# Nested homotopy models of finite metric spaces and their spectral homology

Sergei O. Ivanov  Beijing Institute of Mathematical Sciences and Applications
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

For a real $r\geq 0,$ we consider the notion of $r$-homotopy equivalence in
the category quasimetric spaces, which includes metric spaces and directed
graphs. We show that for a finite quasimetric space $X$ there is a unique (up
to isometry) $r$-homotopy equivalent quasimetric space of the minimal possible
cardinality. It is called $r$-minimal model of $X$. We use this to construct a
decomposition of the magnitude-path spectral sequence of a digraph into a
direct sum of spectral sequences with certain properties.

We also construct an $r$-homotopy invariant ${\rm SH}^{r}_{n,I}(X)$ of a
quasimetric space $X,$ called spectral homology, that generalises many other
invariants: the pages of the magnitude-path spectral sequence, including path
homology, magnitude homology, blurred magnitude homology and reachability
homology.

The work is supported by Beijing Institute of Mathematical Sciences and
Applications

##  1\. Introduction

###  1.1. Background

For over a decade, two theories have been actively developed. The first theory
is the theory of magnitude of metric spaces and their magnitude homology. This
theory was developed by Leinster, Meckes, Shulman, Hepworth, Willerton,
Yoshinaga, Asao and others [21, 24, 20, 16, 23, 18, 3]. It explores an
invariant of a metric space called magnitude. On one hand, the magnitude is a
real number, which is a continuous analogue of the cardinality of a finite
set. On the other hand, it exhibits some properties similar to those of the
Euler characteristic. The concept of magnitude is also related to the notion
of diversity [22], [19]. Subsequently it was discovered that it is possible to
introduce a homology theory for metric spaces, so that for finite metric
spaces the Betti numbers of this theory are related to the magnitude in the
similar way as ordinary Betti numbers are related to the Euler characteristic.
This homology theory is called magnitude homology.

The second theory is the GLMY-theory of path homology of digraphs. This theory
was developed by Grigoryan, Lin, Muranov, Yau and others [7, 8, 14, 12, 9, 13,
11, 10, 17]. It implements a homotopical approach to the theory of digraphs.
Its central concept is the notion of path homology of a digraph, also known as
GLMY-homology. Path homology behave in many ways similar to homology of
spaces. For example, there is an analogue of the Künneth formula; there is a
notion of suspension of a digraph that shifts the path homology; there is a
notion of the fundamental group of a digraph, whose abelianization equals the
first path homology group, and so forth. The theory also has the notion of
homotopy of morphisms in the category of digraphs, and path homology is
homotopy invariant. Moreover, there is a theory of covering digraphs,
analogous to the theory of covering spaces in algebraic topology [5].

Asao later showed that for the case of digraphs, these two theories could be
unified into a single theory [2],[1]. He showed that each digraph $G$ defines
a spectral sequence $E(G)$, whose first page consists of magnitude homology
and the second page contains path homology as the zeroth row. This spectral
sequence is called magnitude-path spectral sequence. He also defines the
concept of $r$-homotopy on the category of digraphs and shows that
$E^{r+1}(G)$ is an invariant under $r$-homotopy. For $r=1$, this concept
reduces to homotopy as defined in the GLMY-theory. Thus, this result
generalizes the fact that the path homology is a homotopy invariant.

For a digraph $G$, one can define the distance between vertices $d_{G}(x,y)$
as the length of the shortest path from $x$ to $y$. If there is no a path from
$x$ to $y$, we assume $d_{G}(x,y)=+\infty.$ Note that this distance is not
necessarily symmetric and can be equal to infinity. However, it satisfies the
triangle inequality and $d_{G}(x,y)=0$ if and only if $x=y.$ Such
generalizations of metric spaces are referred to as quasimetric spaces. We
will work in the category of quasimetric spaces where morphisms are short
maps.

The aim of our work is to study the concept of $r$-homotopy, introduced by
Asao, for the case of quasimetric spaces and any real $r\geq 0.$ We study this
concept from two perspectives. First, we study finite quasimetric spaces that
have the least cardinality among quasimetric spaces in a given $r$-homotopy
equivalence class. Secondly, we introduce a new invariant of the $r$-homotopy
equivalence for any quasimetric spaces, and study this invariant. Both parts
are related to the magnitude-path spectral sequence: in the first part we
obtain its decomposition into a direct sum of some smaller spectral sequences;
in the second part we generalise its pages to the case of any quasimetric
spaces.

###  1.2. Nested $r$-minimal models

In this theory, after the $r$-homotopy equivalence of short maps is defined,
$r$-homotopy equivalences and $r$-deformation retracts are defined similarly
to how it is defined in the classical algebraic topology. We say that a finite
quasimetric space $M$ is $r$-minimal, if it has no proper $r$-deformation
retracts. A quasimetric space $M$ is called $r$-minimal model of a quasimetric
space $X,$ if $M$ is $r$-minimal and it is $r$-homotopy equivalent to $X.$ We
prove the following theorem.

######  Theorem.

Let $r\geq 0$ be a real number. Then the following holds.

  1. (1)

Any finite quasimetric space has an $r$-minimal model, which is unique up to
isometry.

  2. (2)

Two finite quasimetric spaces are $r$-homotopy equivalent if and only if their
$r$-minimal models are isometric.

  3. (3)

Any finite quasimetric space has an $r$-deformation retract, which is its
$r$-minimal model.

The $r$-minimal model of a space $X$ is denoted by $M_{r}(X).$ Note that we do
not claim that $X\mapsto M_{r}(X)$ is a functor, and the inclusion
$M_{r}(X)\hookrightarrow X$ and the retraction $X\to M_{r}(X)$ are not
uniquely defined. It is easy to see that if $r^{\prime}>r,$ then
$M_{r^{\prime}}(M_{r}(X))\cong M_{r^{\prime}}(X).$ Using this we obtain that
there is a finite number of real numbers $0<r_{1}<\dots<r_{n}$ such that
$M_{r_{i}}(X)\not\cong M_{r}(X)$ for $r<r_{i}.$ We call these points homotopy
jumping points and set

(1.1) ${\rm hj}(X)=\\{r_{1},\dots,r_{n}\\}.$

Thus for any finite quasi-metric space $X$ we obtain a nested sequence of
retracts

(1.2) $X\supset M_{r_{1}}(X)\supset M_{r_{2}}(X)\supset\dots\supset
M_{r_{n}}(X).$

Unfortunately, the map $X\mapsto{\rm hj}(X)$ from the set of finite subsets of
$\mathbb{R}^{n}$ to the set of finite subsets of $\mathbb{R}$ is not
continuous with respect to the Hausdorff distance (Example 2.12).

If $X=G$ is a digraph, the sequence (1.2) is a sequence of convex subdigraphs,
such that $M_{r_{i+1}}(G)$ is a retract of $M_{r_{i}}(G).$ Using this we
obtain a decomposition of the magnitude-path spectral sequence $E(G)$ into a
direct sum of spectral sequences

(1.3) $E(G)=\bigoplus_{k=1}^{n}E(G)_{[k]}\oplus E(G)_{[\infty]}$

such that $E^{r_{k}+1}(G)_{[k]}=0$ and for any $1\leq j\leq n$ we have

(1.4) $E(M_{r_{j}}(G))\cong\bigoplus_{k=j+1}^{n}E(G)_{[k]}\oplus
E(G)_{[\infty]}.$

However, we do not claim that this decomposition is natural.

We say that a finite digraph $G$ is homotopy stable, if $M_{r}(G)=G$ for any
$r\geq 0.$ Any finite digraph $G$ defines a homotopy stable digraph by the
formula $M_{\sf stab}(G)=M_{r}(G),$ where $r$ is greater then all homotopy
jumping points of $G$. The decomposition (1.3) shows that the magnitude-path
spectral sequence $E(G)$ contains a summand corresponding to the homotopy
stable digraph $E(M_{\sf stab}(G))$ such that $E^{r}(M_{\sf
stab}(G))=E^{r}(G)$ for all big enough $r.$ This indicates that homotopy
stable digraphs may be an important subject for further study.

###  1.3. Spectral homology

Any non-empty interval $I\subseteq\mathbb{R}$ can be uniquely presented as the
difference of left infinite intervals $I=R\setminus L,$ where left infinite
intervals are intervals of the form
$(-\infty,a],(-\infty,a),\mathbb{R},\emptyset.$ We will work with non-empty
intervals using this presentation. We define an order on the set of non-empty
intervals such that $I\preccurlyeq J$ if and only if $R_{I}\subseteq R_{J}$
and $L_{I}\subseteq L_{J},$ where $I=R_{I}\setminus L_{I}$ and
$J=R_{J}\setminus L_{J}.$

Let $C$ be an $\mathbb{R}$-filtered chain complex. Thus for any real number
$r\in\mathbb{R}$ we have a chain subcomplex $F_{r}C\subseteq C$ such that
$F_{r}C\subseteq F_{r^{\prime}}C$ for $r<r^{\prime}.$ For a left infinite
interval $L\subseteq\mathbb{R}$ we set $F_{L}C=\bigcup_{r\in L}F_{r}C.$ For a
non-empty interval $I=R\setminus L$ we consider a chain complex

(1.5) $C_{I}=F_{R}C/F_{L}C.$

If $I\preccurlyeq J,$ there is a natural morphism of chain complexes $C_{I}\to
C_{J}$ induced by inclusion of the terms of the filtration.

For any $r\geq 0$ we consider two intervals $I_{r}$ and $I^{r}$ that contain
$I,$ which are defined by the formulas $I_{r}=R\setminus(L-r)$ and
$I^{r}=(R+r)\setminus L.$ Then

(1.6) $I_{r}\preccurlyeq I\preccurlyeq I^{r}$

and the $r$-spectral homology of $C$ can be defined as the image

(1.7)
$\mathrm{SH}^{r}_{n,I}(C)\cong\mathrm{Im}(H_{n}(C_{I_{r}})\longrightarrow
H_{n}(C_{I^{r}})).$

For example, $\mathrm{SH}^{r}_{n,\\{\ell\\}}(C)$ is the image of
$H_{n}(C_{[\ell-r,\ell]})\longrightarrow H_{n}(C_{[\ell,\ell+r]}).$ On the
other hand, there is an equivalent definition of $\mathrm{SH}^{r}_{n,I}(C)$ as
a subquotient of $H_{n}(C_{I})$ (Proposition 4.2).

The spectral homology is natural by the interval with respect to the order
defined above. In particular, we obtain a persistent module ${\rm
PSH}^{r}_{n}(C)$ such that

(1.8) ${\rm PSH}^{r}_{n}(C)(\ell)=\mathrm{SH}^{r}_{n,(-\infty,\ell]}(C),$

that we call persistent $r$-spectral homology.

The magnitude-path spectral sequence of a digraph $G$ is defined using a
$\mathbb{Z}$-filtered chain complex $\mathrm{RC}(G),$ which is called
reachability chain complex (see [15]). This complex is defined similarly for
the case of a quasi-metric space $X.$ However, in this case we obtain an
$\mathbb{R}$-filtration on the chain complex $\mathrm{RC}(X).$ Then the
$r$-spectral homology of $X$ is defined as the $r$-spectral homology of the
$\mathbb{R}$-filtered chain complex

(1.9) $\mathrm{SH}^{r}_{n,I}(X)=\mathrm{SH}^{r}_{n,I}(\mathrm{RC}(X)).$

We show that $\mathrm{SH}^{r}_{n,I}(X)$ is an $r$-homotopy invariant. In
particular, if $X$ is finite, we obtain
$\mathrm{SH}^{r}_{n,I}(X)\cong\mathrm{SH}^{r}_{n,I}(M_{r}(X)).$ We also prove
that some other types of homology can be presented as spectral homology.

######  Theorem.

Let $X$ be a quasimetric space and $G$ be a digraph. Then the following holds.

  1. (1)

The magnitude homology can be presented as $0$-spectral homology

${\rm MH}_{n,\ell}(X)\cong\mathrm{SH}^{0}_{n,\\{\ell\\}}(X).$

In particular, it follows that for any $r\geq 0$ the group
$\mathrm{SH}^{r}_{n,\\{\ell\\}}(X)$ is an $r$-homotopy invariant subquotient
of ${\rm MH}_{n,\ell}(X).$

  2. (2)

The persistent module of blurred magnitude homology (see [6]) can be presented
as the persistent $0$-spectral homology

${\rm BMH}_{n}(X)\cong{\rm PSH}^{0}_{n}(X).$

In particular, it follows that for any $r\geq 0$ the persistent module ${\rm
PSH}^{r}_{n}(X)$ is an $r$-homotopy invariant subquotient persistent module of
${\rm BHM}_{n}(X).$

  3. (3)

The reachability homology (see [15]) can be presented as $r$-spectral homology

${\rm RH}_{n}(X)\cong\mathrm{SH}^{r}_{n,\mathbb{R}}(X)$

for any $r\geq 0.$

  4. (4)

For any integer $r\geq 1$ the $r$-th page of the magnitude-path spectral
sequence of $G$ can be presented as $(r-1)$-spectral homology

$E^{r}_{n,\ell-n}(G)\cong\mathrm{SH}^{r-1}_{n,\\{\ell\\}}(G).$

  5. (5)

In particular, the path homology can be described as

${\rm PH}_{n}(G)=\mathrm{SH}^{1}_{n,\\{n\\}}(G).$

In the end of the paper we also give a description of some low-dimensional
spectral homology. In particular, for $\ell>r\geq 0,$ we give a description of
$\mathrm{SH}^{r}_{1,\\{\ell\\}}(X)$ in terms of adjacent pairs, similar to the
description of $\mathrm{MH}_{1,\ell}(X)$ given in [23, Cor. 4.5]. We also work
out the following continuous example. Let $X\subseteq S^{1}$ be an arc in the
circle of radius $1$ (with the Euclidean distance) that does not lie in one
semicircle.

$x$$y$

If we assume that $0<r<\ell<\sqrt{2}/2,$ then

(1.10)
$\mathrm{SH}^{r}_{1,\\{\ell\\}}(X)\cong\begin{cases}\mathbb{Z}^{2},&\ell=d(x,y),\\\
0,&\ell\neq d(x,y).\end{cases}$

So the first spectral homology detects the distance $d(x,y)$ for the “singular
pair” $(x,y).$

###  1.4. Acknowledgments

The author is grateful to Lev Mukoseev for fruitful discussions and Example
2.8.

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 Nested $r$-minimal models
    3. 1.3 Spectral homology
    4. 1.4 Acknowledgments
  2. 2 $r$-minimal models of quasimetric spaces
    1. 2.1 Quasimetric spaces
    2. 2.2 $r$-homotopy category
    3. 2.3 $r$-deformation retracts
    4. 2.4 $r$-minimal spaces
    5. 2.5 $r$-minimal model of a space
    6. 2.6 Homotopy jumping points
  3. 3 Magnitude-path spectral sequence of a digraph
    1. 3.1 $r$-minimal models of digraphs
    2. 3.2 Reminder on the magnitude-path spectral sequence
    3. 3.3 Decomposition of the magnitude-path spectral sequence
  4. 4 Spectral homology
    1. 4.1 Reminder on spectral sequence of a $\mathbb{Z}$-filtered chain complex
    2. 4.2 Spectral homology of $\mathbb{R}$-filtered chain complexes
    3. 4.3 $r$-homotopy of $\mathbb{R}$-filtered chain complexes
    4. 4.4 Spectral homology of quasimetric spaces
    5. 4.5 Relation of spectral homology with other types of homology
    6. 4.6 Low-dimensional spectral homology

##  2\. $r$-minimal models of quasimetric spaces

###  2.1. Quasimetric spaces

A _quasimetric space_ is a set $X$ equipped by a map $d:X\times
X\to[0,+\infty]$ such that the triangle inequality holds $d(x,z)\leq
d(x,y)+d(y,z)$ for any $x,y,z\in X$ and $d(x,y)=0$ if and only if $x=y.$ Note
that we allow the distance to be equal to infinity, and it is not necessary
symmetric, but the distance between distinct points is positive. Two sources
of examples of quasimetric spaces are metric spaces and directed graphs with
the shortest path distance.

A map between two quasimetric spaces $\varphi:X\to Y$ is called short (or
1-Lipschitz, or distance-decreasing), if

(2.1) $d(\varphi(x),\varphi(x^{\prime}))\leq d(x,x^{\prime})$

for any $x,x^{\prime}\in X.$ The set of all short maps from $X$ to $Y$ is
denoted by ${\rm SMap}(X,Y).$ The category of quasimetric spaces with short
maps as morphisms is denoted by ${\sf QMet}.$ Note that isomorphisms in this
category are isometries (distance preserving bijections).

For two short maps $\varphi,\psi:X\to Y$ we set

(2.2) $d(\varphi,\psi)=\underset{x}{\rm sup}\ d(\varphi(x),\psi(x)).$

It is easy to see that this defines a structure of quasimetric space on ${\rm
SMap}(X,Y).$ Note that for any short maps $\beta:Y\to Z$ and $\alpha:W\to X$
we have

(2.3) $d(\beta\varphi\alpha,\beta\psi\alpha)\leq d(\varphi,\psi).$

###  2.2. $r$-homotopy category

Let $r\geq 0$ be a real number. We say that two points $x,x^{\prime}$ of a
quasimetric space $X$ are $r$-homotopic, denoted by $x\sim_{r}x^{\prime},$ if
there is a sequence $x=x_{0},\dots,x_{n}=x^{\prime}$ of points of $X$ such
that for each $0\leq i\leq n-1$ we have either $d(x_{i},x_{i+1})\leq r$ or
$d(x_{i+1},x_{i})\leq r.$ It is easy to see that $\sim_{r}$ is an equivalence
relation on $X.$ We say that two short maps $\varphi,\psi:X\to Y$ are
$r$-homotopic if they are $r$-homotopic as points in ${\rm SMap}(X,Y).$

######  Lemma 2.1.

If $\alpha:W\to X,$ $\varphi,\psi:X\to Y$ and $\beta:Y\to Z$ are short maps
between quasimetric spaces and $\varphi\sim_{r}\psi,$ then
$\beta\varphi\alpha\sim_{r}\beta\psi\alpha.$ In other words, $\sim_{r}$ is a
congruence relation on the category ${\sf QMet}.$

###### Proof.

Follows from (2.3). ∎

Since $\sim_{r}$ is a congruence relation on ${\sf QMet},$ we can define
$r$-homotopy category of quasimetric spaces as the quotient-category ${\sf
QMet}/\sim_{r}$. We say that a short map is an $r$-homotopy equivalence, if
its image in ${\sf QMet}/\sim_{r}$ is an isomorphism, and two quasimetric
spaces are $r$-homotopy equivalent, if they are isomorphic in ${\sf
QMet}/\sim_{r}.$

For a quasimetric space $X$ we denote by ${\rm End}(X)$ the monoid of
endomorphisms of $X$ in ${\sf QMet}.$ We will also denote by ${\rm
End}_{r}(X)$ the submonoid consisting of short maps $\varphi:X\to X$ such that
$\varphi\sim_{r}1_{X}.$

(2.4) ${\rm End}_{r}(X)=\mathrm{Ker}({\rm End}(X)\to{\rm End}_{{\sf
QMet}/\sim_{r}}(X)).$

###  2.3. $r$-deformation retracts

If $X$ is a quasimetric space and $A\subseteq X$ is a subset, $A$ will be
treated as a quasimetric space with the induced metric. We will denote by
$\iota=\iota_{A,X}:A\to X$ the canonical embedding. A _retract_ of a
quasimetric space $X$ is a subset $A\subseteq X$ such that there exists a
short map $\rho:X\to A$, called retraction, such that $\rho\iota=1_{A}$. We
say that the retract is _proper_ , if $A\neq X.$ An _$r$ -deformation retract_
is a retract $A\subseteq X$ with a retraction $\rho$ such that
$\iota\rho\sim_{r}1_{X}.$

######  Proposition 2.2.

Let $X$ be a finite quasimetric space and $\varphi\in{\rm End}_{r}(X)$. Then
there exists a natural $n\geq 1$ such that $\varphi^{n}$ is idempotent, and
for any such $n$ the subset $\varphi^{n}(X)\subseteq X$ is an $r$-deformation
retract of $X.$

###### Proof.

Since $X$ is finite, there exists such $n$ that $\varphi^{n}$ is an
idempotent. Lemma 2.1 implies that
$1\sim_{r}\varphi\sim_{r}\varphi^{2}\sim_{r}\dots\sim_{r}\varphi^{n}.$
Therefore $\varphi^{n}\sim_{r}1_{X}.$ Set $A=\varphi^{n}(X).$ Then we can
present $\varphi^{n}$ as the composition $\varphi^{n}=\iota\rho,$ where
$\iota:A\to X$ is the canonical embedding and $\rho:X\to A$ is some
surjection. Since the quasimetric on $A$ is induced from $X,$ we obtain that
$\rho$ is a short map. Note that $\iota\rho\sim_{r}1_{X}.$ Since $\varphi^{n}$
is idempotent, we have $\iota\rho\iota\rho=\iota\rho.$ Using that $\iota$ is
injective, and $\rho$ is surjective, we obtain $\rho\iota=1_{A}.$ ∎

###  2.4. $r$-minimal spaces

We say that a finite quasimetric space $M$ is $r$-minimal, if it has no proper
$r$-deformation retracts.

######  Proposition 2.3.

Let $X$ be a finite quasimetric space. Then the following statements are
equivalent.

  1. (1)

$X$ is $r$-minimal;

  2. (2)

${\rm End}_{r}(X)\subseteq{\rm Aut}(X)$;

  3. (3)

any $\varphi\in{\rm End}(X)$ such that either $d(1_{X},\varphi)\leq r$ or
$d(\varphi,1_{X})\leq r$ is an automorphism.

###### Proof.

$(1)\Rightarrow(2).$ Take $\varphi\in{\rm End}_{r}(X).$ Proposition 2.2
implies that there exists $n\geq 1$ such that $\varphi^{n}$ is an idempotent
and $\varphi^{n}(X)\subseteq X$ is an $r$-deformation retract. Since $X$ is
$r$-minimal, we obtain $\varphi^{n}(X)=X.$ Using that $X$ is finite, we obtain
that $\varphi^{n}$ is a bijection. Combining this with the fact that $\varphi$
and idempotent, we obtain $\varphi^{n}=1_{X}.$ Therefore $\varphi$ is an
automorphism.

$(2)\Rightarrow(3).$ Obvious.

$(3)\Rightarrow(1).$ Consider an $r$-deformation retract $A\subseteq X$ with a
retraction $\rho:X\to A.$ We need to prove that $A=X.$ Since
$\iota\rho\sim_{r}1_{X},$ there exists a sequence
$1_{X}=\varphi_{0},\dots,\varphi_{n}=\iota\rho$ such that for any $0\leq i\leq
n-1$ either $d(\varphi_{i},\varphi_{i+1})\leq r$ or
$d(\varphi_{i+1},\varphi_{i})\leq r.$ Prove by induction that $\varphi_{i}$ is
an automorphism. The base of induction $i=0$ is obvious. Assume that
$\varphi_{i}$ is an automorphism. If $d(\varphi_{i},\varphi_{i+1})\leq r,$
then $d(1_{X},\varphi_{i}^{-1}\varphi_{i+1})\leq r,$ and hence
$\varphi_{i}^{-1}\varphi_{i+1}$ is an automorphism. If
$d(\varphi_{i+1},\varphi_{i})\leq r,$ then
$d(\varphi_{i+1}\varphi^{-1}_{i},1_{X})\leq r,$ and hence,
$\varphi_{i+1}\varphi^{-1}_{i}$ is an automorphism. It follows that
$\varphi_{i+1}$ is an automorphism. Since $\iota\rho$ is an automorphism,
$A=X.$ ∎

######  Proposition 2.4.

Let $M$ and $M^{\prime}$ be $r$-minimal finite quasimetric spaces. If
$\varphi:M\to M^{\prime}$ is an $r$-homotopy equivalence, then $\varphi$ is an
isometry.

###### Proof.

Since $\varphi$ is an $r$-homotopy equivalence, there is a short map
$\psi:M^{\prime}\to M$ such that $\psi\varphi\sim_{r}1_{M}$ and
$\varphi\psi\sim_{r}1_{M^{\prime}}.$ By Proposition 2.3 we obtain that
$\psi\varphi$ and $\varphi\psi$ are automorphisms. Therefore $\varphi$ is an
isomorphism. ∎

###  2.5. $r$-minimal model of a space

We say that $M$ is an $r$-minimal model of a finite quasimetric space $X,$ if
$M$ is $r$-minimal, and $X$ is $r$-homotopy equivalent to $M.$

######  Theorem 2.5.

Let $r\geq 0$ be a real number. Then the following holds.

  1. (1)

Any finite quasimetric space has an $r$-minimal model, which is unique up to
isometry.

  2. (2)

Two finite quasimetric spaces are $r$-homotopy equivalent if and only if their
$r$-minimal models are isometric.

  3. (3)

Any finite quasimetric space has an $r$-deformation retract, which is its
$r$-minimal model.

###### Proof.

For a finite quasimetric space $X$ we take any $r$-deformation retract
$M\subseteq X$ of the minimal possible cardinality. Denote the retraction by
$\rho:X\to M.$ We claim that $M$ is $r$-minimal. Indeed, if $A\subseteq M$ is
an $r$-deformation retract with a retraction $\rho^{\prime}:M\to A$ then the
equations $\rho\iota_{M,X}=1_{M}$ and $\rho^{\prime}\iota_{A,M}=1_{A}$ imply
$\rho^{\prime}\rho\iota_{A,X}=1_{A},$ and relations
$\iota_{M,X}\rho\sim_{r}1_{X}$ and $\iota_{A,M}\rho^{\prime}\sim_{r}1_{M}$
imply $\iota_{A,X}\rho^{\prime}\rho\sim_{r}1_{X}.$ Hence $A$ is an
$r$-deformation retract of $X,$ which is subset of $M,$ and using the
minimality of the cardinality of $M,$ we obtain $A=M.$

Proposition 2.4 implies that the $r$-minimal model is defined uniquely up to
isometry. It also implies that $r$-homotopy equivalent quasimetric spaces have
isomorphic $r$-minimal homotopy models. ∎

The $r$-minimal model of a quasimetric space $X$ is denoted by $M_{r}(X).$
Note that for $r>r^{\prime}$ we have

(2.5) $M_{r^{\prime}}(M_{r}(X))\cong M_{r^{\prime}}(X).$

######  Example 2.6.

Let $X$ be a finite metric space (with symmetric finite distance). For $r>{\rm
diam}(X),$ the metric space $M_{r}(X)$ is a one-point space. Here we see that
the embedding $\iota:M_{r}(X)\to X$ is not uniquely defined, because we can
take any map from the one-point space to $X$ as the inclusion.

######  Example 2.7.

Consider the following graph $X$

(2.6) $0$$4$$5$$1$$2$$3$

as a metric space with the shortest path distance. Take $r=1.$ Then the
$1$-minimal model $M_{1}(X)$ is the pentagon with the vertex set
$\\{1,2,3,4,5\\}.$ There are two choices of retractions for this
$1$-deformation retract $\rho,\rho^{\prime}:X\to M_{1}(X):$ $\rho(0)=1$ and
$\rho^{\prime}(0)=2.$ In particular, this example shows that the retraction
$\rho:X\to M_{r}(X)$ can’t be defined by a universal property in ${\sf QMet}$.

######  Example 2.8.

The author is grateful to Lev Mukoseev for the following non-trivial but
clarifying example. Consider the following digraph $X$ with four vertices
together with a subdigraph $A\subseteq X$ with tree vertices

(2.7) $X:\ \leavevmode\hbox to51.22pt{\vbox
to45.53pt{\pgfpicture\makeatletter\hbox{\hskip
25.61107pt\lower-22.76387pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
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It is easy to see that $A$ is a retract of $X,$ because there is a retraction
$\rho:X\to A$ such that $\rho(3)=1.$ We claim that $A$ is a $1$-deformation
retract of $X$ and $A=M_{1}(X).$ In particular

(2.8) $X\sim_{1}A.$

In order to prove it consider an endomorphism $\varphi:X\to X$ defined by

(2.9) $0\mapsto 2,\hskip 14.22636pt1\mapsto 0,\hskip 14.22636pt2\mapsto
1,\hskip 14.22636pt3\mapsto 0.$

It can be depicted as follows.

(2.10)
${0}$${1}$${3}$${2}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$

Then $d(\varphi,1_{X})=1$ and $\iota\rho=\varphi^{3}.$ It follows that
$\iota\rho=\varphi^{3}\sim_{1}\varphi^{2}\sim_{1}\varphi\sim_{1}1_{X}.$

######  Remark 2.9.

We say that a finite metric space $X$ is in general position if for any two
pairs of distinct points $x\neq x^{\prime}$ and $y\neq y^{\prime}$ such that
$\\{x,x^{\prime}\\}\neq\\{y,y^{\prime}\\}$ the distances are not equal
$d(x,x^{\prime})\neq d(y,y^{\prime}).$ Then for any finite metric space in
general position, if $M_{r}(X)$ consists of more then one point, then the
embedding $\iota:M_{r}(X)\to X$ is uniquely defined.

###  2.6. Homotopy jumping points

For a finite quasimetric space $X$ we consider a function from non-negative
real numbers to the interval $c_{X}:[0,+\infty)\to[0,1]$ defined as the ratio
of cardinalities

(2.11) $c_{X}(r)=\frac{|M_{r}(X)|}{|X|}.$

######  Lemma 2.10.

The function $c_{X}$ is decreasing, lower semi-continuous and it has a finite
number of values.

###### Proof.

If $r^{\prime}>r,$ then $M_{r^{\prime}}(M_{r}(X))\cong M_{r^{\prime}}(X),$ and
hence $c_{X}(r^{\prime})\leq c_{X}(r).$ Now assume that $r_{n}\to r$ and prove
that $c_{X}(r)\leq\underset{n\to+\infty}{\rm liminf}\>c_{X}(r_{n}).$ Since
$c_{X}$ is decreasing, we can assume that $r_{n}>r$ is an decreasing sequence.
Since $X$ is finite, for $n>\\!>1$ we have that for any two points
$x,x^{\prime}$ either $d(x,x^{\prime})\leq r$ or $d(x,x^{\prime})>r_{n}.$
Therefore, for $n>\\!>1$ the inequality $d(x,x^{\prime})\leq r_{n}$ is
equivalent to the inequality $d(x,x^{\prime})\leq r.$ Hence for $n>\\!>1$ we
have $M_{r}(X)=M_{r_{n}}(X)$ and $c_{X}(r_{n})=c_{X}(r).$ ∎

Lemma 2.10 implies that $c_{X}$ has a finite number of jumping points
$0<r_{1}<\dots<r_{n}$ such that $c_{X}(r_{i})<c_{X}(r)$ for any $r<r_{i}.$ We
will call them homotopy jumping points of $X$ and set

(2.12) ${\rm hj}(X)=\\{r_{1},\dots,r_{n}\\}.$

If we also set $r_{0}=0$ and $r_{n+1}=+\infty,$ then

(2.13) $M_{r}(X)\cong M_{r_{i}}(X),\hskip 28.45274ptr\in[r_{i},r_{i+1}).$

######  Remark 2.11.

Let us call a quasimetric space $X$ homotopy stable, if ${\rm
hj}(X)=\emptyset.$ In other words, $X$ is homotopy stable, if $M_{r}(X)=X$ for
any $r\geq 0.$ If we set ${\rm End}_{\rm fin}(X)=\bigcup_{r}{\rm End}_{r}(X),$
then Proposition 2.4 says that $X$ is homotopy stable if and only if ${\rm
End}_{\rm fin}(X)\subseteq{\rm Aut}(X).$ For example, the quasimetric spaces
defined by the digraphs

(2.14)
${\bullet}$${\bullet}$${\bullet}$${\bullet}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$
${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$

are homotopy stable.

Any finite quasimetric space $X$ with the set of homotopy jumping points
$\\{r_{1}<\dots<r_{n}\\}$ defines a homotopy stable finite quasimetric space
given by

(2.15) $M_{\rm stab}(X):=M_{r_{n}}(X),$

if $n>0,$ and $M_{\rm stab}(X)=X,$ if $n=0.$ For a finite quasimetric space
$X$ with a point $x_{0}\in X$ such that $d(x,x_{0})<+\infty$ for any $x\in X,$
the metric space $M_{\rm stab}(X)$ is always a one-point space.

######  Example 2.12 (Discontinuity of ${\rm hj}$ with respect to the
Hausdorff distance).

For any $0\leq\varepsilon<1/2$ we consider a subset of $\mathbb{R}^{2}$ of six
points

(2.16)
$X_{\varepsilon}=(\\{-2,2\\}\times\\{0,1\\})\cup\\{(-\varepsilon,0),(\varepsilon,1)\\},$

which can be depicted as follows.

(2.17) $\bullet$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\bullet$$\bullet$

Note that $\underset{\varepsilon\to 0+}{\lim}X_{\varepsilon}=X_{0}$ with
respect to the Hausdorff distance. For $\varepsilon=0$ we have
$M_{1}(X_{0})\cong\\{-2,0,2\\}\times\\{0\\},$ because there is a retraction
$\rho:X_{0}\to\\{-2,0,2\\}\times\\{0\\}$ defined by $\rho((a,b))=(a,0)$ such
that $\iota\rho\sim_{1}1_{X}.$ It is also easy to see that $M_{2}(X_{0})$ is a
one-point set. Using this we get

(2.18) ${\rm hj}(X_{0})=\\{1,2\\}.$

However, for $\varepsilon>0$ the direct verification using brute force shows
that for any non-identical short map $\varphi:X_{\varepsilon}\to
X_{\varepsilon}$ we have
$d(\varphi,1_{X})=d(1_{X},\varphi)\geq\sqrt{1+(2-\varepsilon)^{2}}.$ A non-
identical short map $\varphi:X_{\varepsilon}\to X_{\varepsilon}$ with the
minimal possible $d(1,\varphi)$ is the map that sends
$(-2,0),(-2,1),(-\varepsilon,0)$ to $(-\varepsilon,0)$ and sends
$(2,0),(2,1),(\varepsilon,1)$ to $(\varepsilon,1).$ Using this we obtain

(2.19) ${\rm hj}(X_{\varepsilon})=\\{\sqrt{1+(2-\varepsilon)^{2}}\\}.$

Since $\underset{\varepsilon\to
0+}{\lim}\\{\sqrt{1+(2-\varepsilon)^{2}}\\}=\\{\sqrt{5}\\}\neq\\{1,2\\}$, the
map $X\mapsto{\rm hj}(X)$ is not continuous with respect to the Hausdorff
distance.

##  3\. Magnitude-path spectral sequence of a digraph

###  3.1. $r$-minimal models of digraphs

We define a digraph as a pair $G=(G_{0},G_{1}),$ where $G_{1}\subseteq
G_{0}\times G_{0}$ such that $(g,g)\in G_{1}$ for any $g\in G_{0}.$ Elements
of $G_{0}$ are called vertices, elements of $G_{1}$ are called arrows, arrows
of the form $(g,g)$ are called degenerated arrows. A morphism of digraphs
$\varphi:G\to H$ is a map $\varphi:G_{0}\to H_{0}$ such that
$(g,g^{\prime})\in G_{1}$ implies $(\varphi(g),\varphi(g^{\prime}))\in H_{1}.$
The category of digraphs is denoted by ${\sf Digr}.$ A path in a digraph $G$
is a tuple $(g_{0},\dots,g_{n})$ such that $(g_{i},g_{i+1})\in G_{1}.$ Note
that any morphism sends a path to a path.

Any digraph $G$ defines a quasimetric space, that we denote by ${\rm Q}(G),$
whose set of points is $G_{0}$ and the distance from $g$ to $g^{\prime}$ is
defined as the infimum of the lengths of pathes from $g$ to $g^{\prime}.$
Therefore we get a functor

(3.1) ${\rm Q}:{\sf Digr}\longrightarrow{\sf QMet}.$

It is easy to check that the functor is full. So we can identify ${\sf Digr}$
with a full subcategory of ${\sf QMet}.$ So we can define $r$-homotopic maps
for digrphs.

A subdigraph $H$ of a digraph $G$ is called convex, if the shortest path
distance on $H$ coincides with the induced distance:
$d_{H}(h,h^{\prime})=d_{G}(h,h^{\prime})$ for any vertices $h,h^{\prime}$ of
$H.$ A subdigraph $H\subseteq G$ is called a retract, if there is a morphism
of digraphs $\rho:G\to H$ such that $\rho\iota=1_{H},$ where
$\iota:H\hookrightarrow G$ is the canonical embedding.

######  Lemma 3.1.

A retract of a digraph is a convex subdigraph.

###### Proof.

Let $H$ be a retract of $G.$ It is obvious that $d_{H}(h,h^{\prime})\geq
d_{G}(h,h^{\prime})$ for any $h,h^{\prime}\in H_{0}.$ On the other hand, for
any path $(g_{0},\dots,g_{n})$ from $h$ to $h^{\prime}$ in $G$ we have a path
$(\rho(g_{0}),\dots,\rho(g_{n}))$ from $h$ to $h^{\prime}$ in $H.$ Hence
$d_{H}(h,h^{\prime})\leq d_{G}(h,h^{\prime}).$ ∎

######  Lemma 3.2.

Let $G$ be a digraph and $A$ be a retract of ${\rm Q}(G).$ Then $A={\rm Q}(H)$
for a uniquely defined retract $H\subseteq G.$

###### Proof.

Consider a subdigraph $H\subseteq G$ such that $H_{0}=A$ and
$H_{1}=\\{(h,h^{\prime})\mid d_{G}(h,h^{\prime})\leq 1\\}.$ Denote by
$\rho:{\rm Q}(G)\to A$ a retraction. If $(g,g^{\prime})\in G_{1},$ then
$d_{G}(g,g^{\prime})\leq 1,$ and hence $(\rho(g),\rho(g^{\prime}))\in H_{1}.$
So $\rho$ defines a morphism of digraphs $\rho^{\prime}:G\to H.$ Therefore $H$
is a retract of $G.$ In particular $H$ is a convex subdigraph. It follows that
the distance between points in $A$ coincides with the distance in ${\rm
Q}(H).$ ∎

Lemma 3.2 implies that for a finite digraph $G$ and a natural $n$ there is a
digraph $M_{n}(G)$ such that ${\rm Q}(M_{n}(G))=M_{n}({\rm Q}(G)),$ that can
be included into $G$ as an $n$-deformation retract, but the embedding is not
unique. Therefore for a finite digraph we obtain a nested sequence of convex
sudigraphs

(3.2) $G\supseteq M_{1}(G)\supseteq M_{2}(G)\supseteq\dots,$

where the inclusions are not uniquely defined, but the digraphs $M_{n}(G)$ are
defined uniquely up to isomorphism.

###  3.2. Reminder on the magnitude-path spectral sequence

Here we remind the construction and the main properties of the magnitude-path
spectral sequence of a digraph that can be found in [2], [1], [15],[5].

Let $\mathbb{K}$ be a commutative ring. For a set $S$ we will denote by
$\mathbb{K}\cdot S$ the free $\mathbb{K}$-module freely generated by $S.$ For
a digraph $G$ we consider a filtered chain complex ${\rm RC}_{\bullet}(G)$
over $\mathbb{K},$ called reachability complex, whose $n$-th component is
freely generated by tuples of vertices $(g_{0},\dots,g_{n})$ without
consecutive repetitions $g_{i}\neq g_{i+1}$ such that
$d(g_{i},g_{i+1})<+\infty.$

(3.3) ${\rm RC}_{n}(G)=\mathbb{K}\cdot\\{(g_{0},\dots,g_{n})\mid
d(g_{i},g_{i+1})<+\infty,g_{i}\neq g_{i+1}\\}.$

Tuples $(g_{0},\dots,g_{n})$ with a consecutive repetition $g_{i}=g_{i+1}$
will be identified with zero in the chain complex. Then the differential is
defined by the formula

(3.4)
$\partial(g_{0},\dots,g_{n})=\sum_{i=0}^{n}(-1)^{i}(g_{0},\dots,\hat{g}_{i},\dots,g_{n}).$

If we set $L(g_{0},\dots,g_{n})=\sum_{i}d(g_{i},g_{i+1}),$ the filtration on
the chain complex is defined so that for an integer $\ell$ the component
$F_{\ell}{\rm RC}_{n}(G)$ is generated by tuples $(g_{0},\dots,g_{n})$ such
that $L(g_{0},\dots,g_{n})\leq\ell.$

(3.5) $F_{0}{\rm RC}(G)\subseteq F_{1}{\rm RC}(G)\subseteq\dots\subseteq{\rm
RC}(G).$

The spectral sequence associated with this filtered chain complex is called
magnitude-path spectral sequence of $G$ and will be denoted by $E(G).$ Its
first page consists of the magnitude homology

(3.6) $E^{1}_{\ell,n-\ell}(G)={\rm MH}_{n,\ell}(G),$

the zero row of the second page is the path homology

(3.7) $E^{2}_{n,0}(G)={\rm PH}_{n}(G).$

This spectral sequence converges to the homology of the preorder ${\rm
Pre}(G),$

(3.8) $E(G)\Rightarrow H_{*}({\rm Pre}(G))$

which is defined so that its elements are points of $G$ and $g\leq g^{\prime}$
if and only if $d(g,g^{\prime})<+\infty.$ The homology of the preorder are
also known as reachability homology

(3.9) ${\rm RH}_{n}(G)=H_{n}({\rm Pre}(G)).$

The $r+1$-st page of the magnitude-path spectral sequence is an $r$-homotopy
invariant in the following sense. If $\varphi\sim_{r}\psi:G\to H$ two
$r$-homotopy invariant morphisms, then they induce equal morphisms on the
$r+1$-st pages

(3.10) $E^{r+1}(\varphi)=E^{r+1}(\psi):E^{r+1}(G)\longrightarrow E^{r+1}(H)$

(see [1, Th. 4.37]).

###  3.3. Decomposition of the magnitude-path spectral sequence

######  Theorem 3.3.

Let $G$ be a finite digraph and let ${\rm hj}(G)=\\{r_{1}<\dots<r_{n}\\}$ be
the set of homotopy jumping points of $G$. Then the magnitude-path spectral
sequence $E(G)$ can be decomposed into a direct sum of spectral sequences

(3.11) $E(G)=\bigoplus_{k=1}^{n}E(G)_{[k]}\oplus E(G)_{[\infty]}$

such that $E^{r_{k}+1}(G)_{[k]}=0$ and for any $1\leq j\leq n$ we have

(3.12) $E(M_{r_{j}}(G))\cong\bigoplus_{k=j+1}^{n}E(G)_{[k]}\oplus
E(G)_{[\infty]}.$

In particular, $E(M_{\rm stab}(G))=E(G)_{[\infty]}.$

###### Proof.

It is easy to see that, if $E$ is a spectral sequence and $e:E\to E$ is an
idempotent morphism of spectral sequences, then $E$ can be decomposed into a
sum of spectral sequences $E=\mathrm{Ker}(e)\oplus\mathrm{Im}(e).$ In other
words, the category of spectral sequences is Karoubian. Therefore, if
$E^{\prime}$ is a retract of a spectral sequence $E,$ then $E$ is a direct sum
of spectral sequences $E\cong E^{\prime}\oplus E^{\prime\prime}.$

If $n=0,$ then there is nothing to prove. Assume $n\geq 1.$ Chose a sequence
of nested minimal models

(3.13) $G=M_{r_{0}}(G)\supset M_{r_{1}}(G)\supset
M_{r_{2}}(G)\supset\dots\supset M_{r_{n}}(G)$

with retractions $\rho_{j}:M_{r_{j}}(G)\to M_{r_{j+1}}(G)$ for $0\leq j\leq
n-1.$ Since $M_{r_{j+1}}(G)$ is a retract in $M_{r_{j}}(G),$ we obtain that
$E(M_{r_{j+1}})$ is a retract of $E(M_{r_{j}}).$ Hence $E(M_{r_{j}})\cong
E(G)_{[j+1]}\oplus E(M_{r_{j+1}}).$ Then we obtain a decomposition (3.11) with
the property (3.12). Since the embedding $M_{r_{j+1}}(G)\hookrightarrow
M_{r_{j}}(G)$ is an $r_{j+1}$-homotopy equivalence, we obtain that
$E^{r_{j+1}+1}(M_{r_{j+1}}(G))\cong E^{r_{j+1}+1}(M_{r_{j}}(G)).$ Therefore
$E^{r_{j+1}+1}(G)_{[j+1]}=0.$ ∎

######  Remark 3.4.

We do not claim that the decomposition in Theorem 3.3 is natural.

##  4\. Spectral homology

###  4.1. Reminder on spectral sequence of a $\mathbb{Z}$-filtered chain
complex

In this subsection we remind an explicit construction of pages of the spectral
sequence associated with a filtered chain complex that can be found in [4,
Ch.XV,§1]. We will use the following lemma from the book of Cartan-Eilenberg.

######  Lemma 4.1 ([4, Ch.XV,Lemma 1.1]).

For any commutative diagram of $\mathbb{K}$-modules, in which the row is exact

(4.1)
${B}$${A^{\prime}}$${A}$${A^{\prime}}$$\leftarrow$$\rightarrow$$\scriptstyle{\varphi}$$\leftarrow$$\rightarrow$$\scriptstyle{\psi}$$\leftarrow$$\rightarrow$$\scriptstyle{\varphi^{\prime}}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\scriptstyle{\eta}$

there is an inclusion
$\mathrm{Im}\>\varphi\subseteq\mathrm{Im}\>\varphi^{\prime}$ and the map
$\eta$ induces an isomorphism

(4.2)
$\mathrm{Im}\>\psi\cong\mathrm{Im}\>\varphi/\mathrm{Im}\>\varphi^{\prime}.$

Let $\mathbb{K}$ be a commutative ring and $C$ be a $\mathbb{Z}$-filtered
chain complex over $\mathbb{K}$ with an increasing filtration

(4.3) $\dots\subseteq F_{-1}C\subseteq F_{0}C\subseteq
F_{1}C\subseteq\dots\subseteq C.$

For the sake of simplicity we set

(4.4) $F_{\ell}=F_{\ell}C.$

Consider the associated spectral sequence $E.$ Here we will not discuss the
convergence of the spectral sequence, so we do not need any assumptions on the
filtration. The first page of this spectral sequence is described as

(4.5) $E^{1}_{\ell,n-\ell}=H_{n}(F_{\ell}/F_{\ell-1}).$

Let us describe all pages as subquotients of the first page. For any natural
$r\geq 1$ we consider the following short exact sequences induced by the
embeddings

(4.6) $0\longrightarrow F_{\ell-1}/F_{\ell-r}\longrightarrow
F_{\ell}/F_{\ell-r}\longrightarrow F_{\ell}/F_{\ell-1}\longrightarrow 0$

and

(4.7) $0\longrightarrow F_{\ell}/F_{\ell-1}\longrightarrow
F_{\ell+r-1}/F_{\ell-1}\longrightarrow F_{\ell+r-1}/F_{\ell}\longrightarrow
0,$

and the associated long exact sequences of homology. Further we set

(4.8)
$\begin{split}Z^{r}_{\ell,n-\ell}&=\mathrm{Im}(H_{n}(F_{\ell}/F_{\ell-r})\longrightarrow
H_{n}(F_{\ell}/F_{\ell-1}))\\\
&=\mathrm{Ker}(H_{n}(F_{\ell}/F_{\ell-1})\longrightarrow
H_{n-1}(F_{\ell-1}/F_{\ell-r})).\end{split}$

and

(4.9)
$\begin{split}B^{r}_{\ell,n-\ell}&=\mathrm{Ker}(H_{n}(F_{\ell}/F_{\ell-1})\longrightarrow
H_{n}(F_{\ell+r-1}/F_{\ell-1}))\\\
&=\mathrm{Im}(H_{n+1}(F_{\ell+r-1}/F_{\ell})\longrightarrow
H_{n}(F_{\ell}/F_{\ell-1})).\end{split}$

Then the $r$-th page of the spectral sequence can be described as

(4.10) $E^{r}_{\ell,n-\ell}=\frac{Z^{r}_{\ell,n-\ell}}{B^{r}_{\ell,n-\ell}}.$

Using the morphism of short exact sequences,

(4.11)
${0}$${F_{\ell}/F_{\ell-r}}$${F_{\ell+r-1}/F_{\ell-r}}$${F_{\ell+r-1}/F_{\ell}}$${0}$${0}$${F_{\ell}/F_{\ell-1}}$${F_{\ell+r-1}/F_{\ell-1}}$${F_{\ell+r-1}/F_{\ell}}$${0}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\twoheadrightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\twoheadrightarrow$$\leftarrow$$\rightarrow$$\Leftarrow$$\Leftarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$

we obtain a commutative diagram, in which the row is exact.

(4.12)
${H_{n}(F_{\ell}/F_{\ell-r})}$${H_{n+1}(F_{\ell+r-1}/F_{\ell})}$${H_{n}(F_{\ell}/F_{\ell-1})}$${H_{n}(F_{\ell+r-1}/F_{\ell-1})}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$

Applying Lemma 4.1 to this diagram, we obtain another description of the
$E^{r}$ (see [4, Ch.XV,§1,(8)])

(4.13)
$E^{r}_{\ell,n-\ell}\cong\mathrm{Im}(H_{n}(F_{\ell}/F_{\ell-r})\longrightarrow
H_{n}(F_{\ell+r-1}/F_{\ell-1})).$

###  4.2. Spectral homology of $\mathbb{R}$-filtered chain complexes

A subset of the set of real numbers $I\subseteq\mathbb{R}$ is called an
interval if it is convex subset. We say that a interval $L$ is left infinite,
if for any $i\in I$ and $r<i$ we have $r\in L.$ There are four types of left
infinite intervals: $(-\infty,a),(-\infty,a],\emptyset$ and $\mathbb{R}.$ Any
non-empty interval $I\subseteq\mathbb{R}$ can be uniquely presented as the
difference of left infinite intervals $I=R_{I}\setminus L_{I},$ where

(4.14) $R_{I}=\\{r\in\mathbb{R}\mid\exists i\in I\ r\leq i\\},\hskip
28.45274ptL_{I}=\\{r\in\mathbb{R}\mid\forall i\in I\ r<i\\}.$

For example $R_{\\{\ell\\}}=(-\infty,\ell]$ and
$L_{\\{\ell\\}}=(-\infty,\ell).$ We define an order on the set of non-empty
intervals such that $I\preccurlyeq J$ if and only if $R_{I}\subseteq R_{J}$
and $L_{I}\subseteq L_{J}.$

For an interval $I=R\setminus L$ and a real number $r$ we also set
$I+r=\\{i+r\mid i\in I\\}$ and

(4.15) $I_{r}=R\setminus(L-r),\hskip 28.45274ptI^{r}=(R+r)\setminus L.$

Note that

(4.16) $I_{r}\preccurlyeq I\preccurlyeq I^{r}.$

Assume that $C$ is an $\mathbb{R}$-filtered chain complex over a ring
$\mathbb{K}$. So for any $r\in\mathbb{R}$ we have a chain subcomplex
$F_{r}C\subseteq C$ such that $r<r^{\prime}$ implies $F_{r}C\subseteq
F_{r^{\prime}}C.$ When $C$ is fixed we will omit it in notations
$F_{r}=F_{r}C.$ For a left infinite interval $L$ we set

(4.17) $F_{L}=\bigcup_{r\in L}F_{r}$

Further, for a non-empty interval $I=R\setminus L$ we set

(4.18) $C_{I}=F_{R}/F_{L}.$

If $I\preccurlyeq J,$ we have a morphism

(4.19) $C_{I}\longrightarrow C_{J}$

induced by inclusions $F_{R_{I}}C\subseteq F_{R_{J}}C$ and
$F_{L_{I}}C\subseteq F_{L_{J}}C$. Moreover, if $L_{I}=L_{J},$ this morphism is
a monomorphism, and, if $R_{I}=R_{J},$ it is an epimorphism. If we have tree
distinct left inverse intervals $R\supset S\supset L,$ then these morphisms
define a short exact sequence of chain complexes

(4.20) $0\longrightarrow C_{S\setminus L}\longrightarrow C_{R\setminus
L}\longrightarrow C_{R\setminus S}\longrightarrow 0.$

The $0$-spectral homology of $C$ is defined as follows

(4.21) $\mathrm{SH}^{0}_{n,I}(C)=H_{n}(C_{I}).$

Further for any $r\geq 0$ we will define $r$-spectral homology
$\mathrm{SH}^{r}_{n,I}(C)$ as a subquotient of $\mathrm{SH}^{0}_{n,I}(C)$

For any $r\geq 0$ we consider short exact sequences of chain complexes of the
form (4.20) similar to (4.6), (4.7)

(4.22) $0\longrightarrow C_{I_{r}\setminus I}\longrightarrow
C_{I_{r}}\longrightarrow C_{I}\longrightarrow 0,$ (4.23) $0\longrightarrow
C_{I}\longrightarrow C_{I^{r}}\longrightarrow C_{I^{r}\setminus
I}\longrightarrow 0$

and the associated long exact sequences of homology. Then we define
$r$-spectral cycles and $r$-spectral boundaries as

(4.24) $\begin{split}{\rm
SZ}^{r}_{n,I}(C)&=\mathrm{Im}(H_{n}(C_{I_{r}})\longrightarrow H_{n}(C_{I}))\\\
&=\mathrm{Ker}(H_{n}(C_{I})\longrightarrow H_{n-1}(C_{I_{r}}))\end{split}$

and

(4.25) $\begin{split}{\rm
SB}^{r}_{n,I}(C)&=\mathrm{Ker}(H_{n}(C_{I})\longrightarrow
H_{n}(C_{I^{r}}))\\\ &=\mathrm{Im}(H_{n+1}(C_{I^{r}\setminus
I})\longrightarrow H_{n}(C_{I})).\end{split}$

######  Proposition 4.2.

There is an inclusion

(4.26) ${\rm SB}^{r}_{n,I}(C)\subseteq{\rm SZ}^{r}_{n,I}(C).$

Moreover, if we set

(4.27) $\mathrm{SH}^{r}_{n,I}(C)={\rm SZ}^{r}_{n,I}(C)/{\rm SB}^{r}_{n,I}(C),$

then the map $C_{I}\to C_{I^{r}}$ induces an isomorphism

(4.28)
$\mathrm{SH}^{r}_{n,I}(C)\cong\mathrm{Im}(H_{n}(C_{I_{r}})\longrightarrow
H_{n}(C_{I^{r}})).$

###### Proof.

Using that $I^{r}\setminus I=(I^{r}\cup I_{r})\setminus I_{r},$ we obtain that
there is a morphism of short exact sequences of the form (4.20)

(4.29) ${0}$${C_{I_{r}}}$${C_{I^{r}\cup I_{r}}}$${C_{I^{r}\setminus
I}}$${0}$${0}$${C_{I}}$${C_{I^{r}}}$${C_{I^{r}\setminus
I}}$${0}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\twoheadrightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\twoheadrightarrow$$\leftarrow$$\rightarrow$$\Leftarrow$$\Leftarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$

Therefore there is an commutative diagram, in which the row is exact.

(4.30) ${H_{n}(C_{I_{r}})}$${H_{n+1}(C_{I^{r}\setminus
I})}$${H_{n}(C_{I})}$${H_{n}(C_{I^{r}})}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$

Then the assertion follows from Lemma 4.1. ∎

The $\mathbb{K}$-module $\mathrm{SH}^{r}_{n,I}(C)$ defined in Proposition 4.2
is called _$r$ -spectral homology_ of the $\mathbb{R}$-filtered chain complex
$C.$ A morphism of $\mathbb{R}$-filtered chain complexes $f:C\to C^{\prime}$
is a chain map such that $f(F_{\ell}C)\subseteq F_{\ell}C^{\prime}.$ It is
easy to see that the construction is natural and defines a functor

(4.31) $\mathrm{SH}^{r}_{n,I}:\mathbb{R}\text{-}{\sf
FiltCh(\mathbb{K})}\longrightarrow{\sf Mod}(\mathbb{K}).$

######  Corollary 4.3.

There are isomorphisms

(4.32) $\begin{split}{\rm
SZ}^{r}_{n,I}(C)&=\mathrm{Im}(\mathrm{SH}^{0}_{n,I_{r}}(C)\longrightarrow\mathrm{SH}^{0}_{n,I}(C)),\\\
{\rm
SB}^{r}_{n,I}(C)&=\mathrm{Ker}(\mathrm{SH}^{0}_{n,I}(C)\longrightarrow\mathrm{SH}^{0}_{n,I^{r}}(C),\\\
\mathrm{SH}^{r}_{n,I}(C)&\cong\mathrm{Im}(\mathrm{SH}^{0}_{n,I_{r}}(C)\longrightarrow\mathrm{SH}^{0}_{n,I^{r}}(C)).\end{split}$

######  Proposition 4.4.

Let $C$ be a $\mathbb{Z}$-filtered chain complex. We extend the
$\mathbb{Z}$-filtration to an $\mathbb{R}$-filtration on $C$ by the formula

(4.33) $F_{r}C=F_{\lfloor r\rfloor}C.$

Then for a natural number $r\geq 1$ we have

(4.34)
$E^{r}_{n,\ell-n}(C)=\mathrm{SH}^{r-1}_{n,(\ell-1,\ell]}(C)=\mathrm{SH}^{r-1}_{n,\\{\ell\\}}(C).$

###### Proof.

For $I=(\ell-1,\ell]$ we obtain $R=(-\infty,\ell-1]$ and $L=(-\infty,\ell].$
Therefore, $F_{R}=F_{\ell},$ $F_{L}=F_{\ell-1},$ $F_{L-(r-1)}=F_{\ell-r},$
$F_{R+(r-1)}=F_{\ell+r-1}.$ Then the short exact sequences (4.6), (4.7)
coincide with short exact sequences (4.22), (4.23).

Similarly for $I=\\{\ell\\}.$ We obtain $R=(-\infty,\ell]$ and
$L=(-\infty,\ell).$ Therefore $F_{L}=F_{(-\infty,\ell)}=F_{\ell-1},$
$F_{L-(r-1)}=F_{(-\infty,\ell-r+1)}=F_{\ell-r}.$ ∎

######  Proposition 4.5.

Let $C$ be an $\mathbb{R}$-filtered complex such that
$C=\bigcup_{\ell}F_{\ell}C.$ Then for any $r\geq 0$ we have

(4.35) $H_{n}(C)=\mathrm{SH}^{r}_{n,\mathbb{R}}(C).$

###### Proof.

In this case $R=R+r=\mathbb{R}$ and $L=L-r=\emptyset.$ Therefore
$F_{R}=F_{R+r}=C$ and $F_{L}=F_{L-r}=0.$ The assertion follows. ∎

######  Remark 4.6 (Persistent spectral homology).

Note that, if $I\preccurlyeq J,$ then $I^{r}\preccurlyeq J^{r}$ and
$I_{r}\preccurlyeq J_{r}.$ Therefore, the commutative diagram

(4.36)
${H_{n}(C_{I_{r}})}$${H_{n}(C_{I^{r}})}$${H_{n}(C_{J_{r}})}$${H_{n}(C_{J^{r}})}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$

defines a morphism

(4.37)
$f_{I,J}:\mathrm{SH}^{r}_{n,I}(C)\longrightarrow\mathrm{SH}^{r}_{n,J}(C).$

Moreover, it is easy to see that for $I\preccurlyeq J\preccurlyeq K$ we have
$f_{J,K}f_{I,J}=f_{I,K}.$ In particular, it follows that we can consider
persistent module ${\rm PSH}^{r}_{n}(C)$ such that

(4.38) ${\rm PSH}^{r}_{n}(C)(\ell)=\mathrm{SH}^{r}_{n,(-\infty,\ell]}(C).$

###  4.3. $r$-homotopy of $\mathbb{R}$-filtered chain complexes

Similarly to [4, Ch.XV,§3] for $r\geq 0$ we say that two morphisms of
$\mathbb{R}$-filtered chain complexes $f,f^{\prime}:C\to C^{\prime}$ are
$r$-homotopic, if there is a morphism of graded modules of degree one
$h:C_{*}\to C^{\prime}_{*+1}$ such that $\partial h+h\partial=f-f^{\prime}$
and $h(F_{\ell}C)\subseteq F_{\ell+r}C^{\prime}.$

######  Proposition 4.7 (cf. [4, Ch.XV,Prop.3.1]).

Any $r$-homotopic morphisms of $\mathbb{R}$-filtered chain complexes
$f,f^{\prime}:C\to C^{\prime}$ induce equal morphisms on the $r$-spectral
homology

(4.39)
$\mathrm{SH}_{n,I}^{r}(f)=\mathrm{SH}_{n,I}^{r}(f^{\prime}):\mathrm{SH}_{n,I}^{r}(C)\longrightarrow\mathrm{SH}_{n,I}^{r}(C^{\prime}).$

###### Proof.

We present $I$ as the difference of left infinite intervals $I=R\setminus L.$
We set $F_{X}=F_{X}C$ and $F^{\prime}_{X}=F_{X}C^{\prime}.$ Then

(4.40)
$\begin{split}\mathrm{SH}^{r}_{n,I}(C)&\cong\mathrm{Im}(\alpha_{*}:H_{n}(F_{R}/F_{L-r})\longrightarrow
H_{n}(F_{R+r}/F_{L})),\\\
\mathrm{SH}^{r}_{n,I}(C^{\prime})&\cong\mathrm{Im}(\alpha^{\prime}_{*}:H_{n}(F^{\prime}_{R}/F^{\prime}_{L-r})\longrightarrow
H_{n}(F^{\prime}_{R+r}/F^{\prime}_{L})).\end{split}$

Here we denote by $\alpha$ and $\alpha^{\prime}$ the morphisms of chain
complexes induced by embeddings

(4.41) $\alpha:F_{R}/F_{L-r}\to F_{R+r}/F_{L},\hskip
28.45274pt\alpha^{\prime}:F^{\prime}_{R}/F^{\prime}_{L-r}\to
F^{\prime}_{R+r}/F^{\prime}_{L}.$

Set $g=f-f^{\prime}$ and consider the commutative diagram

(4.42)
${H_{n}(F_{R}/F_{L-r})}$${H_{n}(F_{R+r}/F_{L})}$${H_{n}(F^{\prime}_{R}/F^{\prime}_{L-r})}$${H_{n}(F^{\prime}_{R+r}/F^{\prime}_{L}).}$$\leftarrow$$\rightarrow$$\scriptstyle{\alpha_{*}}$$\leftarrow$$\rightarrow$$\scriptstyle{g_{*}}$$\leftarrow$$\rightarrow$$\scriptstyle{g_{*}}$$\leftarrow$$\rightarrow$$\scriptstyle{\alpha^{\prime}_{*}}$

Then it is sufficient to prove that $g_{*}\alpha=\alpha^{\prime}g^{*}=0.$ Take
an element $x\in(F_{R})_{n}$ that represents an element from
$H_{n}(F_{R}/F_{L-r}).$ Then $\partial(x)\in(F_{L-r})_{n-1}.$ Since
$g=\partial h+h\partial,$ we have

(4.43) $g(x)=\partial h(x)+h\partial(x).$

Using that $h$ is an $r$-homotopy and $\partial(x)\in(F_{L-r})_{n-1},$ we
obtain $h\partial(x)\in(F^{\prime}_{L})_{n}.$ Hence $h\partial(x)$ represents
a zero element in $H_{n}(F^{\prime}_{R+r}/F^{\prime}_{L}).$ Using that $h$ is
an $r$-homotopy once again, we obtain $h(x)\in(F^{\prime}_{R+r})_{n+1}.$
Therefore $\partial h(x)$ represents a zero element in
$H_{n}(F^{\prime}_{R+r}/F^{\prime}_{L})$ too. It follows that $g(x)$
represents zero element in $H_{n}(F^{\prime}_{R+r}/F^{\prime}_{L}).$ So we
obtain $\alpha^{\prime}_{*}g_{*}=0.$ ∎

###  4.4. Spectral homology of quasimetric spaces

Let $\mathbb{K}$ be a commutative ring and $X$ be a quasimetric space. Define
the $\mathbb{R}$-filtered reachability chain complex ${\rm RC}(X)$ similarly
to the case of digraphs. Its $n$-th component is freely generated by tuples
$(x_{0},\dots,x_{n})$ without consecutive repetitions $x_{i}\neq x_{i+1}$ such
that $d(x_{i},x_{i+1})<+\infty.$

(4.44) ${\rm RC}_{n}(X)=\mathbb{K}\cdot\\{(x_{0},\dots,x_{n})\mid
d(x_{i},x_{i+1})<+\infty,x_{i}\neq x_{i+1}\\}.$

In this complex we identify sequences with consecutive repetitions with zero.
Then the differential is defined by the formula

(4.45)
$\partial(x_{0},\dots,x_{n})=\sum_{i=0}^{n}(-1)^{i}(x_{0},\dots,\hat{x}_{i},\dots,x_{n}).$

If we set $L(x_{0},\dots,x_{n})=\sum_{i}d(x_{i},x_{i+1}),$ the
$\mathbb{R}$-filtration is defined so that for a real number $r$ the component
$F_{r}{\rm RC}_{n}(G)$ is generated by tuples $(x_{0},\dots,x_{n})$ such that
$L(x_{0},\dots,x_{n})\leq r.$

For a non-empty interval $I,$ a real number $r\geq 0,$ and natural $n$ the
spectral homology of $X$ is defined as the spectral homology of this
$\mathbb{R}$-filtered chain complex

(4.46) $\mathrm{SH}^{r}_{n,I}(X)=\mathrm{SH}^{r}_{n,I}({\rm RC}(X)).$

It is easy to see that these constructions are natural and they define
functors

(4.47) $\mathrm{SH}^{r}_{n,I}:{\sf QMet}\longrightarrow{\sf Mod}(\mathbb{K}).$

By Remark 4.6 the spectral homology is natural by the interval. In particular,
we can consider the persistent spectral homology ${\rm PSH}^{r}_{n}(X)$ which
is a persistent module such that ${\rm
PSH}^{r}_{n}(X)=\mathrm{SH}^{r}_{n,[0,\ell]}(X).$

######  Theorem 4.8.

Any $r$-homotopic morphisms of quasimetric spaces $\varphi,\psi:X\to Y$ induce
equal morphisms on $r$-spectral homology

(4.48)
$\mathrm{SH}^{r}_{n,I}(\varphi)=\mathrm{SH}^{r}_{n,I}(\psi):\mathrm{SH}^{r}_{n,I}(X)\longrightarrow\mathrm{SH}^{r}_{n,I}(Y)$

for any $n$ and $I$.

###### Proof.

The differential in the reachability complex is defined by the formula
$\partial_{n}=\sum_{i}(-1)^{i}\partial_{n,i},$ where

(4.49)
$\partial_{n,i}(x_{0},\dots,x_{n})=(x_{0},\dots,\hat{x}_{i},\dots,x_{n}).$

Set $f=\mathrm{RC}(\varphi)$ and $f^{\prime}=\mathrm{RC}(\psi).$ For $0\leq
j\leq n$ we consider the map
$h_{n,j}:\mathrm{RC}_{n}(X)\to\mathrm{RC}_{n+1}(Y)$ given by

(4.50)
$h_{n,j}(x_{0},\dots,x_{n})=(\varphi(x_{0}),\dots,\varphi(x_{j}),\psi(x_{j}),\dots,\psi(x_{n}))).$

These maps satisfy simplicial homotopy relations

(4.51) $\partial_{n+1,0}h_{n,0}=f^{\prime}_{n},\hskip
28.45274pt\partial_{n+1,n+1}h_{n,n}=f_{n},$ (4.52)
$\partial_{n+1,i}h_{n,j}=\begin{cases}h_{n-1,j-1}\partial_{n,i},&i<j,\\\
\partial_{n,i}h_{n,i-1},&i=j\neq 0,\\\
h_{n-1,j}\partial_{n,i-1},&i>j+1.\end{cases}$

Therefore, if we set $h_{n}=\sum_{j}(-1)^{j}h_{n,j},$ we obtain
$\partial_{n+1}h_{n}+h_{n-1}\partial_{n}=f^{\prime}_{n}-f_{n}.$

If $d(\varphi,\psi)\leq r,$ then

(4.53) $L(h_{n,j}(x_{0},\dots,x_{n}))\leq L(x_{0},\dots,x_{n})+r.$

It follows that $h:\mathrm{RC}_{*}(X)\to\mathrm{RC}_{*+1}(Y)$ is an
$r$-homotopy between morphisms $\mathbb{R}$-filtered chain complexes. Hence
the statement follows from Proposition 4.7. ∎

######  Corollary 4.9.

If $X$ and $Y$ are $r$-homotopy equivalent quasimetric spaces, then

(4.54) $\mathrm{SH}^{r}_{n,I}(X)\cong\mathrm{SH}^{r}_{n,I}(Y).$

In particular, if $X$ is finite, then
$\mathrm{SH}^{r}_{n,I}(X)\cong\mathrm{SH}^{r}_{n,I}(M_{r}(X)).$

###  4.5. Relation of spectral homology with other types of homology

Let $X$ be a quasimetric space. Then in our notation its magnitude homology
can be defined as

(4.55)
$\mathrm{MH}_{n,\ell}(X)=H_{n}\left(\frac{F_{\ell}\mathrm{RC}(X)}{F_{(-\infty,\ell)}\mathrm{RC}(X)}\right),$

and the blurred magnitude homology [6] are defined as a persistent module such
that

(4.56) ${\rm BMH}_{n}(X)(\ell)=H_{n}(F_{\ell}\mathrm{RC}(X)).$

######  Theorem 4.10.

Let $X$ be a quasimetric space and $G$ be a digraph. Then the following holds.

  1. (1)

The magnitude homology can be presented as $0$-spectral homology

${\rm MH}_{n,\ell}(X)\cong\mathrm{SH}^{0}_{n,\\{\ell\\}}(X).$

In particular, it follows that for any $r\geq 0$ the group
$\mathrm{SH}^{r}_{n,\\{\ell\\}}(X)$ is an $r$-homotopy invariant subquotient
of ${\rm MH}_{n,\ell}(X).$

  2. (2)

The persistent module of blurred magnitude homology (see [6]) can be presented
as persistent $0$-spectral homology

${\rm BMH}_{n}(X)\cong{\rm PSH}^{0}_{n}(X).$

In particular, it follows that for any $r\geq 0$ the persistent module ${\rm
PSH}^{r}_{n}(X)$ is an $r$-homotopy invariant subquotient persistent module of
${\rm BHM}_{n}(X).$

  3. (3)

The reachability homology (see [15]) can be presented as $r$-spectral homology

${\rm RH}_{n}(X)\cong\mathrm{SH}^{r}_{n,\mathbb{R}}(X)$

for any $r\geq 0.$

  4. (4)

For any integer $r\geq 1$ the $r$-th page of the magnitude-path spectral
sequence of $G$ can be presented as $(r-1)$-spectral homology

$E^{r}_{n,\ell-n}(G)\cong\mathrm{SH}^{r-1}_{n,\\{\ell\\}}(G).$

  5. (5)

In particular, path homology can be described as

${\rm PH}_{n}(G)=\mathrm{SH}^{1}_{n,\\{n\\}}(G).$

###### Proof.

(1) and (2) obviously follow from the definitions equation (4.21) and Remark
4.6. (3) follows from Proposition 4.5. (4) follows from Proposition 4.4. ∎

###  4.6. Low-dimensional spectral homology

In this section we fix a quasimetric space $X$ and we will omit it in some
notations.

######  Proposition 4.11.

Let $I$ be an interval and $r\geq 0$ be a real number. Consider an equivalence
relation $\sim$ on $X$ such that $x\sim y$ if and only if $x\sim_{s}y$ for
some $s\in I^{r}.$ Then, assuming $0\in I,$ we have

(4.57) $\mathrm{SH}^{r}_{0,I}(X)\cong\mathbb{K}\cdot(X/\sim).$

In particular,

(4.58) $\mathrm{SH}^{r}_{0,\\{0\\}}(X)\cong\mathbb{K}\cdot(X/\sim_{r}).$

If $0\notin I,$ then $\mathrm{SH}^{r}_{0,I}(X)=0.$

###### Proof.

For the proof we denote by $\sim_{I}$ the equivalence relation on $X$ such
that $x\sim_{I}y$ if and only if there is $s\in I$ such that $x\sim_{s}y.$ We
set $C_{I}=F_{R_{I}}\mathrm{RC}(X)/F_{L_{I}}\mathrm{RC}(X).$ If $0\notin I,$
then $(C_{I})_{0}=0$ and $\mathrm{SH}^{r}_{0,I}(X)=0.$ If $0\in I,$ we have
$(C_{I})_{0}=\mathbb{K}\cdot X$ and $(C_{I})_{1}=\mathbb{K}\cdot\\{(x,y)\mid
d(x,y)\in I\\}.$ Therefore
$\mathrm{Im}(\partial_{1}:(C_{I})_{1}\to(C_{I})_{0})$ is spanned by
differences $x-y,$ where $d(x,y)\in I.$ It follows that
$\mathrm{SH}^{0}_{0,I}(X)=\mathbb{K}\cdot(X/\sim_{I}).$ Hence
$\mathrm{SH}^{r}_{0,I}(X)=\mathrm{Im}(\mathbb{K}\cdot(X/\sim_{I_{r}})\to\mathbb{K}\cdot(X/\sim_{I^{r}}))=\mathbb{K}\cdot(X/\sim_{I^{r}}).$
∎

A pair of points $(x,y)$ of a quasimetric space $X$ is called $\ell$-adjacent,
if $d(x,y)=\ell$ and there is no a point $a$ such that $d(x,a)+d(a,y)=\ell$
and $a\neq x,y.$ The set of $\ell$-adjacent pairs is denoted by ${\rm
Adj}_{\ell}.$ Then it is known [23, Cor. 4.5] that

(4.59) $\mathrm{MH}_{1,\ell}(X)\cong\mathbb{K}\cdot{\rm Adj}_{\ell}.$

We will prove a generalization of this statement describing
$\mathrm{SH}^{r}_{1,\\{\ell\\}}(X)$ for $\ell>r\geq 0.$

Let $\ell>r\geq 0.$ Denote by $P_{\leq\ell+r}$ the set of all pairs of points
$(x,y)$ such that $d(x,y)\leq\ell+r.$

(4.60) $P_{\leq\ell+r}=\\{(x,y)\in X^{2}\mid d(x,y)\leq\ell+r\\}.$

We consider the minimal equivalence relation $\sim$ on $P_{\leq\ell+r}$ such
that the following holds

  * •

$(x,y)\sim(x^{\prime},y),$ if $d(x,x^{\prime})\leq r$ and
$d(x,x^{\prime})+d(x^{\prime},y)\leq\ell+r,$

  * •

$(x,y)\sim(x,y^{\prime}),$ if $d(y^{\prime},y)\leq r$ and
$d(x,y^{\prime})+d(y^{\prime},y)\leq\ell+r.$

We say that a pair $(x,y)\in P_{\leq\ell+r}$ is $(\ell,r)$-trivial, if there
exists a point $a$ such that

(4.61) $d(x,a)+d(a,y)\leq\ell+r,\hskip 14.22636ptd(x,a)<\ell,\hskip
14.22636ptd(a,y)<\ell.$

In particular, if $d(x,y)<\ell,$ then $(x,y)$ is $(\ell,r)$-trivial. We denote
the set of $(\ell,r)$-trivial pairs by

(4.62) $T_{\ell,r}\subseteq P_{\leq\ell+r}.$

We say that a pair $(x,y)$ is homotopy $(\ell,r)$-trivial, if it is equivalent
to a trivial pair with respect to the equivalence relation defined above. The
set of homotopy $(\ell,r)$-trivial pairs is denoted by $\overline{T}_{\ell,r}$

(4.63) $T_{\ell,r}\subseteq\overline{T}_{\ell,r}\subseteq P_{\leq\ell+r}.$

A pair $(x,y)\in P_{\leq\ell+r}$ is called $(\ell,r)$-adjacent, if it is not
homotopy $(\ell,r)$-trivial

(4.64) ${\rm Adj}_{\ell,r}=P_{\leq\ell+r}\setminus\overline{T}_{\ell,r}.$

We consider the restricted equivalence relation on ${\rm Adj}_{\ell,r}.$

######  Proposition 4.12.

Let $X$ be a quasimetric space and $\ell>r\geq 0.$ Then

(4.65) $\mathrm{SH}^{0}_{1,[\ell,\ell+r]}(X)\cong\mathbb{K}\cdot({\rm
Adj}_{\ell,r}/\sim).$

###### Proof.

Set $C=F_{[\ell,\ell+r]}\mathrm{RC}(X).$ Then $C_{0}=0,$

(4.66) $C_{1}=\mathbb{K}\cdot\\{(x,y)\mid\ell\leq d(x,y)\leq\ell+r\\},$ (4.67)
$C_{2}=\mathbb{K}\cdot\\{(x,a,y)\mid\ell\leq d(x,a)+d(a,y)\leq\ell+r\\}.$

Consider the map $\theta:C_{1}\to\mathbb{K}\cdot({\rm Adj}_{\ell,r}/\sim)$
defined so that it sends pairs from $\overline{T}_{\ell,r}$ to zero, and all
other pairs $(x,y)\in{\rm Adj}_{\ell,r}$ to their equivalence classes
$\theta(x,y)=[(x,y)].$ It is easy to see that $\theta$ is surjective. We need
to prove that

(4.68) $\mathrm{Ker}(\theta)=\mathrm{Im}(\partial:C_{2}\to C_{1}).$

Let us prove that $\mathrm{Im}(\partial:C_{2}\to
C_{1})\subseteq\mathrm{Ker}(\theta).$ If $(x,a,y)\in C_{2},$ then either
$d(x,a)<\ell$ or $d(a,y)<\ell,$ because if $d(x,a)\geq\ell$ and
$d(a,y)\geq\ell,$ then $d(x,a)+d(a,y)\geq 2\ell>\ell+r.$ Assume first that
$d(x,a)<\ell.$ Then we have four cases.

Case 1.1: $d(a,y)<\ell$ and $d(x,y)<\ell.$ Then $\partial(x,a,y)=0.$

Case 1.2: $d(a,y)<\ell$ and $d(x,y)\geq\ell.$ Then $\partial(x,a,y)=-(x,y).$
In this case $(x,y)\in T_{\ell,r}$ and hence $\theta(\partial(x,a,y))=0.$

Case 1.3: $d(a,y)\geq\ell$ and $d(x,y)<\ell.$ Then $d(x,a)\leq r$ and
$\partial(x,a,y)=(a,y).$ In this case $(x,y)\sim(a,y)$ and $(x,y)\in
T_{\ell,r}.$ In particular, $(a,y)\in\overline{T}_{\ell,r}$ and
$\theta(\partial(x,a,y))=0.$

Case 1.4: $d(a,y)\geq\ell$ and $d(x,y)\geq\ell.$ Then $d(x,a)\leq r$ and
$\partial(x,a,y)=-(x,y)+(a,y).$ In this case $(x,y)\sim(a,y),$ and
$\theta(\partial(x,a,y))=0.$

If $d(a,y)<\ell,$ we have symmetric cases. So we proved that
$\mathrm{Im}(\partial:C_{2}\to C_{1})\subseteq\mathrm{Ker}(\theta).$

Let us prove that $\mathrm{Ker}(\theta)\subseteq\mathrm{Im}(\partial:C_{2}\to
C_{1}).$ First we need to check that if $(x,y),(x^{\prime},y^{\prime})\in
P_{\leq\ell+r}$ and $(x,y)\sim(x^{\prime},y^{\prime}),$ then
$(x,y)-(x^{\prime},y^{\prime})\in\mathrm{Im}(\partial:C_{2}\to C_{1}).$ Here
we assume that a pair $(x,y)\in P_{\leq\ell+r}$ such that $d(x,y)<\ell$
represents the zero element in $C_{1},$ and a triple $(x,a,y)$ such that
$d(x,a)+d(a,y)<\ell$ represents the zero element in $C_{2}.$

Let $(x,y),(x^{\prime},y)\in P_{\leq\ell+r}$ such that $d(x,x^{\prime})\leq r$
and $d(x,x^{\prime})+d(x^{\prime},y)\leq\ell+r.$ Then $(x,x^{\prime},y)\in
C_{2}$ and $\partial(x,x^{\prime},y)=-(x,y)+(x^{\prime},y).$ Therefore
$(x,y)-(x^{\prime},y)\in\mathrm{Im}(\partial).$ The case of pairs
$(x,y),(x,y^{\prime})\in P_{\leq\ell+r}$ such that $d(y^{\prime},y)\leq r$ and
$d(x,y^{\prime})+d(y^{\prime},y)\leq\ell+r$ is similar. It follows that the
differences of equivalent pairs from $P_{\leq\ell+r}$ are in the image of
$\partial.$

Now assume that $(x,y)\in T_{\ell,r}.$ Then there exists $a$ such that
$d(x,a)+d(a,y)\leq\ell+r$ and $d(x,a),d(a,y)<\ell.$ Therefore $(x,a,y)\in
C_{2}$ and $\partial(x,a,y)=-(x,y).$ It follows that
$\mathrm{Ker}(\theta)\subseteq\mathrm{Im}(\partial:C_{2}\to C_{1}).$ ∎

######  Theorem 4.13.

Let $X$ be a quasimetric space and $\ell>r\geq 0.$ Then there is an
isomorphism

(4.69) $\mathrm{SH}^{r}_{1,\\{\ell\\}}(X)\cong\mathbb{K}\cdot(({\rm
Adj}_{\ell}\cap{\rm Adj}_{\ell,r})/\sim),$

where $\sim$ is the equivalence relation induced from the equivalence relation
on ${\rm Adj}_{\ell,r}.$

###### Proof.

We set $C_{I}=F_{R}\mathrm{RC}(X)/F_{L}\mathrm{RC}(X).$ Since $r<\ell$ we have
$(C_{[\ell-r,\ell]})_{0}=0.$ Then we obtain that any element in
$(C_{[\ell-r,\ell]})_{1}$ is a cycle. It follows that

(4.70)
$\mathrm{SH}^{r}_{1,\\{\ell\\}}(X)=\mathrm{Im}((C_{[\ell-r,\ell]})_{1}\longrightarrow\mathrm{SH}^{0}_{1,[\ell,\ell+r]}(X)).$

Therefore, $\mathrm{SH}^{r}_{1,\\{\ell\\}}(X)$ is a submodule of
$\mathrm{SH}^{0}_{1,[\ell,\ell+r]}(X)$ generated by pairs $(x,y)$ such that
$d(x,y)=\ell.$ Then the statement follows from Proposition 4.12. ∎

Assume that $x,y\in\mathbb{R}^{n}.$ Then we consider the following filled
ellipsoid of revolution with foci $x,y$

(4.71) ${\rm Ell}_{r}(x,y)=\\{a\in\mathbb{R}^{n}\mid d(x,a)+d(a,y)\leq
d(x,y)+r\\}.$

We will also consider open balls of radius $\ell$

(4.72) $B^{\circ}_{\ell}(x)=\\{a\in\mathbb{R}^{n}\mid d(x,a)<\ell\\}.$

We define the _$r$ -thick open interval_ between $x,y$ as

(4.73) $I_{r}(x,y)=B^{\circ}_{d(x,y)}(x)\cap{\rm Ell}_{r}(x,y)\cap
B^{\circ}_{d(x,y)}(y).$

It can be depicted as the following gray area between $x$ and $y$.

$y$$x$

So for $x,y\in X$ such that $d(x,y)=\ell$ we have $(x,y)\in T_{\ell,r}$ if and
only if $I_{r}(x,y)\cap X\neq\emptyset.$

######  Corollary 4.14.

Let $\ell>r\geq 0$ and $X\subseteq\mathbb{R}^{n}.$ Assume that for each
$x,y\in X$ such that $d(x,y)=\ell,$ the intersection of $I_{r}(x,y)$ with $X$
is not empty. Then

(4.74) $\mathrm{SH}^{r}_{1,\\{\ell\\}}(X)=0.$

###### Proof.

In this case all elements from ${\rm Adj}_{\ell}$ are in $T_{\ell,r},$ and
${\rm Adj}_{\ell}\cap{\rm Adj}_{\ell,r}=\emptyset.$ ∎

Assume that $X\subseteq\mathbb{R}^{n}.$ We say that a (continuous) path
$\gamma:[0,1]\to X$ from $x=\gamma(0)$ reduces the distance to $y\in X,$ if
$d(x,y)>d(\gamma(t),y)$ for any $0<t\leq 1$ (we do not assume that
$\gamma(1)=y$). In other words, $\gamma$ reduces the distance to $y,$ if

(4.75) $\gamma((0,1])\subseteq B^{\circ}_{d(\gamma(0),y)}(y).$

For example, the path from $x$

$x$$y$

reduces the distance to $y.$ Note that, if $r>0,$ and $\gamma$ is a path from
$x$ reducing the distance to $y$ we have $\gamma(\varepsilon)\in I_{r}(x,y)$
for small enough $\varepsilon>0.$

######  Corollary 4.15.

Let $\ell>r>0$ and $X\subseteq\mathbb{R}^{n}.$ If $x,y\in X$ such that
$d(x,y)=\ell$ and there is a path from $x$ reducing the distance to $y,$ then
$(x,y),(y,x)\in T_{\ell,r}.$ In particular, if for any distinct $x,y\in X$
there is either a path from $x$ reducing the distance to $y,$ or a path from
$y$ reducing the distance to $x,$ then $\mathrm{SH}^{r}_{1,\\{\ell\\}}(X)=0.$

######  Example 4.16.

Consider the standard circle $S^{1}\subseteq\mathbb{R}^{2}.$ By Corollary 4.15
for any $0<r<\ell$ we have $\mathrm{SH}^{r}_{1,\\{\ell\\}}(S^{1})=0.$

######  Example 4.17.

Let $X\subseteq S^{1}$ be an arc lying in one open semicircle.

$y$$x$

In other words the anticlockwise directed distance from $x$ to $y$ is strictly
less than $\pi.$ Then by Corollary 4.15 we obtain
$\mathrm{SH}^{r}_{1,\\{\ell\\}}(X)=0$ for any $\ell>r>0.$

######  Example 4.18.

Now assume that $X\subseteq S^{1}$ is an arc that does not lie in one
semicircle.

$x$$y$

So the anticlockwise directed distance from $x$ to $y$ is strictly greater
than $\pi.$ Moreover, let us assume that the points are close enough
$0<d(x,y)<\sqrt{2}/2.$ Then we claim that

(4.76)
$\mathrm{SH}^{r}_{1,\\{\ell\\}}(X)=\begin{cases}\mathbb{Z}\cdot\\{(x,y),(y,x)\\},&\ell=d(x,y),\\\
0,&\ell\neq d(x,y).\end{cases}$

Let us prove this.

If $\ell\neq d(x,y),$ then for any $x^{\prime},y^{\prime}\in X$ such that
$d(x^{\prime},y^{\prime})=\ell$ there is either a path from $x$ reducing the
distance to $y,$ or a path from $y$ reducing the distance to $x.$ So
$\mathrm{SH}^{r}_{1,\ell}(X)=0.$

Now assume that $\ell=d(x,y).$ Then all pairs $(x^{\prime},y^{\prime})\in{\rm
Adj}_{\ell}$ except $(x,y)$ and $(y,x)$ lie in $T_{\ell,r}$ because there is a
path from $x^{\prime}$ reducing the distance to $y^{\prime}.$ We claim that
$(x,y),(y,x)\notin\overline{T}_{\ell,r}.$ For this we write everything in
coordinates in a symmetric form $y=(a,b)$ and $x=(-a,b).$ If
$(x,y)\sim(x^{\prime},y^{\prime})\in P_{\leq\ell+r},$ using that
$\ell+r<\sqrt{2}$ and $r<\ell,$ one can show that $x^{\prime}$ is in the
Quadrant two and $y^{\prime}$ is in the Quadrant one.

$x$$y$$x^{\prime}$$y^{\prime}$

Then there is no a point $a\in X$ such that $d(x^{\prime},a)<\ell$ and
$d(a,y^{\prime})<\ell.$ Hence $(x,y)\notin\overline{T}_{\ell,r}.$ This also
implies that $(x,y)$ is not equivalent to $(y,x).$ Therefore $(({\rm
Adj}_{\ell}\cap{\rm Adj}_{\ell,r})/\sim)\cong\\{(x,y),(y,x)\\}.$

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