVO/STO. c Photoemission spectra near the $\Gamma$ point as
indicated by the orange line in panel b. d The corresponding Momentum
distribution curves (MDCs) taken at [$E_{\mathrm{F}}-0.8$ eV,
$E_{\mathrm{F}}$] of the spectra in c. e Energy distribution curves (EDCs)
along cut in (c). f Electron energy loss spectra of 42 UC SVO/STO and bare STO
substrate.

SVO is a perovskite-type transition-metal oxide whose lattice constant (3.8425
$\AA$) is comparable to that of STO (3.905 $\AA$) 25. It exhibits metallic
behavior when the thickness exceeds 5 UC 26, 27. Thick SVO films with VO2
termination were epitaxially grown on the commercial Nb-doped STO (001)
substrates at 750 ∘C under an oxygen pressure of 1.9 $\times 10^{-7}$ mbar by
molecular beam epitaxy (see details in Supplementary Note 1). The low energy
electron diffraction (LEED) patterns of SVO films (Figure 1a) shows sharp
diffraction spots, indicating a high-quality two-dimensional surface. Two
distinct surface reconstructions, $\sqrt{5}\times\sqrt{5}$ R$26.6^{\circ}$ and
$\sqrt{2}\times\sqrt{2}$ R$45^{\circ}$, are observed. The complex surface
reconstruction does not affect the subsequent growth of tetragonal FeSe, which
is in line with the previous report on the growth of FeSe on the reconstructed
STO surfaces 28.

The Fermi surface (FS) of thick SVO films, measured using 21.2 eV photons,
mainly consists a rounded-square pocket around the $\Gamma$ point, and weak
parallel-line spectral weight extending along $\Gamma$-X and $\Gamma$-Y
directions (Figure 1b). The ARPES spectra along $\Gamma$-M direction (Figure
1c) and the corresponding momentum distribution curves (MDCs) (Figure 1d) show
an electron-like dispersion which is contributed by the $d_{xy}$ orbital of
vanadium according to previous reports 29, 30. The parallel features could be
contributed by the spectral weight from dxz/dyz orbitals 29 with finite kz
broadening or $\sqrt{2}\times\sqrt{2}$ R$45^{\circ}$ reconstruction-induced
band folding 31, 32. The energy distribution curves (EDCs) display pronounced
Fermi cut-offs at $E_{\mathrm{F}}$ (Figure 1e), demonstrating the metallic
nature of the SVO films. Considering three cylindrical FSs of $d_{xy},d_{yz}$
and $d_{xz}$ 29, 30, the three-dimensional carrier density of SVO films
estimated by the Luttinger theorem 33 is approximately $2\times 10^{22}$ /cm3.
Such a large density of itinerant electrons in thick SVO films is expected to
provide strong screening effects to the dipolar field of FK phonons 34.

Figure 2: Interfacial atomic structure of Se capped 1.5 UC FeSe/SVO/STO. a
Schematic illustration of the FeSe/SVO/STO heterostructure. b High angle
annular dark field (HAADF) image of amorphous Se capped 1.5 UC FeSe/SVO/STO
along the [100] direction of the SVO lattice. c Zoom-in HAADF image near the
FeSe/SVO interface as indicated by the orange box in (b), and the side view of
the corresponding atomic structure is overlaid. d The same as c but along
[110] direction. e Element-resolved maps based on the EELS data at the
Sr-$L_{2,3}$, V-$L_{2,3}$, Fe-$L_{2,3}$ and Se-$L_{2,3}$ absorption edges. The
first image is the region where the measurement is applied.

To assess the screening effects on interfacial phonons, high-resolution
electron energy loss (HR-EELS) measurements were performed on SVO/STO to study
the phonon spectra and compare them to those at the STO surface (Figure 1f).
The HR-EELS spectra of STO surface show two prominent peaks at $\sim$ 60 meV
and $\sim$ 92 meV, which correspond to the FK surface phonons of STO 35, 36.
However, these two peaks disappear in SVO/STO. Given that the incident
electrons are highly sensitive to the dipole electric field 37, 17, signals of
the STO FK phonons should be detected if they could penetrate through the top
SVO films17, 38. Moreover, signals of the SVO phonons are also weak in the
energy range of 80 $\sim$ 100 meV. Overall, the FK phonons either from SVO or
STO in the SVO/STO heterostructure get screened by the mobile electrons in
SVO.

FeSe films were grown on the thick SVO films subsequently at 490 ∘C by co-
evaporating Fe and Se and then post-annealed at 520 ∘C for 3 hours (Figure
2a). The cross-sectional atomic structure of amorphous-Se-capped 1.5 UC
FeSe/SVO/STO was resolved at the atomic level using scanning transmission
electron microscopy (STEM) (Figure 2b-2d). The first layer of FeSe shows clear
atomic structure and no intermixing with the capped Se.

An atomically-resolved intermediate layer between the SVO films and the first
layer of FeSe is observed. By analyzing the elementary-resolved maps (Figure
2e), it was observed that the top layer of $\mathrm{VO_{x}}$ double layers 39,
as well as some Fe and Se atoms diffusing downward, contribute to the
formation of this intermediate layer

(see details in Supplementary Note 2). The first layer of FeSe and the
intermediate layer form a sharp interface, indicating that FeSe can be well
grown on the VOx-terminated SVO. Our results provide a new interfacial system
of monolayer FeSe grown on non-TiOx terminated oxides after FeSe/MgO 40,
FeSe/NdGaO3 41 and FeSe/LaFeO3 (FeSe/LFO) 38.

Figure 3: Electronic structure of 1 UC FeSe/SVO/STO. a Sketch of FeSe/SVO/STO
heterostructures and the screening of phonons by itinerant electrons. b
Photoemission intensity map of 1 UC FeSe/SVO/STO integrated over [$E_{F}-20$
meV, $E_{F}+20$ meV] using 21.2 eV photons. c-e Photoemission spectra along
cut #1 in b (c), the second derivative of the photoemission intensity with
respect to energy (d), EDC integrated over the momentum range indicated by the
blue dashed rectangle in c (e). f-h The same as c-e, but along cut #2 in panel
b. The EDC of 1 UC FeSe/STO from ref. 23 with the same integration range is
appended in h. i Electron energy loss spectra of 1 UC FeSe/STO and 1 UC
FeSe/SVO/STO.

In-situ ARPES measurements were conducted on 1 UC FeSe/SVO/STO
heterostructures to study the electronic structure. The FS of 1 UC FeSe/SVO is
composed of two elliptical electron pockets located at the Brillouin zone
corner, while no hole pocket at the Brillouin zone center (Figure 3b),
indicating that the monolayer FeSe is heavily electron-doped by the interface,
similar to 1 UC FeSe/STO and 1 UC FeSe/LFO. The photoemission spectra along
high symmetric cuts #1 (Figure 3c-e) exhibit two parabolic bands (noted as
$\alpha$ and $\beta$) located near the $\Gamma$ point with their band top at
$E-E_{\mathrm{F}}\sim-59$ meV. Along high symmetric cuts #2 (Figure 3f-h),
two-electron bands (noted as $\gamma_{1}$ and $\gamma_{2}$) are resolved
around the M point with their band bottom at $E-E_{\mathrm{F}}\sim-51$ meV.
According to Luttinger theorem 33, the doping level was estimated to be about
0.088 $e^{-}$/Fe based on the Fermi surface volume, slightly lower than the
optimal doping (0.12 $e^{-}/\mathrm{Fe}$) of 1 UC FeSe/STO 8, but almost
identical to that of 1 UC FeSe/LFO (0.087 $e^{-}/\mathrm{Fe}$) 38, indicating
that 1 UC FeSe/SVO and 1 UC FeSe/LFO should locate at the same position in the
phase diagram when only considering the effect of electron doping 8.

Figure 4: Superconducting gap of 1 UC FeSe/SVO/STO. a Symmetrized
photoemission spectra with respect to the corresponding $E_{\mathrm{F}}$ at 6
K of 1 UC FeSe/LFO/STO and 1 UC FeSe/SVO/STO. b Symmetrized EDCs at the
corresponding Fermi momentum $k_{\mathrm{F}}$ of 1 UC FeSe/SVO, 1 UC FeSe/STO
23, and 1 UC FeSe/LFO 38. The fittings to a superconducting function based on
the simplified Bardeen-Cooper-Schrieffer (BCS) self-energy are appended. c, d
Temperature dependence of the symmetrized EDC at the $k_{\mathrm{F}}$ (c), and
the determined superconducting gap (d) of 1 UC FeSe/SVO/STO. The error bars of
the gap are from the standard deviation of the fitting process and the
measurement uncertainty. e Superconducting gap close temperatures
($T_{\mathrm{g}}$s) of 1 UC FeSe/SVO/STO, 1 UC FeSe/STO, and 1 UC FeSe/LFO/STO
are plotted as a function of the interfacial EPC constant $\lambda$.

As the small-$q$ FK phonons of STO are screened by the itinerant electrons in
thick SVO layers (Figure 3a), the replica bands, which are the signature of
interfacial EPC 4, 14, 42, 43, 44, were expected to be absent. Consistently,
in the ARPES spectra of 1 UC FeSe/SVO/STO, replica bands for all the main
bands of monolayer FeSe ($\alpha$, $\beta$, $\gamma_{1}$, and $\gamma_{2}$)
are not observed [Figs. 3c-h], in contrast with the ARPES spectra of 1 UC
FeSe/STO and 1 UC FeSe/LFO, in which the prominent band replicas are observed
at $\sim$ 100 meV, and $\sim$ 80 meV below the main bands, respectively 4, 23,
38. Compared with the high-quality 1 UC FeSe/STO 23, the momentum-integrated
EDCs near the M point show a comparable signal-to-noise ratio and similar main
band intensity (Figure 3h), and the replica should be observed if exists.
However, the lineshape at the replica band region is dramatically different,
with no replica band intensity in 1uc FeSe/SVO. The analysis of the
quasiparticle lifetime also suggests a similar scattering rate in 1 UC
FeSe/SVO/STO.

Therefore, the absence of replica bands in 1 UC FeSe/SVO/STO is not an effect
of the sample quality. It should be noted that the absence of replica bands is
consistent across multiple ARPES measurements and analyses conducted on
several 1 UC FeSe/SVO/STO samples. Consistent with ARPES results, the HR-EELS
measurements show the significantly suppressed FK phonon peaks in 1 UC
FeSe/SVO/STO in comparison with that of 1 UC FeSe/STO (Figure 3i). All these
findings suggest that the dipole field generated by the FK phonons is
incapable of penetrating through the thick SVO layers and the interfacial EPC
is well-screened in 1 UC FeSe/SVO/STO.

FeSe/oxides | doping level ($e^{-}$/Fe) |  $\Delta_{1}$ (meV) |  $\Gamma_{1}$ (meV)
---|---|---|---
1 UC FeSe/SVO | 0.088 | 9.6 | 21.7
1 UC FeSe/STO | 0.12 | 12.1 | 20.8
1 UC FeSe/LFO | 0.087 | 17 | 20
Table 1: Summary of doping level, superconducting gap size $\Delta_{1}$, and
quasiparticle lifetime $\Gamma_{1}$ of 1 UC FeSe/SVO, 1 UC FeSe/STO 23, and 1
UC FeSe/LFO 38.

As shown in Fig.4a, the electron bands ($\gamma_{2}/\gamma_{1}$) exhibit an
evident back bending near $E_{\mathrm{F}}$ at low temperature, which is a
characteristic behavior of the Bogoliubov quasiparticle dispersion. With
almost the same electron doping level, 1 UC FeSe/LFO has a significantly
larger superconducting gap (see the right panel of Figure 4a). To
quantitatively compare the size of the superconducting gap, the EDC at Fermi
momentum ($k_{\mathrm{F}}$) is fitted by the superconducting spectral function
with the simplified BCS self-energy
$\Sigma(\boldsymbol{k},\omega)=-\mathrm{i}\Gamma_{1}+\Delta^{2}/\left[\omega+\epsilon(k)\right]$
45, 46, 23 (Figure 4b), where $\Delta$ is the superconducting gap,
$\Gamma_{1}$ is the energy broadening caused by the quasiparticle scattering
off impurities, and hence an indication of the quasiparticle lifetime. The
three films show similar values of $\Gamma_{1}$, where $\Gamma_{1}$(FeSe/SVO)
= 21.7 meV, $\Gamma_{1}$(FeSe/STO) = 20.8 meV, and $\Gamma_{1}$(FeSe/LFO) = 20
meV, suggesting similar sample quality. On the other hand, the superconducting
gap size of $\gamma_{1}$ band ($\Delta_{1}$) of FeSe/SVO ($\Delta_{1}$ = 9.6
meV) is significantly smaller than that of 1 UC FeSe/STO ($\Delta_{1}$=12.1
meV) and FeSe/LFO ($\Delta_{1}$=17 meV) 23, 38. These values are summarized in
Table 1 to visually compare these three systems. Since the sample qualities
are comparable, the reduced gap size indicates a weaker pairing strength in 1
UC FeSe/SVO/STO.

Temperature-dependent measurements to determine the $T_{\mathrm{g}}$ were
conducted on the same sample (Figure 4c). The gap gradually decreases with
increasing temperature and eventually closes at $T_{\mathrm{g}}$ = 48 K, which
is well fitted by the BCS formula (Figure 4d).

###  2.1 Discussion

Monolayer FeSe grown on SVO/STO demonstrates a high-quality heterostructure
with a well-ordered interfacial atomic structure. The quasiparticle scattering
rate is similar to that in 1 UC FeSe/STO and FeSe/LFO. The well-screened
interfacial EPC makes it a prototypical system to investigate the role of the
interfacial EPC in FeSe/oxide interfacial systems at the 1 UC limit. The gap-
closing temperature $T_{\mathrm{g}}$, which represents the onset of the
superconducting pairing, can be directly compared among the 1 UC FeSe/SVO,
FeSe/STO, and FeSe/LFO as a function of interfacial EPC strength while
preserving the identical thickness of FeSe. A monotonic suppression of the
$T_{\mathrm{g}}$ is observed as the interfacial EPC strength decreases from
the 1 UC FeSe/LFO, 1 UC FeSe/STO to 1 UC FeSe/SVO (Figure 4e). Therefore, our
observations emphasize the crucial role of interfacial EPC in the enhancement
of the superconducting pairing in 1 UC FeSe/oxide systems.

It is also noticeable that the $T_{\mathrm{g}}$ of 1 UC FeSe/SVO/STO is quite
similar to the highest $T_{\mathrm{g}}$ of surface potassium-dosed multilayer
FeSe, in spite of their different FeSe thicknesses, suggesting a minimal
influence of dimensionality on the superconducting pairing in these systems.
In regard to the higher $T_{\mathrm{g}}$ in organic molecules intercalated
FeSe 20, the contribution of the coupling between FeSe electrons and certain
vibrational modes of organic molecules may need to be investigated and taken
into account.

Moreover, the discovery of the high-quality FeSe/VOx interface and its
superconductivity provide a new playground for studying the FeSe/oxide
interfaces. Perovskite vanadates, typically the family of $R\mathrm{VO_{3}}$
($R$ being a rare earth or yttrium atom), possess highly tunable orbital and
spin orderings with external stimuli 47. For example, in PrV$\mathrm{O_{3}}$
thin films, tunable magnetic properties have been achieved by epitaxial strain
48, 49, or chemical strain through the control of oxygen vacancies 50, 51. 1
UC FeSe/SVO/STO, as a new realization of FeSe grown on the non-TiOx terminated
substrate, inspires the future designing of the heterostructure in
investigating strongly correlated physics, such as bandwidth control,
relationship between superconductivity and magnetic orders, and electron-boson
interactions.

##  3 Method

Sample fabrication. SVO films were epitaxially grown on the TiOx terminated
SrTi$\mathrm{O_{3}}$ (001) substrates (Crystech GmbH) by using atomic layer-
by-layer molecular beam epitaxy (MBE) method, at a substrate temperature of
750 ∘C and an oxygen partial pressure of 1.9 $\times 10^{-7}$ mbar. The
thickness of SVO was confirmed by the real-time intensity oscillation of
reflective high-energy electron diffraction (RHEED) (Figure S1). FeSe films
were grown on SVO subsequently at 490 ∘C by co-evaporation method and annealed
at 520 ∘C for 3 h before ARPES measurement.

ARPES measurements. In-situ ARPES measurements were performed on FeSe/SVO/STO
using a Scienta DA30 electron analyzer and a Fermi Instruments 21.2 eV helium
discharge lamp, under a typical vacuum of 2 $\times 10^{-11}$ mbar. The total
energy resolution is 6 meV and the angular resolution is 0.3 ∘.

HR-EELS measurements. For the in-situ HR-EELS measurements, the samples were
annealed at $\sim$ 520 ∘ in an ultra-high vacuum of 5 $\times$ 10-9 mbar to
get the clean surface. RHEED patterns were collected to verify the surface
quality before HR-EELS measurements. The incident electron beam in HR-EELS
measurements is produced by an electron gun of Model LK5000M (LK
Technologies). The incident electron energy was 13.6 eV and the incident angle
with respect to the surface normal was 45∘. Data were collected by an analyzer
A-1 (MBS) under an ultra-high vacuum of 5 $\times$ 10-11 mbar. The
measurements were conducted at 6 K.

STEM measurements. For STEM measurements, a 1.5 UC FeSe/SVO/STO sample was
capped with Se and taken out of the vacuum chamber to protect the FeSe layers
from being oxidized. Then the sample was prepared for STEM measurements by
using Focused Ion Beam (FIB) milling. Cross-sectional lamella was thinned down
to 100 nm at an accelerating voltage of 30 kV with a decreasing current from
the maximum 2.5 nA, followed by fine polish at an accelerating voltage of 2 kV
with a small current of 40 pA. The atomic structures of the FeSe/SVO/STO films
were characterized using an ARM-200CF (JEOL, Tokyo, Japan) transmission
electron microscope operated at 200 kV and equipped with double spherical
aberration (Cs) correctors. HAADF images were acquired at acceptance angles of
90-370 mrad.

This work is supported in part by the National Science Foundation of China
under the grant Nos. 11922403, 12274085, 12074074, the National Key R&D
Program of the MOST of China (2023YFA1406300), the New Cornerstone Science
Foundation, the Innovation Program for Quantum Science and Technology (Grant
No. 2021ZD0302803), and Shanghai Municipal Science and Technology Major
Project (Grant No.2019SHZDZX01).

Details for sample preparation and experimental method, additional data from
cross-sectional STEM.

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