SEMI-COARSE SPACES HOMOTOPY AND HOMOLOGY ANTONIO RIESER AND JONATHAN TREVI NO-MARROQU IN Abstract. We begin the study the algebraic topology of semi-coarse spaces

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SEMI-COARSE SPACES, HOMOTOPY AND HOMOLOGY

ANTONIO RIESER AND JONATHAN TREVI ˜

NO-MARROQU´

Abstract. We begin the study the algebraic topology of semi-coarse spaces,

which are generalizations of coarse spaces that enable one to endow non-trivial

‘coarse-like’ structures to compact metric spaces, something which is impossi-

ble in coarse geometry. We ﬁrst study homotopy in this context, and we then

construct homology groups which are invariant under semi-coarse homotopy

equivalence. We further show that any undirected graph G= (V, E) induces

a semi-coarse structure on its set of vertices VG, and that the respective semi-

coarse homology is isomorphic to the Vietoris-Rips homology. This, in turn,

leads to a homotopy invariance theorem for the Vietoris-Rips homology of

undirected graphs.

Contents

1. Introduction 1

2. Semi-Coarse Spaces 2

2.1. Fundamental Concepts and Examples 2

2.2. Subspaces 6

2.3. Product Semi-Coarse Spaces 7

2.4. Quotient Spaces 10

3. Homotopy 13

3.1. Homotopy and Homotopy Equivalence for Semi-Coarse Spaces 13

3.2. Homotopy Groups 16

3.3. Connectedness 26

3.4. The Semi-Coarse Fundamental Group of Cyclic Graphs 28

3.5. Long Exact Sequence in Homotopy 32

4. Homology 35

4.1. Simplicial Homology 36

4.2. Graphs and Semi-Coarse Spaces 38

5. Discussion and Future Work 40

Appendix A. Additional Results on Point-Set Topology of Semi-Coarse

Spaces and Their Relations with Semi-Uniform and Coarse

Spaces 42

This material is based in part upon work supported by the US National Science Foundation

under Grant No. DMS-1928930 while the authors participated in a program supported by the

Mathematical Sciences Research Institute. The program was held in the summer of 2022 in part-

nership with the the Universidad Nacional Aut´onoma de M´exico. This work was also supported

by the CONACYT Investigadoras y Investigadores por M´exico Project #1076, the CONACYT

Ciencia de Fronteras grant CF-2019-217392, and by the grant N62909-19-1-2134 from the US Of-

ﬁce of Naval Research Global and the Southern Oﬃce of Aerospace Research and Development

of the US Air Force Oﬃce of Scientiﬁc Research. The second author was also supported by the

CONACYT postgraduate studies scholarship number 839062.

arXiv:2210.02569v3 [math.AT] 28 Sep 2024

SEMI-COARSE SPACES, HOMOTOPY AND HOMOLOGY 1

A.1. Semi-uniform Spaces, Graphs and Roofed Semi-coarse Spaces 42

A.2. Product Structure in Semi-coarse Spaces 48

A.3. Coarse Space Induced by a Semi-Coarse Space 51

Acknowledgements 55

References 55

1. Introduction

Coarse geometry [14] is often referred to as ‘geometry in the large’, or ‘large scale

geometry’, for the reason that, on a metric space endowed with the natural coarse

structure, all bounded phenomena are trivial from the coarse point of view. The

study of coarse geometry has led to a number of important advances, in particular

in group theory, where it has enabled many questions about groups to be addressed

geometrically via the analysis of their Cayley graphs (see, for instance, [6] for a

recent book-length discussion of this approach). Nonetheless, the large-scale nature

of coarse geometry forces all ﬁnite-diameter spaces to be coarsely-equivalent to a

point. It is not diﬃcult, however, to imagine scenarios where one might like to

‘coarsen’ a space up to a certain scale, but where one does not want to erase all

bounded phenomena at every scale. Many problems of topological data analysis,

for instance, in which one would like to deduce the topological invariants of a space

from algebraic invariants built from a ﬁnite subset of the space, present themselves

naturally as problems of ‘medium-scale’ coarsening. Several possibilities for a formal

context for ‘coarsening’ a space up to a scale were developed by the ﬁrst author in

[12] and [13], where he studied the problem from the point of view of ˇ

Cech closure

spaces and semi-uniform spaces, respectively. The connection to coarse geometry in

these earlier works, however, remained unclear, and this question forms the primary

motivation for the work presented here.

Whereas the closure structures and semi-uniform structures studied in [12] and

[13] are generalizations of topological structures and uniform structures, respec-

tively, in this article, we focus instead on a generalization of coarse structures,

modifying the axioms of coarse geometry [14] to allow ‘coarsenings’ only up to a

preferred size. Doing so links the homotopy of spaces coarsened up to a ﬁxed scale

to the study of coarse spaces, and, in particular, the development of homotopy on

semi-coarse spaces that we have begun here has been used by the second author

in [17] to construct new homotopical invariants on coarse spaces themselves, some-

thing which is unclear how to accomplish through either closure or semi-uniform

spaces directly. Technically, this ‘coarsening up to a scale’ is achieved by eliminat-

ing the product axiom in the deﬁnition of the coarse structure, which, indeed, has

a similar ﬂavor to the elimination of the idempotence axiom in the passage from

Kurotowski (topological) closure structures to more general ˇ

Cech closure structures,

or to the elimination of the product axiom in the passage from uniform spaces to

semi-uniform spaces. While removing the product axiom eﬀectively destroys the

notion of coarse equivalence, we observe in the following that coarse equivalence

can be proﬁtably generalized to a useful notion of homotopy equivalence in this

setting. The resulting structures and spaces are called semi-coarse structures and

semi-coarse spaces, respectively, and were ﬁrst introduced in [18]. Many exam-

ples of semi-coarse spaces exist. The examples of most interest to topological data

SEMI-COARSE SPACES, HOMOTOPY AND HOMOLOGY 2

analysis may be built from pseudo-metric spaces and a preferred scale r > 0, but

many others exist as well, in particular those constructed from semi-uniform spaces.

Furthermore, when specialized to graphs on lattices, the semi-coarse homotopy in-

troduced here is also similar to the digital homotopy studied in [2–4, 7–11, 15].

The outline of this article is as follows. In Section 2, we give the basic deﬁni-

tions, point-set properties, and principal examples of semi-coarse spaces. We will

also show in that section how to obtain a coarse space from a semi-coarse space

through a certain limiting process. In Section 3, we begin the study of homotopy

invariants in the semi-coarse category. Unfortunately, the interval does not appear

to have a natural non-trivial semi-coarse structure, signiﬁcantly complicating the

construction. We are able circumvent this shortcoming by adapting the homotopy

construction from [1], using ﬁnite-length subsequences of Zin place of the inter-

val to construct cylinders. While technically delicate, this nonetheless allows us

to deﬁne homotopy groups, prove a long-exact sequence of pairs, and compute the

fundamental group of a ‘semi-coarse circle’ with four points. Finally, in Section 4,

we construct homology groups for semi-coarse spaces, inspired by the Vietoris-Rips

construction now commonly used in topological data analysis. We give a construc-

tion for any semi-coarse space, demonstrate the invariance of the homology groups

with respect to the homotopy introduced in Section 3, and show that, for a count-

able semi-coarse space, this homology is exactly the Vietoris-Rips homology of an

associated graph.

Many of the results in this paper were ﬁrst presented in the Master’s thesis of

the second author [16], written under the supervision of the ﬁrst author, where

semi-coarse spaces and structures were called pseudo-coarse.

2. Semi-Coarse Spaces

In this section, we deﬁne semi-coarse spaces, bornologous functions, and we

give examples of semi-coarse structures constructed from a metric space and a

scale parameter r > 0. We then deﬁne semi-coarse quotients, disjoint unions, and

products, and we show how to build an induced coarse structure from a semi-coarse

structure.

2.1. Fundamental Concepts and Examples. We begin by setting some nota-

tion which we will use throughout the article.

Deﬁnition 2.1.1. Let Xbe a set. We denote by P(X) the collection of all subsets

of X, and

(1) Xnwill denote

n−times

z }| {

X×X× · · · × X.

(2) ∆X:= {(x, x)∈X×X|x∈X}will be called the diagonal of X.

(3) For V∈ P(X×X), we deﬁne

V−1:= {(y, x)∈X×X: (x, y)∈V},

which we call the inverse of V.

(4) For V, W ∈ P(X×X), we deﬁne

V◦W:= {(x, y)∈X×X| ∃z∈X, (x, z)∈Vand (z, y)∈W},

which will be called the set product of Vand W.

SEMI-COARSE SPACES, HOMOTOPY AND HOMOLOGY 3

Given an integer n≥2, we write V◦nfor

n−times

z }| {

V◦. . . ◦V.

(5) Let Xand Ybe sets, f:X→Ya set function, and V∈ P(X×X). Then

(f×f)(V) := {(f(x), f(x′)) ∈Y×Y|(x, x′)∈V},

and we call (f×f)(V)the image of Vunder f×f.

We now deﬁne semi-coarse spaces, our principal object of interest.

Deﬁnition 2.1.2 (Semi-coarse space).Let Xbe a set, and let V ⊂ P(X×X) be

a collection of subsets of X×Xwhich satisﬁes

(sc1) ∆X∈ V,

(sc2) If B∈ V and A⊂B, then A∈ V,

(sc3) If A, B ∈ V, then A∪B∈ V,

(sc4) If A∈ V, then A−1∈ V.

We call Vasemi-coarse structure on X, and we say that the pair (X, V) is a

semi-coarse space.

If, in addition, Vsatisﬁes

(sc5) If A, B ∈ V, then A◦B∈ V.

then Vwill be called a coarse structure, and (X, V) will be called a coarse space, as

in [14].

The elements of Vwill be called controlled sets. Moreover, if there exist a, b such

that {(a, b)} ∈ V we will say that aand bare adjacent. If Vand V′are semi-coarse

structures on Xsuch that V ⊂ V′, then we say that V′is ﬁner than Vand Vis

coarser than V′. Finally, when the structure Vis unambiguous, we will sometimes

refer to the semi-coarse space only by X.

The functions of interest between semi-coarse spaces will be those which preserve

the semi-coarse structure.

Deﬁnition 2.1.3. We will say that f:X→Yis a (V,W)-bornologous function,

or simply bornologous, if f×fmaps each controlled set V∈ V to a controlled set

(f×f)(V)∈ W.

Since the composition of set maps is associative, the composition of bornologous

maps is bornologous, and the identity is bornologous for every semi-coarse space

(X, V), we have

Theorem 2.1.4. Semi-coarse spaces and bornologous functions form a category.

We denote the category of semi-coarse spaces and bornologous functions by

SCoarse.

For our ﬁrst collection of examples, we show how undirected graphs induce semi-

coarse structures. We begin with a pair of lemmas.

Lemma 2.1.5. Let Xbe a set, and let W⊂X×Xsuch that

(1) ∆X⊂W,

(2) W=W−1.

Then (X, P(W)) is a semi-coarse space.

Proof. We verify directly that the axioms (sc1) - (sc4) from Deﬁnition 2.1.2 are

satisﬁed.

SEMI-COARSE SPACES, HOMOTOPY AND HOMOLOGY 4

(sc1) ∆X⊂W, so ∆X∈ P(W).

(sc2) P(W) is closed under taking subsets by deﬁnition.

(sc3) Since any two sets A, B ∈ P(W) are subsets of W, the union A∪B⊂W,

and therefore A∪B∈ P(W).

(sc4) Suppose A∈ P(W). Then A⊂W. However, by hypothesis on W, if

(a, b)∈Wthen (b, a)∈W. Therefore A−1⊂W, so A−1∈ P(W) as well.

It follows from the above that (X, P(W)) is a semi-coarse space, as desired. □

It will sometimes be convenient to use this lemma in the following alternative

form.

Lemma 2.1.6. Let Xbe a set and suppose that U ⊂ P(X×X)is a collection of

subsets of X×Xsuch that

(1) U∈ U =⇒U−1∈ U, and

(2) ∆X∈ U.

Let W:=SU∈U U. Then (X, P(W)) is a semi-coarse space.

Proof. By construction, Wsatisﬁes the hypotheses of Lemma 2.1.5. The conclusion

follows. □

We now use these lemmas to give several important examples of semi-coarse

spaces.

Example 2.1.6.1. Let G= (V, E) be an undirected graph (i.e. (u, v)∈E⇐⇒

(v, u)∈E). Deﬁne V ⊂ P(V×V) to be

VG:=P(E∪∆V),

where ∆Vis the diagonal of V×V. By Lemma 2.1.6 above, the pair (V, VG) is a

semi-coarse space.

Deﬁnition 2.1.7. Given a graph G= (V, E), we say that the semi-coarse space

(V, VG) constructed in Example 2.1.6.1 is the semi-coarse space generated by the

graph G.

Another important class of examples may be constructed from metric spaces

combined with a positive scale parameter r > 0.

Example 2.1.7.1. (a) Let (X, d) be a metric space. Let r > 0 be a positive real

number and deﬁne

Ur:={(x, x′)∈X×X|d(x, x′)≤r},

and let Vr:=P(Ur). Then (X, Vr) is a semi-coarse space by Lemma 2.1.5.

(b) Similarly, deﬁning U<

rby

r:={(x, x′)∈X×X|d(x, x′)< r},

and let V<

r:=P(U<

r). Lemma 2.1.5 gives that (X, V<

r) is a semi-coarse space.

For the next example, which generalizes the ones in Example 2.1.7.1 above, we

introduce semi-pseudometric spaces.

Deﬁnition 2.1.8 (Semi-Pseudometric; [5], 18 A.1.).Let Xbe a set and d:X×

X→Rbe a function, we will say dis a semi-pseudometric on Xif they satisﬁes

the next conditions

(m1) For each x∈X,d(x, x) = 0.

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SEMI-COARSESPACES,HOMOTOPYANDHOMOLOGYANTONIORIESERANDJONATHANTREVI˜NO-MARROQU´INAbstract.Webeginthestudythealgebraictopologyofsemi-coarsespaces,whicharegeneralizationsofcoarsespacesthatenableonetoendownon-trivial‘coarse-like’structurestocompactmetricspaces,somethingwhichisimpossi-bleincoarsegeometry...

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