Homological Methods in Coarse Geometry A

The Pennsylvania State University

The Graduate School

The Eberly College of Science

HOMOLOGICAL METHODS IN COARSE GEOMETRY

A Dissertation in Mathematics by Steven Hair

c 2010 Steven Hair

Submitted in Partial Fulﬁllment of the Requirements for the Degree of

Doctor of Philosophy

December 2010 The dissertation of Steven Hair was reviewed and approved* by the following:

John Roe Professor of Mathematics Dissertation Advisor Chair of Committee Head of the Department of Mathematics

Nathanial Brown Associate Professor of Mathematics

Nigel Higson Evan Pugh Professor of Mathematics

Darrell Velegol Professor of Chemical Engineering

Chun Liu Professor of Mathematics Director of Graduate Studies in the Department of Mathematics

*Signatures are on ﬁle in the Graduate School.

ii Abstract

The ﬁeld of coarse algebraic topology involves the application of principles of algebraic topology to coarse spaces, particularly the calculation of the so-called “coarse homology” and “coarse cohomology” of a space. In this dissertation, we present new techniques in coarse algebraic topology, and give applications to studying coarse spaces.

We first develop an analogue of the Mayer-Vietoris theorem for coarse homology, discuss a coarse excision principle, and give a “coarse mapping cone” construction for relative coarse cohomology. Next, the notion of homological degree for a coarse map is introduced, and is used to prove a coarse surjectivity theorem as well as a coarse fixed point theorem. Then, we develop an algebraic formalization of underlying concepts in coarse algebraic topology in the form of “geometric modules”. These modules are subsequently utilized to formulate a coarse analogue of the Künneththeorem and Universal

Coeﬃcient theorem; these theorems are then applied to compute the coarse homology and cohomology of speciﬁc spaces.

Finally, we discuss coarse versions of topological CW-complexes and ﬁbrations, use this discussion to introduce a coarse analogue of the Serre spectral sequence, and apply spectral sequence calculations to compute the coarse cohomology of certain nilpotent groups.

iii Contents

List of Figures vii

Acknowledgements viii

1 Introduction 1

1.1 Coarse Spaces ...... 2

1.2 Examples ...... 3

1.3 Controlled and Bounded Sets ...... 4

1.4 Coarse Maps, Equivalence, Homotopy ...... 5

1.4.1 Gluing of Coarse Spaces ...... 8

1.5 Coarse Algebraic Topology ...... 10

1.5.1 Coarse Cohomology ...... 10

1.5.2 Examples ...... 11

1.6 Covers and Nerves ...... 15

1.6.1 Coarse Homology ...... 18

1.7 Organization ...... 21

2 Coarse Excision, Relative Cohomology 22

2.1 The Mayer-Vietoris Sequence ...... 22

2.2 Deﬁnition of Relative Coarse Cohomology ...... 27

2.3 Coarse Excision ...... 29

iv 2.4 The Mapping Cone Construction ...... 34

3 Coarse Degree Theory 37

3.1 Degree and Essential Surjectivity ...... 38

3.2 A Combinatorial Coarse Degree Formula ...... 42

3.3 Examples ...... 44

3.4 The Coarse Brouwer Fixed Point Theorem ...... 46

3.4.1 The Weak Fixed Point Property ...... 51

4 Geometric Modules 53

4.1 The Controlled and Bounded Coarse Categories ...... 53

4.2 Geometric Modules ...... 54

4.2.1 Morphisms of Geometric Modules ...... 57

4.3 Controlled and Cocontrolled Modules ...... 59

4.4 Generalized Morphisms ...... 60

4.4.1 Composition of Morphisms ...... 67

4.5 Partial Modules and Closures ...... 70

4.6 Total Modules ...... 73

4.7 Cocontrolled and Controlled Tensor Products ...... 76

4.8 Tensor Products ...... 77

4.8.1 The Cross Product Map ...... 83

4.8.2 Decomposing elements of CXn(X × Y ) ...... 87

4.8.3 The Shuﬄe Map ...... 89

4.8.4 Construction of Homotopy Equivalences ...... 91

5 Coarse Universal Coeﬃcient and K¨unnethTheorems 101

5.1 A Coarse Universal Coeﬃcient Theorem ...... 101

5.2 A Coarse K¨unnethTheorem ...... 105

5.3 Examples ...... 117

v 6 Coarse CW-Complexes and Fibrations 119

6.1 Coarse CW-Complexes ...... 119

6.2 Coarse Cellular Cochain Complexes ...... 125

6.3 Coarse Fibrations ...... 128

6.4 The Main Theorem ...... 137

6.4.1 Construction of the Isomorphism ...... 138

6.4.2 Construction of the Spectral Sequence ...... 141

6.4.3 Examples ...... 144

Bibliography 145

vi List of Figures

3.1 Construction of similar triangles...... 50

4.1 Path corresponding to the shuﬄe µ = {1, 4, 5, 6}, ν = {2, 3, 7}...... 92

vii Acknowledgements

This dissertation would have never seen the light of day without the contributions of many important people. First, I would like to thank my advisor, John Roe, for his patience and motivation over the years. His influence helped to shape me from an unfocused graduate student to someone who can hopefully contribute positively to the world of mathematics as an educator and researcher. Thanks also to my committee members, Nate Brown, Nigel Higson, and Darrell Velegol, for donating their time. I would also like to thank Becky Halpenny and the rest of the office staff, without whom

I would have been lost over the years.

Additionally, appreciation goes out to my Master’s thesis advisor at Virginia Tech,

Peter Linnell, for inspiring me to follow this path in the ﬁrst place, and my fellow graduate students, whose mathematical conversations proved invaluable over the years.

Finally, special thanks go out to Mom and Dad, Jolene and Scott, Kerry, and the rest of the family, friends, and dogs in my life, for their unfailing support and love.

viii Chapter 1

Introduction

There exists in the current literature a number of algebraic topological methods for analyzing properties of coarse spaces. These methods include calculations of the coarse homology and cohomology groups of these spaces, with associated notions of excision, relative (co)homology, and Mayer-Vietoris sequences. In this dissertation, we introduce additional methods in coarse algebraic topology in order to perform these calculations for a greater number of coarse spaces. These methods will mirror some of the more well-known techniques in classical algebraic topology.

In particular, we shall give a coarse mapping cone construction of the coarse cohomology of a space relative to a subspace, and verify, via coarse excision, that the relative coarse cohomology groups are isomorphic to the absolute coarse cohomology groups of the mapping cone . We also introduce the notion of degree for coarse maps, prove a geometric result for maps of nonzero degree, and provide a combinatorial technique by which degree can be computed, with computations for speciﬁc examples.

Additionally, by applying results on direct and inverse limits of modules in homological algebra, we shall develop coarse notions of the Universal Coeﬃcient Theorem and

K¨unnethFormula. Then we shall discuss the concept of coarse ﬁbrations: maps between coarse spaces with associated Serre-like spectral sequences in coarse cohomology. This

1 discussion is aided by the presentation of coarse CW-complexes: coarse spaces with a structure analogous to CW-complexes in classical topology. The spectral sequence results will then be applied to compute the coarse cohomology of speciﬁc spaces, most notably semidirect products of groups.

1.1 Coarse Spaces

Coarse geometry is often thought of as the study of the “large-scale” geometry or the

“geometry at infinity” of a space, i.e. the study of what structure survives as one zooms infinitely far away from the space. More precisely, coarse geometry is the study of coarse spaces. Such spaces are the fundamental objects in coarse geometry, analogous to topological spaces in classical topology. A coarse space is a set equipped with a so-called coarse structure, as defined below, which is analogous to the topology on a topological space.

Deﬁnition 1.1.1. A coarse space (X, E) is a set X accompanied by a collection E of subsets of X × X (the so-called controlled subsets or entourages of X × X) with the following properties:

• the diagonal ∆X×X is a member of E;

• if E is a member of E, then so is every subset of E;

• if E,F ∈ E, then so is E ∪ F ;

• if E is a member of E, then so is the inverse

E−1 := {(y, x)|(x, y) ∈ E};

• if E and F are members of E, then so is the product

E ◦ F := {(x, y)| there exists z ∈ X such that (x, z) ∈ E, (z, y) ∈ F }.

2 The collection E is called the coarse structure of X.

Generally, we will use X to refer to the coarse space (X, E). A collection S of subsets of X × X containing ∆X×X is said to generate a coarse structure E if E is the smallest coarse structure containing S.

1.2 Examples

1. If X is a metric space, the metric coarse structure is the structure generated by

the set of all metric neighborhoods of the diagonal, i.e. the sets of pairs (x, y) in

X × X which are distance at most R apart, for all R ≥ 0. It is well-known (cf.

[18], Thm. 2.55) that a coarse structure E on X is countably generated if and

only if it is metrizable, i.e. there exists a metric on X for which E is the metric

coarse structure.

2. The trivial or indiscrete coarse structure on a set X is the structure consisting of

all subsets of X × X.

3. The discrete coarse structure on X is generated by all sets of the form ∆X×X ∪ F for ﬁnite F ⊂ X × X.

4. If A is a subset of a coarse space X, we may equip A with the subspace coarse

structure {E ∩ (A × A) | E ⊂ X × X controlled}. (In the future, when we refer to

such a subset A as being a subspace of X, we mean that A is equipped with this

particular coarse structure.)

5. If X and Y are coarse spaces, we may equip the set X ×Y with the product coarse

structure, consisting of those subsets of (X × Y ) × (X × Y ) whose projections to

X × X and Y × Y are controlled under the respective coarse structures.

6. If X is a coarse space, A is a set, and p : X → A is a surjective map, the quotient

coarse structure on A relative to p is generated by the images (p × p)(E) ⊂ A × A

3 for all controlled E ⊂ X × X. Then p, as a map from the coarse space X to the

coarse space A, is bornologous in the sense of Deﬁnition 1.4.1. One important

example of this particular coarse structure arises from gluing of coarse spaces via

coarse maps, as deﬁned in Section 1.4.1.

7. If X and Y are coarse spaces, the union coarse structure on X ∪Y is the structure

generated by EX ∪ EY .

To simplify many of the calculations that we shall perform in this dissertation, we shall assume that all coarse spaces we work with are coarsely connected:

Deﬁnition 1.2.1. A coarse space X is coarsely connected if for every x, y ∈ X, the set

{(x, y)} is controlled.

Related to this idea is the notion of coarsely disjoint spaces.

Deﬁnition 1.2.2. Let X be a coarse space and A, B be disjoint subspaces of X. Then, A and B are said to be coarsely disjoint if, for all a ∈ A, b ∈ B, {(a, b)} is not controlled.

One example of coarsely disjoint spaces is as follows: if X and Y are disjoint and the disjoint union X t Y is given the union coarse structure, then X and Y are coarsely

disjoint in X t Y . Just as is the case in classical algebraic topology, every coarse space

can be decomposed as a “coarsely disjoint union” of coarsely connected subspaces, and

results on coarsely connected components extend to the entire space in a way which is

analogous to connected components of a topological space.

1.3 Controlled and Bounded Sets

Let X be a coarse space and the maps πi, 1 ≤ i ≤ n be the canonical projections from Xn to X.

Deﬁnition 1.3.1. If n > 1, a subset E of Xn is controlled iﬀ the set

{(πi(x), πj(x)) | x ∈ E, 1 ≤ i ≤ n}

4 is an entourage. In the case n = 1, we declare every subset of X, including X itself, to be controlled.

Note that when n = 2, this coincides with our previous deﬁnition of a controlled set.

Deﬁnition 1.3.2. A subset B of Xn is bounded iﬀ there exists y ∈ X for which the set

{(πi(x), y)|x ∈ B, 1 ≤ i, j ≤ n}

is an entourage.

Observe that a set B ⊂ X is bounded iﬀ B × B is controlled. If X is a bounded subset of itself, we call X a bounded coarse space.

1.4 Coarse Maps, Equivalence, Homotopy

When presented with two coarse spaces, we want to study the relevant maps between

them. In this case, such maps are called coarse, and are analogous to continuous maps in topology. Alongside this notion, we have relationships between coarse spaces which are analogous to homeomorphism and homotopy equivalence. In this way, we may discuss so-called coarse invariants and coarse homotopy invariants in coarse geometry.

Deﬁnition 1.4.1. Let X and Y be coarse spaces.

A map f : X → Y is called coarse if:

• for all bounded B ⊂ Y , the set f −1(B) is bounded in X (f is proper);

• for all controlled E ⊂ X × X, the set (f × f)(E) is controlled in Y × Y (f is

bornologous).

Two coarse maps f, g : X → Y are close if the set {(f(x), g(x))|x ∈ X} is controlled.

Two coarse spaces X,Y are coarsely equivalent, denoted X 'c Y , if there exist coarse maps f : X → Y , g : Y → X such that g ◦ f and f ◦ g are close to the identity

maps on X and Y respectively.

5 For example, the spaces R and Z, equipped with the obvious metric coarse structures, are coarsely equivalent. This equivalence is realized by the floor function from R to Z and the inclusion map from Z to R. The following definition is due Higson, Roe, and others, and refined in the dissertation of Viet-Trung Luu ([14]):

Deﬁnition 1.4.2. Let X and Y be coarse spaces. A coarse homotopy from X to Y is a map ht : X × [0, 1] → Y such that:

• for all bounded B ⊂ Y , the set

[ −1 ht (B) t∈[0,1]

is bounded (the family {ht} is uniformly proper);

• for all controlled E ⊂ X × X, the set

[ (ht × ht)(E) t∈[0,1]

is controlled (the family {ht} is uniformly bornologous);

• there is a controlled set F ⊂ Y × Y such that for all x ∈ X, t ∈ [0, 1], there is an

0 open neighborhood Ix,t ⊂ [0, 1] of t such that t ∈ Ix,t implies (ht(x), ht0 (x)) ∈ F

(the family {ht} is uniformly pseudocontinuous).

Deﬁnition 1.4.3. Two coarse maps f, g : X → Y are coarsely homotopic if there is a

coarse homotopy ht : X × [0, 1] → Y such that h0 = f, h1 = g.

Two coarse spaces X,Y are coarse homotopy equivalent, written X 'ch Y if there exist coarse maps f : X → Y , g : Y → X such that g◦f and f ◦g are coarsely homotopic

to the identity maps on X and Y respectively.

6 Deﬁnition 1.4.4. If X is a coarse space and A ⊂ X, a coarse deformation retraction of

X onto A is a coarse homotopy ht : X × [0, 1] → X such that h0 is the identity on X

and h1(X) = A.

We should note that coarse homotopy equivalence is an equivalence relation. As in

the classical case, transitivity is a consequence of two facts:

• If ht : X × [0, 1] → Y is a coarse homotopy and f : Y → Z and g : W → X are coarse maps, then

0 f ◦ ht : X × [0, 1] → Z, ht ◦ (g × id[0,1]): X × [0, 1] → Y

are coarse homotopies;

• If ft : X × [0, 1] → Y and gt : X × [0, 1] → Y are coarse homotopies, with f1 = g0;

then ht : X × [0, 1] → Y deﬁned by

  f2t(x) if 0 ≤ t ≤ 1/2; ht(x) =  g2t−1(x) if 1/2 ≤ t ≤ 1;

is a coarse homotopy.

The ﬁrst part is Lemma 3.2.6 in [14]. To verify the homotopy in the second part is

uniformly proper and bornologous, observe that

    [ −1 [ −1 [ −1 ht (B) =  ft (B) ∪  gt (B) t∈[0,1] t∈[0,1] t∈[0,1]

and     [ [ [ (ht × ht)(E) =  (ft × ft)(E) ∪  (gt × gt)(E) , t∈[0,1] t∈[0,1] t∈[0,1]

and the union of two bounded (controlled) sets are bounded (controlled). From these

7 two facts, it follows that if f : X → Y and g : Y → Z are coarse maps with homotopy inverses f 0, g0, then g ◦ f : X → Z has homotopy inverse f 0 ◦ g0.

An example of two coarse spaces which are coarsely homotopy equivalent but not

2 coarsely equivalent are the Euclidean plane R and the hyperbolic plane H. As discussed in the remarks following Lemma 9.9 in [20], the homotopy equivalence is given in part

2 by composing the (distance-increasing) exponential map exp: R → H with a “radial shrinking” which turns exp into a coarse map.

1.4.1 Gluing of Coarse Spaces

Let X and Y be coarse spaces, A a subspace of X, and f : A → Y be coarse. We may impose an equivalence relation ∼ on the disjoint union X t Y by letting x ∼ x for all x ∈ X \ A, y ∼ y for all y ∈ Y \ im(f), and a ∼ f(a) for all a ∈ A. Consider the set of equivalence classes

X ∪f Y := X t Y/ ∼;

since the map p sending each element of X t Y to its equivalence class is surjective,

there exists a natural coarse structure on X ∪f Y , namely the quotient structure with respect to p (here, the relevant coarse structure on X t Y is the union structure). We

call f the gluing map of A to Y which produces the above coarse structure.

Remark 1.4.5. Note that the original coarse structure E of Y coincides with the coarse

0 structure E inherited by Y as a subspace of X ∪f Y . By deﬁnition of the quotient structure, E0 is generated by E and all (f × f)(E), E controlled in A × A. Because f is

bornologous, every such (f × f)(E) lies in E. Therefore, E = E0.

Remark 1.4.6. Additionally, if X and Y are path metric spaces, the coarse structure of

0 X as a subspace of X ∪f Y is induced by the metric d deﬁned by letting d(x, x ) be the

0 inﬁmum of the lengths of paths from x to x in X ∪f Y . To see this, let C be the coarse

structure of X in X ∪f Y and Cd be the structure induced by d. Then C is generated

8 by the union of the controlled subsets of (X \ A) × (X \ A), all (f × f)(E), E ⊂ A × A controlled, and {(x, f(x0)) | (x, x0) ∈ E} for all E ⊂ (X \ A) × A controlled. Suppose

0 0 x, x ∈ X ⊂ X ∪f Y with d(x, x ) = R and consider the following cases:

• Suppose that x, x0 6∈ A and, for every > 0, there exists a path from x to x0 in

X \ A with length at most R + . If dX is the original path metric on X, it follows

0 0 0 that d(x, x ) = dX (x, x ) = R and (x, x ) lies in the radius R neighborhood of the diagonal of (X \ A) × (X \ A).

• Suppose x, x0 are equivalence classes of points a, a0 ∈ A and, for every > 0, there

exists a path from x to x0 in f(A) with length at most R + . Since f is proper,

0 0 there exists S > 0 such that dX (a, a ) ≤ S, regardless of the choice of a and a . Therefore, (x, x0) lies in the image under (f × f) of the radius S neighborhood of

the diagonal in A × A.

• Suppose x 6∈ A and x0 is the equivalence class of a0 ∈ A. Again, since f is proper,

0 0 there exists S > 0 such that dX (x, a ) ≤ S, regardless of the choice of a .

Since every path in X ⊂ X ∪f Y can be decomposed into path components whose endpoints satisfy one of the three cases above, it follows that Cd ⊂ C. Conversely, if x, x0 ∈ X\A with at least one of x, x0 6∈ A, then, since f is bornologous, there exists C > 0 such that every path from x to x0 in X of length R has image of

0 length at most R·max{C, 1} in X ∪f Y . Similarly, if x, x ∈ A, then every path from x to

0 0 x in X of length R has image in X ∪f Y (a path from f(x) to f(x )) with length at most R · max{C, 1}. Since C is generated by the controlled subsets of (X \ A) × (X \ A), all

(f × f)(E), E ⊂ A × A controlled, and {(x, f(x0)) | (x, x0) ∈ E} for all E ⊂ (X \ A) × A

controlled, it is that case that C ⊂ Cd.

9 1.5 Coarse Algebraic Topology

1.5.1 Coarse Cohomology

Deﬁnition 1.5.1. Let X be a (coarsely connected) coarse space and G an abelian group.

The group of coarse p-cochains on X with coeﬃcients in G, written CXp(X; G), is the

group of all maps ψ : Xp+1 → G such that the support of ψ,

Supp(ψ) := {x ∈ Xp+1|ψ(x) is a non-identity element}, has bounded intersection with every controlled subset of Xp+1. (Such a map ψ is said to have cocontrolled support).

We then equip the groups CX∗(X; G) with the Alexander-Spanier coboundary map

∂: p+1 X i ∂ψ(x0, . . . , xp+1) := (−1) ψ(x0,..., xˆi, . . . , xp+1). i=0

Observe that if ψ ∈ CXp(X; G), then ∂ψ indeed lies in CXp+1(X; G): let E ⊂ Xp+2

be controlled. Then, for all 0 ≤ i ≤ p + 1,

Ei := {(x0,..., xˆi, . . . , xp+1) |(x0, . . . , xp+1) ∈ E}

p+1 is a controlled subset of E . Since ψ is a coarse p-cochain, for each Ei there exists yi ∈

X and a controlled F ⊂ X ×X such that (xj, yi) ∈ F for all (x0, . . . , xp) ∈ Supp(ψ)∩Ei and all 0 ≤ j ≤ p. Therefore, there exists A ⊂ X × X controlled and y ∈ X such that (xj, y) ∈ A for all (x0, . . . , xp) ∈ ∪i (Supp(ψ) ∩ Ei) and all 0 ≤ j ≤ p. But since

(x0, . . . , xp+1) ∈ Supp(∂ψ) ∩ E implies that (x0,..., xˆi, . . . , xp+1) ∈ Supp(ψ) ∩ Ei for some i, it follows that for y and A deﬁned above, (xj, y) ∈ A for all (x0, . . . , xp+1) ∈ Supp(∂ψ) ∩ E and 0 ≤ j ≤ p + 1. Therefore, Supp(∂ψ) ∩ E is bounded.

10 Note also that ∂2 = 0; for each i, j with 0 ≤ i < j ≤ p + 2 there are exactly two

ψ(x0,..., xˆi,..., xˆj, . . . , xp+2)

terms in the sum representing ∂∂ψ(x0, . . . , xp+2). Consider the coeﬃcients of each of these terms; the the case that the xj term was removed ﬁrst (i.e. by the “inner” ∂), xi is

th j i the i coordinate in (x0,..., xˆj, . . . , xp+2) since i < j; so the coeﬃcient is (−1) (−1) .

st For the other such term, the xi term was removed ﬁrst; therefore xj is the (j − 1)

j−1 i coordinate in (x0,..., xˆj, . . . , xp+2) and so the coeﬃcient of this term is (−1) (−1) . Therefore the two terms cancel in the sum, and since ∂∂ψ can be decomposed into pairs of terms as above, the entire sum vanishes.

The subgroups ker ∂ and im ∂ are called the coarse cocycle and coboundary groups of X respectively, and are denoted ZX∗(X; G) and BX∗(X; G). The resulting cohomol-

ogy groups ZX∗(X; G)/BX∗(X; G) are called the coarse cohomology groups of X with coeﬃcients in G, denoted HX∗(X; G).

1.5.2 Examples

1. For a coarse space X,

  Z if X is bounded, 0 ∼ HX (X; Z) =  0 if X is unbounded.

Observe by the deﬁnitions above that a coarse 0-cochain ψ is a map ψ : X → Z such that the restriction of Supp(ψ) to every controlled subset of X (including X

itself!) is bounded, and ψ(y) − ψ(x) = ∂ψ(x, y) = 0 for all x, y ∈ X. In other

words, ψ is a constant map on X with bounded support, and the above claim

follows.

11 2.   Z if p = n, p n ∼ HX (R ; Z) =  0 otherwise.

A detailed proof of this result can be found in [19], Example 2.26; we shall

give an element that we claim to be a cohomology generator. Such a genera-

n n n n+1 tor for HX (R ; Z) is represented by the cocycle ψ :(R ) → Z such that

ψ(x0, . . . , xn) = ±1 if and only if {x0, . . . , xn} span an (n + 1)-simplex which contains the origin (sign depending on the orientation of the simplex).

For many calculations we will be examining the coarse cohomology of spaces with co-

∗ eﬃcients in Z, in which case we shall denote the cohomology groups as simply HX (X). Given two coarse spaces X,Y , a coarse map f : X → Y induces a chain homomorphism f ∗ from CXp(Y ) to CXp(X), in the obvious way; for ψ ∈ CXp(Y ), deﬁne f ∗ψ ∈ CXp(X) by

∗ (f ψ)(x0, . . . , xp) := ψ(f(x0), . . . , f(xp)).

(More generally, a coarse map f : Xp+1 → Y p+1 induces a homomorphism f ∗ :

p p ∗ CX (Y ) → CX (X) by (f ψ)(x0, . . . , xp) := ψ(f(x0, . . . , xp)).) As one may expect, this induced map passes to a homomorphism (which we shall also call f ∗) from HXp(Y ) to HXp(X), and:

Theorem 1.5.2. If two coarse maps are close (or coarsely homotopy equivalent), then they induce the same homomorphism on coarse cohomology.

The proof of Proposition 5.12 in [18] covers the case of close maps f, g : X → Y .

We may deﬁne a map h : CXq(Y ) → CXq−1(X) by

q−1 X i hψ(x0, . . . , xq−1) = (−1) ψ(f(x0), . . . , f(xi), g(xi), . . . , g(xq−1); i=0

12 the fact that f and g are close ensures that hψ has cocontrolled support. By a combinatorial calculation, h is a chain homotopy between f ∗ and g∗.

The proof of the analogous result for coarsely homotopic maps is somewhat more complicated, and typically involves a number of concepts which will be discussed later in this section (namely ﬂasqueness and anti-Cechˇ sequences of covers). An example of such an argument can be found in [9].

The identity map on a coarse space clearly induces the identity isomorphism on coarse cohomology, so one immediate result is the following:

Theorem 1.5.3. If X and Y are coarsely equivalent (or coarse homotopy equivalent), then HX∗(X) =∼ HX∗(Y ).

Another less immediate result deals with the coarse cohomology of so-called ﬂasque spaces (in the deﬁnition below, we shall abuse notation somewhat; if f : X → X is a map, we will also denote by f the map (f × · · · × f): Xq → Xq and f n to be the n-fold composition of (f × · · · × f) with itself):

Deﬁnition 1.5.4. An unbounded coarse space X is ﬂasque if it is equipped with a coarse map f : X → X such that for each q > 0:

1. For every bounded subset B ⊂ Xq, there exists N such that f n(Xq) ∩ B = ∅ for

all n ≥ N.

2. Given any controlled set E ⊂ Xq, there exists another controlled E0 ⊂ Xq such

that f n(E) ⊂ E0 for all n.

3. f induces the identity isomorphism on HX∗(X) (this follows if, for instance, f is

close to the identity map on X).

We shall now show that

Proposition 1.5.5. If X is ﬂasque, then HXp(X) = 0 for all p.

13 Proof. Let [ψ] ∈ HXp(X) be represented by a coarse cocycle ψ. The map f induces the identity isomorphism on coarse cohomology, so

[ψ − f ∗ψ] = [ψ] − f ∗[ψ] = 0 ∈ HXp(X);

∗ p p−1 ∗ therefore ψ − f ψ ∈ BX (X). Thus there exists φ0 ∈ CX (X) with ∂φ0 = ψ − f ψ. If we deﬁne

∗n ∗n ψ0 := ψ, ψn := f ψ, φn := f (φ0),

∗n ∗n ∗ p+1 note that ∂φn = f ∂φ0 = f (ψ − f ψ) = ψn − ψn+1. For ﬁxed x ∈ X , there exist controlled sets E,E0 ⊂ Xp+1 with x ∈ E and f n(E) ⊂ E0 for all n. Since

Supp(ψ) ∩ E0 = B is bounded and f n(x) ∈ B for only ﬁnitely many n by condition (1)

on ﬂasqueness, it follows that ψn(x) 6= 0 for only ﬁnitely many n as well. Repeating

this argument for the φn gives an analogous result. It now makes sense to deﬁne

∞ X φ(x) := φn(x), n=0

since for each ﬁxed x, only ﬁnitely many terms in the above sum are nonzero.

We wish to show that φ has cocontrolled support. Let E be a controlled subset of

Xp and choose a controlled E0 such that f n(E) ⊂ E0 for all n. Then there exists B0

0 n 0 bounded such that φ0 = 0 on E ∩ (X \ B ). By condition (1) on ﬂasqueness, there

n 0 exists N such that f (E) ∩ B = ∅ for all n > N; thus φn(E) = 0 for all such n. So Supp(φ) ∩ E ⊂ B0 ∪ f(B0) ∪ · · · ∪ f N (B0) is bounded and φ ∈ CXp−1(X).

Finally, for every x ∈ Xp+1 there exists a maximal M with respect to the property

that ψM (x) 6= 0, and so

∞ M+1 X X ∂φ(x) = ψn(x) − ψn−1(x) = ψn(x) − ψn+1(x) = ψ0(x) = ψ(x); n=0 n=0

14 therefore ψ is a coboundary.

The most elementary example of a ﬂasque space is the ray X = [0, ∞) in R equipped with the map f : X → X which sends t ∈ X to t + 1. Clearly f is close to the identity

n n map; also, for all n ∈ N, d(f (x), f (y)) = d(x+n, y+n) = d(x, y) for all x, y ∈ X. This shows that X equipped with f satisﬁes the second and third properties for ﬂasqueness.

q q To check the ﬁrst property, let B ∈ X be bounded. If, for 1 ≤ i ≤ q, πi : X → R+ is the canonical projection map, there exists N ∈ R+ such that πi(b) < N for all i and all b ∈ B. Then, f n(X) ∩ B = ∅ for all n ≥ N.

The notion of ﬂasqueness will be revisited in this dissertation, with all relevant examples relatable back to the space [0, ∞).

1.6 Covers and Nerves

One alternative formulation of coarse cohomology (and coarse homology, as deﬁned in the next subsection) which is particularly amenable to classical topological methods involves so-called anti-Cechˇ systems of covers. In the following discussion, a cover U of a coarse space X will be a set of subsets U of X such that every x ∈ X is contained in

some U (that is to say, a cover is not necessarily composed of open sets).

Deﬁnition 1.6.1. Let X be a coarse space. A cover U of X is uniform if:

• each bounded subset of X intersects only ﬁnitely many members of U;

S • U∈U U × U is controlled.

In the metric case, uniform covers are ones which are locally ﬁnite and whose mem-

bers have uniformly bounded radii.

Deﬁnition 1.6.2. If E ⊂ X × X is controlled, a uniform cover U of X has appetite E if,

for all x ∈ X, every set of the form

0 0 x 0 0 Ex := {x ∈ X|(x , x) ∈ E} or E := {x ∈ X|(x, x ) ∈ E}

15 is contained in some member of U.

Deﬁnition 1.6.3. A collection of uniform covers of X comprises an anti-Cechˇ system if it contains a cover of appetite E for every controlled set E.

In the case of metric spaces, the above condition is equivalent to saying that for every R > 0 there exists a cover in the system with Lebesgue number at least R.

We turn an anti-Cechˇ system of covers into a directed set by declaring that U ≤ V iﬀ every element of U is contained in some element of V.

It is known that every proper metric space has such a system of covers – see Lemma

3.15 in [19] – however, we would like to also show that anti-Cechˇ systems exist for all boundedly ﬁnite coarse spaces.

Definition 1.6.4. A coarse space is boundedly finite iff every bounded subset is finite.

We ﬁrst prove the following:

Lemma 1.6.5. Let X be a boundedly ﬁnite coarse space and E ⊂ X × X controlled.

x Then, if E[x] := Ex ∪ E , there exists a subset Y of X such that:

• For all distinct x, y ∈ Y , x 6∈ E[y];

• The collection of sets E[y] for all y ∈ Y cover X.

Also, the cover UE := {E[y] | y ∈ Y } is uniform.

Proof. The class of all subsets of X with the property that distinct points x, y have x 6∈ E[y] is partially ordered by inclusion, and every chain has an upper bound. By

Zorn’s Lemma, there exists a maximal such subset, which we shall call Y . The collection

UE = {E[y] | y ∈ Y } covers X; otherwise, there would exist some x 6∈ Y such that x 6∈ E[y] for all y ∈ Y , violating the maximality of Y .

To show that UE is uniform, let B ⊂ X be bounded. Then, B is ﬁnite, as is

{y ∈ Y | E[y] ∩ B 6= ∅}, so UE is locally ﬁnite. For each E[y] ∈ UE, E[y] × E[y] =

16 {(x, x0) | (x, y), (y, x0) ∈ E ∪ E−1}, so E[y] × E[y] ⊂ (E ∪ E−1) ◦ (E ∪ E−1) for all y.

Thus [ E[y] × E[y] ⊂ (E ∪ E−1) ◦ (E ∪ E−1), y∈Y

a controlled set.

Lemma 1.6.6. Let E and Y be deﬁned as in Lemma 1.6.5 and for F controlled, let

UF := {F [y] | y ∈ Y }. Then, the collection

{UF |F controlled, E ⊂ F } forms an anti-Cechˇ system.

Proof. Let A ⊂ X × X be controlled and let A0 = E ∪ A. Then, since every x ∈ X lies

0 0x x 0 −1 in E[y] for some y ∈ Y , Ax and A (and therefore Ax and A ) lie in (E ◦ A ◦ E )[x].

So UE◦A0◦E−1 has appetite A.

Recall that, given a cover U of a space, there is an associated topological space called the nerve of U, denoted |U|. It is deﬁned to be a simplicial complex, with vertices corresponding to the members of U, and where {U0,...Un} span an n-simplex if and only if the set intersection U0 ∩U1 ∩· · ·∩Un is nonempty. As the radii of the elements in an anti-Cechˇ system of covers of X increase, points of X which are farther and farther apart from each other are contained in the same element of the cover. By examining the

“limit at inﬁnity” of this process, we can make observations about the coarse structure of X.

With these preliminary deﬁnitions established, we may give another deﬁnition of

∗ our coarse cochain groups. For a cover U of a coarse space X, let Cc (|U|) be a simplicial complex with compact supports on the locally ﬁnite complex |U|. One such

compactly supported complex, described in Chapter 3 of [19], is the compactly sup-

ported Cechˇ complex of U; i.e. the simplicial cochain complex of |U|. If Γ is a presheaf

17 of abelian groups on |U|, we may deﬁne the qth Cechˇ cochain group with coeﬃcients in

∗ Γ, Cc (|U|; Γ), to be the group of functions ψ which send each (q + 1)-tuple (U0,...,Uq)

to an element of Γ(U0 ∩ · · · ∩ Uq) and such that ψ is nonzero for only ﬁnitely many such q-tuples. Then, CX∗(X) is chain equivalent to

lim C∗(|U |), ←− c i i

where the inverse limit is taken over all covers in an anti-Cechˇ system Ui (cf. [19], Section 3.2).

Remark 1.6.7. One advantage to this deﬁnition is that it relates a coarse object (the

coarse cochain complex of a space) to topological ones (the compactly supported cochain

complexes of the nerves). However, one must avoid a pitfall here; when presented with a

cochain complex C∗ which is itself an inverse limit of countably many cochain complexes

∗ {Ci }, there exists a “Milnor lim-1 sequence” as in [15], which is a short exact sequence of the form

0 - lim 1Hp−1(C∗) - Hp(C∗) - lim Hp(C∗) - 0. (1.1) ←− i ←− i

Care must be therefore taken with respect to the lim 1 term when attempting to relate ←− classical results in algebraic topology to analogous results in coarse algebraic topology via this inverse limit method.

There are a number of other alternative formulations of coarse cochain groups (all of which in some way involve inverse limits); some of these will be examined in further detail in later chapters.

1.6.1 Coarse Homology

Dual to the notion of coarse cohomology is that of coarse homology, deﬁned in the following way:

18 Deﬁnition 1.6.8. Let X be a coarse space and G an abelian group. The group of coarse

p-chains on X with coeﬃcients in G, written CXp(X; G), is the group of all controlledly supported, locally ﬁnite linear combinations of points of Xp+1. That is to say, an element of CXp(X) is a formal sum

X σ = αxx, αx ∈ G x∈Xp+1

p+1 such that there exists a controlled E ⊂ X , called the support of σ, for which αx = 0

for all x 6∈ E; additionally, for all bounded B ⊂ E, all but ﬁnitely many αx, x ∈ B are zero.

We then equip the chain groups CX∗(X; G) with the boundary map b deﬁned by extending p X i b(x0, . . . , xp) := (−1) (x0,..., xˆi, . . . , xp+1) i=0 to formal sums as given above. (We shall not explicitly show that b is a boundary

map, but the proof is directly analogous to the one in Section 1.5.1). The resulting

homology groups are called the coarse homology groups of X with coeﬃcients in G,

written HX∗(X; G).

Analogously to the case of coarse cochains, a coarse map f : X → Y (or f : Xp+1 →

p+1 Y ) induces homomorphisms f∗ : CXp(X) → CXp(Y ) and f∗ : HXp(X) → HXp(Y ) in the obvious way.

n For example (see Chapter 2 in [20]), R has coarse homology

  Z if p = n, n ∼ HXp(R ) =  0 otherwise.

n n If N is a neighborhood of the diagonal in R of radius at least 1, HXn(R ) is generated

19 by the class of the cycle X (x0, . . . , xn). n xi∈Z ∩N As before, there exist alternative formulations of coarse chains which involve taking

limits; they are “dual” to our limit deﬁnitions of coarse cochains in the sense that direct

lf limits are dual to inverse limits. For example, if U is a cover of X, let Hp (|U|) be the locally ﬁnite homology group of the nerve |U|. Recall that the group of locally

lf ﬁnite p-chains of a simplicial complex X is the group Cp (X) consisting of formal linear combinations of p-simplices σi, X riσi, ri ∈ Z such that for all compact subsets K of X, the sum

X riσi

σi∩K6=∅

lf is ﬁnite. The locally ﬁnite homology of X is the homology of the complex C∗ (X) with respect to the obvious chain map.

Given an anti-Cechˇ system {U} of covers of X, we have that

HX (X) ' lim Hlf (|U|); p −→ ∗ U the direct limit taken over all covers in the system. (This is, in fact, the deﬁnition of

“coarse homology” that is given in Section 5.5 of [18].)

Remark 1.6.9. It may not be immediately obvious that the explicit descriptions of coarse chain and cochain groups given above agree with the direct and inverse limit formulations. The discussion of so-called controlled and cocontrolled geometric modules in Section 4.2 of this dissertation provides one method by which the claimed agreement can be proven explicitly.

20 Since taking direct limits commutes with homology, we may take the limit on the level of homology as opposed to chains. A major advantage in computing homology as opposed to cohomology using this sort of limit technique is that the lim 1 terms referred ←− to in Remark 1.6.7 do not appear when taking direct limits.

1.7 Organization

The remainder of this dissertation is organized as follows. In Chapter 2, we discuss coarse excision, Mayer-Vietoris sequences, and relative coarse cohomology. This chapter, in part, reviews the results of [6]. In Chapter 3, we introduce the notion of degree for coarse maps, with some basic results and calculations. Chapter 4 deals with geometric modules, in which we define so-called “controlled” and “cocontrolled” modules and establish an algebraic framework for the following chapter. In Chapter 5, we introduce some machinery to compute coarse (co)homology; first, applying results from homological algebra to analyze tensor products of coarse cochain complexes, then establishing coarse versions of the Künnethformula and universal coefficient theorem. Finally, Chap- ter 6 deals with computing the coarse homology and cohomology of specific spaces – for example, group extensions – by first defining “coarse CW-complexes” and “coarse

ﬁbrations”, then forming an coarse analogue to the Serre spectral sequence.

21 Chapter 2

Coarse Excision, Relative Cohomology

The goal of this section is to establish a notion of “coarse excisiveness” for coarse spaces, with associated notions of Mayer-Vietoris sequences and relative coarse (co)homology.

Once these ideas are established, we shall show that there exists a “mapping cone” construction of relative cohomology analogous to a construction in classical algebraic topology.

2.1 The Mayer-Vietoris Sequence

Let X be a coarse space, with A and B subspaces of X such that X = A∪B. In general, there exists no Mayer-Vietoris long exact sequence relating the coarse (co)homology of

X to that of A, B, and A ∩ B. For such a sequence to exist, we need additional excisiveness condition, based upon neighborhoods of the subspaces A and B in X:

Deﬁnition 2.1.1. If A ⊂ X and E ⊂ X ×X is a controlled subset containing the diagonal

22 in A × A, then deﬁne the E-neighborhood of A in X to be

NE(A) := {x ∈ X|(a, x) ∈ E for some a ∈ A}.

Deﬁnition 2.1.2. Let X = A ∪ B as before. A and B form a coarsely excisive decomposition of X if, for each controlled E ⊂ X × X containing δ(A∪B)×(A∪B) there exists

F ⊂ X × X, controlled and containing δ(A∩B)×(A∩B), such that

NE(A) ∩ NE(B) ⊂ NF (A ∩ B).

In the case that X has a metric coarse structure, the notion of coarsely excisive decomposition is easier to visualize. If A ⊂ X and R > 0, let NR(A) denote the radius R neighborhood of A in X. Then:

Deﬁnition 2.1.3. If X has a metric coarse structure, the union A ∪ B = X is a coarsely excisive decomposition if, for every R > 0 there exists S > 0 such that

NR(A) ∩ NR(B) ⊂ NS(A ∩ B).

Example 2.1.4. Let X = R and A = (−∞, 0], B = [0, ∞). Then clearly A ∪ B = X is coarsely excisive, since for every R > 0, NR(A) ∩ NR(B) = (−∞,R) ∩ (−R, ∞) = (−R,R), a radius R neighborhood of {0} = A ∩ B.

Example 2.1.5. An example of a decomposition which is not coarsely excisive is as

follows. Let X be the union of two points x and y which are distance 1 apart. Then

X = {x} ∪ {y} is not coarsely excisive: for example, N2({x}) = N2({y}) = X, so

N2({x}) ∩ N2({y}) = X, while NS({x} ∩ {y}) = ∅ for all S.

It is not surprising that this sort of additional property is needed in order to con-

struct a Mayer-Vietoris sequence. In classical algebraic topology, we need a similar sort

of excisiveness for decompositions of topological spaces, the analogous condition (in sin-

23 gular cohomology) being that A◦ ∪ B◦ = X. As was observed in [11], the diﬀerence

between the topological and coarse notions of excisiveness arises from the fact that in

coarse theory, the role of small open sets in topology is replaced by large neighborhoods.

In [11], Roe, Higson, and Yu prove that if X has a metrizable coarse structure and

A∪B = X is a coarsely excisive decomposition, there exists an associated Mayer-Vietoris

long exact sequence in coarse cohomology; i.e., a sequence of the form

· · · → HXp(X) → HXp(A) ⊕ HXp(B) → HXp(A ∩ B) → HXp+1(X) → · · · .

The proof of this result will not be replicated here in its entirety; however, a key part

of it involves forming the coarse cochain groups of neighborhoods

∗ ∗ ∗ CX (NE(A)),CX (NE(B)),CX (NE(A) ∩ NE(B)),

and observing that the (not necessarily exact) sequence

∗ ∗ ∗ ∗ 0 → CX (X) → CX (NE(A)) ⊕ CX (NE(B)) → CX (NE(A) ∩ NE(B)) → 0

passes to an exact sequence of coarse cochain groups

0 → CX∗(X) → CX∗(A) ⊕ CX∗(B) → CX∗(A ∩ B) → 0

via the inverse limit. A similar argument is used later in this chapter to prove excision

for coarse cohomology. We shall attempt to replicate the spirit of this argument in

forming an analogous statement about coarse homology.

Proposition 2.1.6 (Mayer-Vietoris sequence in coarse homology). Let A ∪ B = X be a

coarsely excisive decomposition. Then there exists a Mayer-Vietoris sequence in coarse

24 homology, of the form

· · · → HXp(A ∩ B) → HXp(A) ⊕ HXp(B) → HXp(X) → HXp−1(A ∩ B) → · · · .

Proof. Let {Ui} be an anti-Cechˇ system of covers of X, and deﬁne for each i

A B Ui := {U ∈ Ui|U ∩ A 6= ∅}, Ui := {U ∈ Ui|U ∩ B 6= ∅},

A∩B Ui := {U ∈ Ui|U ∩ (A ∩ B) 6= ∅},

A ˇ As a consequence of the subspace coarse structure, {Ui } is an anti-Cech system of B A∩B covers of A (similarly {Ui } and {Ui } for B and A ∩ B). Observe also that for all i, A B |Ui | (similarly |Ui |) is a subcomplex of |Ui|. lf lf Let H∗ be a locally ﬁnite simplicial homology theory with chain groups C∗ . By classical topological results, we have a short exact sequence of locally ﬁnite chain groups

lf A B lf A lf B lf A B 0 → C∗ (|Ui | ∩ |Ui |) → C∗ (|Ui |) ⊕ C∗ (|Ui |) → C∗ (|Ui | ∪ |Ui |) → 0 for each i. One way that this can be seen is by using the locally ﬁnite simplicial chain

A B groups of the simplicial complexes |Ui|, |Ui |, and |Ui |. By bounded ﬁniteness of the lf covers Ui, the above simplicial complexes are locally ﬁnite; therefore, the group Cn (|Ui|) consists of all formal sums of simplices in |Ui| (similarly for A and B). Then, the maps in the above sequence are the usual ones; namely, the map

lf A B lf A lf B in ⊕ jn : Cn (|Ui | ∩ |Ui |) → Cn (|Ui |) ⊕ Cn (|Ui |)

A B sending a formal linear combination of n-simplices in |Ui | ∩ |Ui | to the direct sum of

25 A B its images under inclusion in |Ui | and |Ui |, and the map

lf A lf B lf lf A B kn : Cn (|Ui |) ⊕ Cn (|Ui |) → Cn (|Ui|) = Cn (|Ui | ∪ |Ui |)

induced by the diﬀerence of inclusions. Clearly in ⊕ jn is injective. Furthermore kn is surjective for the same reason that its analogue for ordinary simplicial chains is surjective; a formal linear combination of simplices in |Ui| can be decomposed as a sum

A of a formal linear combination of simplices in |Ui | plus a formal linear combination B of simplices in |Ui |. To show this, let σ = [U0,...Uk] be a k-simplex in |Ui|. Then there exist open sets U0,...,Uk ∈ Ui such that U := U0 ∩ · · · ∩ Uk is nonempty. Since A∪B = X, U has nonempty intersection with at least one of A, B. Assume without loss

A A A A of generality that U ∩A 6= ∅. Then Uj := Uj ∩A 6= ∅ for 0 ≤ j ≤ k, so σ = U0 ,...Uk A is a k-simplex in {Ui }. Using this fact, a linear combination

X lf riσi ∈ Cn (|Ui|) has a decomposition

X 0 00A X 00 00B lf A B riσi , ri σi ∈ Cn (|Ui | ∪ |Ui |).

A Since A and B are coarsely excisive, and since Ui is a neighborhood of A, it follows A B that Ui ∩ Ui is contained in some neighborhood of A ∩ B. By the fact that {Ui} is an anti-Cechˇ system, for every i there exists j such that

A B A∩B A B A∩B Ui ∩ Ui ⊂ Uj , i.e., |Ui | ∩ |Ui | ⊂ |Uj |.

Since the direct limit functor is exact and commutes with the boundary map in homol-

26 ogy, we may take the direct limit of the chain short exact sequence

lf A B lf A lf B lf A B 0 → C∗ (|Ui | ∩ |Ui |) → C∗ (|Ui |) ⊕ C∗ (|Ui |) → C∗ (|Ui | ∪ |Ui |) → 0 to obtain a short exact sequence of chain complexes, which is chain equivalent to

0 → CX∗(A ∩ B) → CX∗(A) ⊕ CX∗(B) → CX∗(A ∪ B) → 0.

The homomorphisms in this sequence are direct limits of chain maps, and therefore are themselves chain maps. Thus we have an associated long exact homology sequence, which is the Mayer-Vietoris sequence in coarse homology.

2.2 Deﬁnition of Relative Coarse Cohomology

Let X be a coarse space and A a subspace of X. Observe that, by the deﬁnition of the subspace coarse structure, the inclusion map from A to X is coarse. Therefore, by the discussion in the previous chapter, inclusion induces a map j : CXp(X) → CXp(A) deﬁned by restricting each coarse p-cochain on X to Ap+1. Furthermore, j is surjective:

p+1 if φ : A → Z is a coarse p-cochain on A, we may extend φ to a coarse p-cochain φ on X by deﬁning φ to be zero outside Ap+1. Therefore, we have a short exact sequence of groups

i j 0 - ker(j) - CXp(X) - CXp(A) - 0.

here, i is the inclusion map.

Deﬁnition 2.2.1. The group of p-cochains X relative to A, written CXp(X,A), is ker(j);

i.e.,

p p CX (X,A) := {ψ ∈ CX (X) such that ψ|Ap+1 = 0}.

27 The group homomorphisms i and j are clearly chain maps (since inclusions and restrictions commute with coboundary maps), and we therefore have an associated long exact cohomology sequence involving the induced homomorphisms i∗ and j∗:

i∗ j∗ δ ··· - HXp(X,A) - HXp(X) - HXp(A) - HXp+1(X,A) - ··· .

Recall that one deﬁnes the homomorphism δ as follows: let [ψ] ∈ HXp(A) be represented

by the cocycle ψ. Since j is surjective, ψ = jφ for some φ ∈ CXp(X). Since j(∂φ) =

∂(jφ) = ∂ψ = 0, ∂φ ∈ ker(j) = im(i). Therefore ∂φ = iψ0 for some ψ0 ∈ CXp+1(X,A).

Note that ψ0 is a cocycle, since i is injective and i∂ψ0 = ∂iψ = ∂∂φ = 0. Deﬁne δ[ψ] to

be [ψ0].

Remark 2.2.2. A straightforward generalization is the long exact sequence of a triple

(X, A, B); if X is a coarse space and B ⊂ A ⊂ X, then there obviously exists a

surjection j from CXp(X,B) to CXp(A, B) given by restricting every cochain on Xp+1

which is zero on Bp+1 to Ap+1. The kernel of j is then the group of cochains on Xp+1

which are zero on Ap+1; that is, ker j = CXp(X,A). Therefore (referring to the previous

argument for pairs to ensure the homomorphisms involved are cochain maps), we have

a short exact sequence of relative cochain complexes

0 - CX∗(X,A) - CX∗(X,B) - CX∗(A, B) - 0

giving rise to a long exact sequence of relative cohomology groups

··· - HXp(X,A) - HXp(X,B) - HXp(A, B) - HXp+1(X,A) - ··· .

28 2.3 Coarse Excision

With relative coarse cohomology thus deﬁned, we now wish to establish a coarse excision principle; that is to say,

Proposition 2.3.1. When A ∪ B = X is a coarsely excisive decomposition,

HXp(X,A) =∼ HXp(B,A ∩ B)

for all p.

Proof. We begin by deﬁning an additional cochain complex:

Deﬁnition 2.3.2. Let X be a coarse space and A a subspace of X. The group of p-

cochains of A subordinate to X, written CXp(A ⊂ X), is the group of maps ψ : Xp+1 →

p+1 p Z such that ψ restricted to NE(A) is an element of CX (NE(A)) for all E. Equipping CX∗(A ⊂ X) with the Alexander-Spanier coboundary map as before, we

write the cohomology groups of this complex as HX∗(A ⊂ X).

For controlled subsets E,F ⊂ X × X such that E ⊂ F , there exists a (surjective)

p p map from CX (NF (A)) to CX (NE(A)) induced by the inclusion NE(A) ⊂ NF (A). Thus the group

lim CXp(N (A)), ←− E E the inverse limit taken over all controlled subsets of X × X, is well-deﬁned. Recalling

the deﬁnition of the inverse limit, note that lim CXp(N (A)) consists of sequences of ←−E E p+1 coarse cochains (ψ0, ψ1, ψ2,...), such that the restrictions of ψi, ψi+1,... to NEi (A)

agree. In other words, for each such (ψ0, ψ1, ψ2,...) there exists a unique map ψ on

p+1 p+1 X whose restriction to NEi (A) is ψi. Therefore the map θ :(ψ0, ψ1, ψ2,...) 7→ ψ deﬁnes a homomorphism θ : lim CXp(N (A)) → CXp(A ⊂ X). By deﬁnition of ←−E E

θ, θ(ψ0, ψ1, ψ2,...) = 0 iﬀ every ψi = 0; in which case, (ψ0, ψ1, ψ2,...) is the zero element, so θ is injective. To show θ is surjective, let ψ ∈ CXp(A ⊂ X). By deﬁnition

29 p p+1 of CX (A ⊂ X), the restriction ψi of ψ to NEi (A) for each i is an element of p p+1 CX (NEi ), and clearly for all Ej containing Ei, the restriction of ψj to NEi (A)

agrees with ψi. This shows that θ is surjective, and so

lim CXp(N (A)) ∼ CXp(A ⊂ X). ←− E = E

∗ ∗ Letting CE := CX (NE(A)), we claim that standard results in homological algebra [15] give a short exact sequence

0 - lim 1 Hp−1(C∗ ) - Hp(lim C∗ ) - lim Hp(C∗ ) - 0, ←− E ←− E ←− E E E E

where the inverse limit is taken over all controlled E.

Remark 2.3.3. When X is metrizable, the directed systems we have deﬁned previously

are countable, and the above exact sequence follows from the fact that all derived

functors of the inverse limit higher than lim 1 will vanish. In the non-metrizable case, ←− the above still holds, but a few additional technical lemmas are required; they will not

be repeated here, but the curious reader may refer to the proof of Proposition 4.6 in [6].

Additionally, for each controlled E, the spaces A and NE(A) are coarsely equivalent,

the associated coarse maps being inclusion and a map sending each x ∈ NE(A) to an x0 ∈ A such that (x, x0) ∈ E. By Theorem 1.5.3,

Hp(lim CXp(N (A)) ∼ HXp(A), ←− E =

and therefore HXp(A ⊂ X) =∼ HXp(A). An isomorphism from HXp(A ⊂ X) to HXp(A) can be constructed explicitly; it induced by the map which sends each cocycle

representative to its restriction on Ap+1.

The next step in the proof of coarse excision is to show that there exists a quotient

complex whose cohomology is isomorphic to relative cohomology; in particular:

30 p p p p ∗ ∼ p+1 Lemma 2.3.4. Let CA := CX (A ⊂ X)/CX (X). Then H (CA) = HX (X,A).

Proof. First, observe that the above quotient complex is well deﬁned. Indeed, CX∗(X)

is a subcomplex of CX∗(A ⊂ X); therefore, the short exact sequence

- p i-0 p j0 - p - 0 CX (X) CX (A ⊂ X) CA 0

yields a long exact cohomology sequence

∗ ∗ - p i-0 p j0- p ∗ δ-0 p+1 - ··· HX (X) HX (A ⊂ X) H (CA) HX (X) ··· ,

with δ0 deﬁned analogously to δ in the relative long exact sequence. By a previous argument, HXp(A ⊂ X) =∼ HXp(A), so we have a diagram

∗ ∗ ∗ - p i-0 p j0 - p ∗ δ-0 p+1 - ··· HX (X) HX (A ⊂ X) H (CA) HX (X) ···

=∼ =∼ =∼ ? ? j∗ ? δ i∗ ··· - HXp(X) - HXp(A) - HXp+1(X,A) - HXp+1(X) - ··· .

p ∗ p+1 By constructing a map β from H (CA) to HX (X,A) which makes the above diagram p ∗ commute, we shall obtain the desired result from the ﬁve lemma. Let [ψ] ∈ H (CA) with p cocycle representative ψ. There exists φ ∈ CX (A ⊂ X) such that j0φ = ψ; we may assume that the restriction of φ to Ap+1 is zero (since for any such φ, the restriction

p+1 p of φ to A is an element of CX (X), thus φ and φ − φ|Ap+1 are both sent to ψ). 0 p+1 0 Then ∂φ ∈ ker(j0) = im(i0), so there exists ψ ∈ CX (X) such that ∂φ = i0ψ and ∂ψ0 = ∂∂φ = 0; i.e. ψ0 ∈ ZXp+1(X). Since φ vanishes on Ap+1, ∂φ vanishes on Ap+2,

so ψ0 vanishes on Ap+2 as well. Therefore ψ0 ∈ ZXp+1(X,A), and we deﬁne

β[ψ] := [ψ0].

31 It remains to show that this construction is independent of the choices made. Suppose

p p−1 ψ = ∂α ∈ CA is a coboundary. By the above argument, there exists β ∈ CX (A ⊂

X) such that j0(β) = α; since the quotient map j0 is a cochain map, it follows that

0 p+1 j0(∂β) = ∂α = ψ. In this case, ψ ∈ CX (X) as described above is zero, and therefore β[ψ] = 0. Therefore, β is independent of the cocycle representative ψ chosen.

0 p 0 0 Similarly, suppose φ, φ ∈ CX with j0(φ) = j0(φ ) = ψ. Then φ − φ ∈ ker j0, so

p 0 0 there exists α ∈ CX (X) with i0(α) = φ − φ . So ψ as deﬁned above is the coboundary ∂α and it follows that β[ψ] is also independent of the representative φ ∈ CXp(A ⊂ X

chosen.

∗ ∗ By construction, i β = δ0, so we need only show that βj0 = δ to prove commutativity of our diagram. A cocycle ψ ∈ ZXq(A ⊂ X) can be mapped to a cocycle ψ0 ∈ ZXq(A)

such that [ψ] and [ψ0] are isomorphic by letting ψ0 be the restriction of ψ to Aq+1. Now

∗ q q note that we begin to calculate βj0 [ψ] by ﬁrst mapping ψ to CX (A ⊂ X)/CX (X), then pulling back to φ ∈ CXq(A ⊂ X) such that the restriction of φ to Aq+1 is zero. If

we map ψ0 isomorphically to an element of CXq(A ⊂ X) via extending by zero outside

Aq+1, we may let φ = ψ0 − ψ. Then ∂(ψ0 − ψ) = ∂ψ0 − ∂ψ = ∂ψ0. It follows that

∗ 0 βj0 [ψ] = δ[ψ ].

The purpose of this “subordinate cohomology” construction is to restate the coarse excision theorem in a more easily computable way. We claim that the result can be reduced to the following statement about cochain groups:

Lemma 2.3.5. Let A ∪ B = X be a coarsely excisive decomposition. Then

CXp(A ⊂ X)/CXp(X) =∼ CXp(A ∩ B ⊂ X)/CXp(B ⊂ X).

Proof. The proof is by a Mayer-Vietoris type construction. For each controlled E ⊂

p p p X × X the groups CX (NE(A)), CX (NE(B)) and CX (NE(A) ∩ NE(B)) ﬁt into a

32 sequence

- p i-E p p j-E p - 0 CX (X) CX (NE(A)) ⊕ CX (NE(B)) CX (NE(A) ∩ NE(B)) 0,

where iE is a direct sum of restriction maps and jE is a diﬀerence of two restriction maps. The sequence is not exact in general; however, passing to the inverse limit, we

obtain

i j 0 - CXp(X) - CXp(A ⊂ X) ⊕ CXp(B ⊂ X) - CXp(A ∩ B ⊂ X) - 0,

and we claim that this sequence is exact. Let ψ ∈ CXp(X); then i(ψ) = (ψ, ψ) ∈

CXp(A ⊂ X) ⊕ CXp(B ⊂ X). Therefore i is injective. Similarly, if (φ, ψ) ∈ CXp(A ⊂

X) ⊕ CXp(B ⊂ X), then j(φ, ψ) = φ − ψ. The map φ − ψ has cocontrolled support on

(A ∩ B)p+1, since φ and ψ have cocontrolled support on Ap+1 and Bp+1 respectively.

Also, j is surjective; if α ∈ CXp(A ∩ B ⊂ X), we may deﬁne a map φ on Xp+1 to be

equal to α on (A∩B)p+1 and on the complement of Ap+1 and zero otherwise. Then φ has

cocontrolled support on Ap+1. Similarly, deﬁne ψ to be equal to −α on Ap+1 \(A∩B)p+1

and zero otherwise. Then ψ ∈ CXp(B ⊂ X) and j(φ, ψ) = α.

Obviously im i ⊂ ker j, since j ◦ i(ψ) = j(ψ, ψ) = 0. To show the opposite inclusion,

observe that an element of ker j is a pair (φ, ψ) ∈ CXp(A ⊂ X) ⊕ CXp(B ⊂ X) such

that φ − ψ = 0, that is, φ = ψ. We therefore must show that ψ ∈ CXp(X). Let E be a

controlled subset of Xp+1; we shall construct a controlled subset F of Ap+1 as follows.

p+1 0 0 0 p For all x = (x1, . . . , xp) ∈ E which is not in B , construct x = (x1, . . . , xp) ∈ A 0 0 by letting xi = xi if xi ∈ A and by choosing xj ∈ A and letting xi = xj if xi 6∈ A. Then let F be the set of all such x0. We build a controlled F 0 ⊂ Bp analogously. Then,

Supp(ψ) ∩ F and Supp(ψ) ∩ F 0 are bounded. That is, there exists a ∈ A and controlled

0 0 0 0 G ⊂ A × A such that, for all (x1, . . . , xp) ∈ F ,(a, xi) ∈ G for all i, and there exists 0 00 00 0 00 0 b ⊂ B and controlled G ⊂ B × B such that, for all (x1, . . . , xp) ∈ F ,(b, xi ) ∈ G for

33 0 all i. Then, for all (x1, . . . , xp) ∈ E,(a, xi) ∈ F or (a, xi) ∈ {(a, b)} ◦ G . Therefore Supp(ψ) ∩ E is bounded.

It follows that the sequence is short exact.

Now, consider the homomorphism θ : CXp(A ⊂ X) → CXp(A ∩ B ⊂ X)/CXp(B ⊂

X) given by composing the inclusion CXp(A ⊂ X) ,→ CXp(A ∩ B ⊂ X) with the

quotient map. First we shall show θ is surjective; let the class [α] ∈ CXp(A ∩ B ⊂

X)/CXp(B ⊂ X) be represented by α ∈ CXp(A ∩ B ⊂ X). By the above discussion,

there exist ψ ∈ CXp(A ⊂ X) and φ ∈ CXp(B ⊂ X) such that ψ − φ = α. Therefore,

the equivalence class of ψ in CXp(A ∩ B ⊂ X)/CXp(B ⊂ X) is equal to the class of

ψ − φ; i.e. θ(ψ) = [α]. A map ψ on Xp+1 is an element of ker θ if and only if ψ is

an element of CXp(A) and CXp(B). Since A ∩ B = X, for every controlled subset E

of Xp+1 there exist controlled subsets F,G of X × X and controlled subsets F 0,G0 of

p+1 p+1 0 0 0 NF (A) , NG(B) such that E ⊂ F ∪ G . Since the intersection of Supp(ψ) with F and G0 are bounded, Supp(ψ) ∩ E is bounded as well. It follows that ker θ = CXp(X),

and therefore the lemma holds.

The above quotient maps of groups are cochain maps, therefore the group isomorphisms pass to isomorphisms on cohomology. Also, HXp(A∩B ⊂ X) =∼ HXp(A∩B) and

HXp(B ⊂ X) =∼ HXp(B), so repeating the previous ﬁve-lemma argument shows that the right-hand chain complex had coarse cohomology isomorphic to HX∗(B,A∩B).

2.4 The Mapping Cone Construction

As stated earlier, the above exercises are intended to lead us towards a “mapping cone construction” of relative cohomology. The desired construction is the following:

Deﬁnition 2.4.1. Let X be a coarse space and A be a subspace of X, and let R+ be the ray [0, ∞) with the obvious metric coarse structure. The coarse cone on A, denoted

CA, is the space A × R+, with the product coarse structure.

34 The coarse mapping cone of A into X, denoted X ∪c CA, is the coarse space X ∪f CA with respect to the gluing map f : A → C(A) deﬁned by f(a) = (a, 0).

We may now state the main result.

Proposition 2.4.2. For X and A deﬁned as above,

p ∼ p HX (X,A) = HX (X ∪c CA).

Proof. The initial step will be to apply coarse excision to the subspaces CA, X of X ∪c CA. These two spaces indeed form a coarsely excisive decomposition; a neighborhood

0 0 N of X in X ∪c CA is of the form X ∪ N where N is a neighborhood of A × 0 in CA.

0 0 Similarly, a neighborhood M of CA in X ∪c CA is of the form CA ∪ M where M is a neighborhood of A in X. Therefore, N ∩ M = N 0 ∩ M 0, a neighborhood of X ∩ CA = A.

From the discussion of gluing maps in Section 1.4.1 the coarse structures of X and A as subspaces of X ∪c CA agree with the original coarse structures; so by coarse excision,

p ∼ p Lemma 2.4.3. HX (X ∪c CA, CA) = HX (X,A).

The next step is to recall the long exact sequence involving the cohomology of X∪cCA relative to CA:

p−1 p p p · · · → HX (CA) → HX (X ∪c CA, CA) → HX (X ∪c CA) → HX (CA) → · · · , and then recall from the introductory chapter that the coarse cohomology of ﬂasque spaces vanishes in all dimensions. We aim to show that:

Lemma 2.4.4. The space CA is ﬂasque.

Proof. Clearly CA is unbounded, since it contains as an embedded subspace the unbounded space {a}×R+ for any ﬁxed a ∈ A. Therefore, we shall verify that there exists a map f : CA → CA such that:

35 1. For every bounded subset B ⊂ Xq, there exists N such that f n(Xq) ∩ B = ∅ for

all n ≥ N.

2. Given any controlled set E, there exists another controlled E0 such that f n(E) ⊂

E0 for all n.

3. f is close to the identity map on X.

Consider the map f sending each (a, t) to (a, t + 1); it is indeed close to the identity map, since ((a, t), (a, t + 1)) ⊂ ∆A×A × N1(∆R+×R+ ) for all a ∈ A, t ≥ 0. q q Next, let B ⊂ (CA) be bounded. The projection of B to (R+) must then also be bounded and therefore is a subset of [0,N] for some N > 0. For n > N, the image of B under f n must have empty intersection with B.

q Finally, let E ⊂ (CA) be controlled. The projections E1,E2 of E to A, R+ respectively must be controlled, and therefore E is a subset of the controlled set E1 ×E2 (using

q q q the obvious identiﬁcation (A × R+) = A × (R+) ). Then, if g is the “ﬂasqueness map” on R+ sending t to t + 1,

∞ ∞ ∞ ∞ [ n [ n [ n [ n f (E) ⊂ f (E1 × E2) ⊂ E1 × g (E2) ⊂ E1 × g (E2). n=1 n=1 n=1 n=1

S∞ n q S∞ n Since the space n=1 g (E2) is controlled in (R+) , E1 × n=1 g (E2) is controlled in (CA)q

p ∼ p ∼ From the preceding lemmas, it now follows that HX (X,A) = HX (X ∪c CA, CA) =

p HX (X ∪c CA) for all p.

36 Chapter 3

Coarse Degree Theory

n As stated in the introductory chapter, the coarse space R has coarse homology equal to   Z if p = n, n ∼ HXp(R ) =  0 otherwise.

lf n ∼ In fact, the above holds for locally ﬁnite homology: Hp (R ) = Z for p = n and is zero otherwise. In this sense, Euclidean n-space takes a role in locally ﬁnite homology

analogous to that of the n-sphere Sn in ordinary homology. Since the locally ﬁnite and

n lf n n coarse homologies of R are isomorphic, the isomorphism from H∗ (R ) to HX∗(R ) realized by sending each locally ﬁnite cycle to its truncation on some ﬁxed neighborhood

n of the diagonal, one may think of R as a “coarse n-sphere” as well. If we have a coarse n n map f : R → R , it therefore makes sense to discuss the degree of f:

n n Deﬁnition 3.0.5. Let f : R → R be a coarse map. Then, the induced homomorphism n f∗ : Z → Z on HXn(R ; Z) is of the form f∗(z) = d · z for some integer d. The degree of f, denoted deg f, is d.

In this section, we shall make a number of general observations about the degree of

coarse maps, as well as provide a combinatorial method for determining degree.

37 3.1 Degree and Essential Surjectivity

One well-known result in classical algebraic topology is the following: if a continuous map f : Sn → Sn has nonzero degree, then f is necessarily surjective. We shall show that there is a coarse analogue of this phenomenon:

n n Proposition 3.1.1. Let f : R → R be a coarse map. If f has nonzero coarse degree, n then there exists r > 0 for which every x ∈ R lies in the radius r neighborhood of im(f) (we shall call such a map essentially surjective).

Proof. In the case n = 1, the proof is straightforward. Suppose the coarse map f :

R → R is not essentially surjective; we shall show that the image of f is a subset of a ﬂasque space. Since f is bornologous, there exists R > 0 such that, if |x − y| ≤ 1,

|f(x) − f(y)| ≤ R for all x, y ∈ R. Since f is not essentially surjective, there exists

z ∈ R such that no element of im(f) lies in the interval (z − R, z + R). Then, either f(0) < z − R or f(0) > z + R; we shall assume without loss of generality that f(0) >

z + R. Furthermore, for any x such that f(x) > z + R and y ∈ [x − 1, x + 1], since im(f) ∩ (z − R, z + R) = ∅, it follows that f(y) > z + R as well. By induction, im(f) is a subset of [z + R, ∞), which is coarsely equivalent to R+. Since the coarse cohomology of a ray vanishes in all dimensions, and f factors as a map to a ray composed with an inclusion map, f necessarily has degree 0.

The proof for n ≥ 2 will generalize this idea. If f is not essentially surjective, we shall show that im(f) lies in a subset A with HXn(A) = 0. Suppose f is not essentially

n surjective; we claim there exists a subset B of R composed of a disjoint union of balls

Bi such that:

• im(f) is a subset of X \ B;

• the radii of the Bi become arbitrarily large;

• the distance between any distinct Bi and Bj is greater than or equal to the max-

38 imum of their diameters.

We shall prove this via an inductive process. Let {r1, r2,...} be a sequence of positive real numbers such that ri < ri+1 and limi→∞ ri = ∞. Since f is not essentially sur-

n jective, we may select an x1 ∈ R such that the ball B1 of radius r1 centered at x1 is 0 disjoint from the image of f. Now suppose we have a ﬁnite collection B = {B1,...,Bk}

0 of balls with im(f) ⊂ X \ B , the radius of Bi equal to ri, and the distance between distinct Bi and Bj at least equal to the maximum of their diameters. We then add

0 a ball Bk+1 to B in the following way: choose d > 0 such that there exists a ball C

0 0 centered at the origin of radius d with B ⊂ C. There exists a ball C of radius d+3rk+1

0 0 such that im(f) ∩ C = ∅ and there necessarily exist xk+1 ∈ C such that the distance

0 between xk+1 and the center of C is at most rk+1 and the distance between xk+1 and

0 C is at least rk+1. Let Bk+1 be the ball of radius rk+1 centered at xk+1; Bk+1 ⊂ C , so

0 im(f) ∩ Bk+1 = ∅, and the distance between Bk+1 and B is at least rk+1.

If Si is the boundary sphere of each Bi, let

A := (X \ B) ∪ tiSi;

clearly im(f) ⊂ A. It suﬃces to show that HXn(A) vanishes; since f factors as a map

to A composed with an inclusion, this would prove that f∗ is the zero homomorphism on

n HXn(R ). Note that (A, B) form a coarsely excisive pair with A∩B = tSi, the union of

the boundary spheres; by the construction of A and B, a radius R neighborhood NR(A)

of A is the union of A with the radius R neighborhood of A ∩ B; similarly, NR(B) is

the union of B with the radius R neighborhood of A ∩ B. Therefore, NR(A) ∩ NR(B) =

NR(A ∩ B).

We introduce a system of “coarsenings” {Xr} of A∩B by ﬁrst deﬁning an equivalence

relation ∼r on A ∩ B as follows; for each i, ﬁx a point xi on the sphere Si ⊂ A ∩ B.

Then, say that for x, y ∈ A ∩ B, x ∼r y if and only for some i, x and y both lie in the

39 intersection of Si with the open ball of radius r centered at xi, or if x = y. Finally, let

Xr := (A ∩ B)/ ∼r .

There exists an anti-Cechˇ system of covers {Ur} of A ∩ B such that the nerve Ur is

th homeomorphic to the r coarsening Xr. We may select each Ur to be the intersection √ n with A ∩ B of a cover V of R consisting of open balls of radius r 2 such that, for each xi as deﬁned above, there exist balls V ∈ V centered at xi and its antipodal point on

Si. Then, the nerve |Ur| consists of a disjoint union of points corresponding to each sphere with radius less than r/2 along with a disjoint union of boundaries of (n − 1)- dimensional polyhedra, whose faces are (n − 1)-simplices, corresponding to each sphere with radius at least r/2. Therefore |Ur| and Xr are homeomorphic. By the discussion of coarse homology in the introduction, HX (A ∩ B) ∼ lim Hlf (X ). p = −→ p r

Lemma 3.1.2. HXn−1(A ∩ B) is nontrivial.

th Proof. The r coarsening Xr of A ∩ B identiﬁes points which are distance at most r

from a ﬁxed xi; since the diameters of the spheres in A ∩ B become arbitrarily large, Xr is homeomorphic to a disjoint union of points plus a disjoint union of inﬁnitely many

(n − 1)-spheres. By topological methods, (for example, Proposition 3.12(ii) of [12]) we

observe that lf ∼ Y Hn−1(Xr) = Z,

a copy of Z for each (n − 1)-sphere in the coarsening. lf The group homomorphisms in the directed system {Hp (Xr)} are described on the

individual copies of Z composing the direct product as follows. Let zi be an element

of the copy of Z corresponding to the sphere Si ⊂ Xr. For r < s, the homomorphisms lf lf Hp (Xr) → Hp (Xs) are induced by maps φi on the individual spheres Si of Xr as

follows. If the image of Si in Xs is a sphere, then φi is a degree one continuous map; if

40 the image of Si in Xs is a point, then φi maps Si to that point. Therefore, the induced

homomorphism φi∗ : Z → Z on homology sends zi to zi if the image of Si in Xs is a

sphere, and sends zi to zero if the image of Si in Xs is a point. The homomorphisms in lf the directed system {Hp (Xr)} are those whose restrictions to each copy of Z are the

φi∗ above. lf An element in any Hn−1(Xr) is equivalent to the zero element in the direct limit if and only if it is eventually mapped to zero in the directed system. In particular, an Q ∼ lf element (z1, z2,...) ∈ Z = Hn−1(X0) is equivalent to zero in the direct limit if and only if all but ﬁnitely many of the z are zero. It follows that lim Hlf (X ) is nonzero i −→ n−1 r (and is in fact inﬁnitely generated); one particular nonzero element is represented by

the element

lf (1, 1,..., 1) ∈ Hn−1(X0)

consisting of the sequence of generators of the fundamental classes for each (n−1)-sphere

in B = X0.

By coarse Mayer-Vietoris, we have a long exact sequence

· · · → HXn(A ∩ B) → HXn(A) ⊕ HXn(B) → HXn(A ∪ B) → HXn−1(A ∩ B) ··· .

We wish to show that the generator of HXn(A∪B) maps to the element of HXn−1(A∩B) represented by

lf (1, 1,..., 1) ∈ Hn−1(X0)

via the M-V boundary.

Let {Ur} be an Anti-Cechˇ sequence of covers on A ∪ B consisting of closed balls of √ radius r 2 as described before. Let Yr be the nerve |Ur| and Ar,Br be the nerves of the restrictions of {Ur} to A and B as described in the proof of Proposition 2.1.3. For

41 lf each r, the fundamental class of Hn (Yr) is represented by the sum

X σ = σi

of all n-simplices in Yr. As discussed in Section 2.1, the decomposition Ar ∪ Br = Yr is excisive, and the image of σ under the Mayer-Vietoris boundary map is

  X ∂σ = b  σi . σi∈Br

Since the boundary of the fundamental class of each ball in Br is the fundamental class lf of its boundary sphere, it follows that ∂σ is the image of (1, 1,...) ∈ Hn−1(X0) in lf th Hn−1(Xr), where Xr is the r coarsening of A ∩ B as described earlier. Passing to the direct limit, we see that the Mayer-Vietoris boundary indeed sends the generator of

lf HXn(A ∪ B) to the element of HXn−1(A ∩ B) represented by (1, 1,..., 1) ∈ Hn−1(X0); in particular, it is injective. Thus, HXn(A) = 0.

3.2 A Combinatorial Coarse Degree Formula

We begin with the following known result for classical degree (cf. Chapter 4, Exercise

F.1 in [23]):

Lemma 3.2.1. Let X and Y be n-dimensional oriented triangulated manifolds and f : X → Y be a simplicial map. If σ is an n-simplex in Y and

−1 + + − − f (σ) = {σ1 , . . . σm, σ1 , . . . σk }, where the σ+’s are positively oriented and the σ−’s negatively oriented, then deg f = m − k.

The goal of this section is to show that there exists an analogue of this combinatorial

42 n n formula for coarse maps from R to R . n n n Let f : R → R be a coarse map and ∆ be a triangulation of R formed by n tessellating a triangulation of an n-torus. Then, the cover U1 of R consisting of all the stars of the vertices of ∆ has nerve homeomorphic to ∆. The homeomorphism is realized

by the bijective simplicial map from ∆ to |U1| which sends each vertex of ∆ to the vertex

n corresponding to its star. Then, deﬁne a system of covers {Ur} on R by scaling U1 by a factor of r for each r. By the Lebesgue number lemma, the tessellated triangulation of

torus has positive Lebesgue number δ; therefore, U1 has Lebesgue number δ as well. It

follows that the Lebesgue numbers of the system {Ur} become arbitrarily large; i.e. the system is anti-Cech.ˇ The nerve of each of these covers provides a triangulation of the

n n-manifold R . Since f is a coarse map, there exists R such f sends each ball of radius

C2 in {U1} inside a ball of radius at most R. Furthermore, as the Lebesgue numbers

of covers in the system are arbitrarily large, we can ﬁnd a cover {US} for which each

member of {U1} is sent inside a member of {US} via f.

Thus we may deﬁne a simplicial map f : |U1| → |US| by f(U) = V , where U is a member of U1 and V a member of US for which f(U) ⊂ V . By the previous lemma, orienting |U1| and |US| in the obvious way, we may combinatorially determine the degree of f in the context of locally ﬁnite homology. Also, there exists an isomorphism

lf n c : Hn (|Ur|) → HXn(R ) induced by the map sending each ball in the open cover to its center (see, for example, the discussion of the so-called “character map” on page 31 of [19]), and c makes the following diagram commute:

lf f-∗ lf Hn (|U1|) Hn (|US|)

c c

? ? n f-∗ n HXn(R ) HXn(R ).

It follows that deg f = deg f. We therefore obtain the following result:

43 n n Theorem 3.2.2. Let f : R → R be a coarse map, let the family {Ur}, r ∈ N be an ˇ n anti-Cech system of covers of R , and let S ∈ N be such that for every U ∈ U1, f(U) is

a subset of an element of US. If f : |U1| → |US| is the induced simplicial map, σ ∈ |US| is a positively oriented n-simplex, and

−1 + + − − f (σ) = {σ1 , . . . , σm, σ1 , . . . , σk }, where the σ+ have positive orientation and the σ− have negative orientation, then

deg f = m − k.

Thus counting the number of n-simplices in the preimage under f of an oriented n-simplex in |US|, taking signs into account, is suﬃcient to ﬁnd the coarse degree of the map f.

3.3 Examples

1. This method provides an alternative proof of Proposition 3.0.5. Let f be a coarse

n n map from R to R which is not essentially surjective. Then for every R > 0 n −1 there exists a ball B in R of radius R such that f (B) = ∅. Then, for S > 0 as n deﬁned earlier, there is a cover US of R and members U0,U1,...Un of US such −1 that U0 ∩ U1 ∩ · · · ∩ Un 6= ∅ and f (U0 ∪ U1 ∪ · · · ∪ Un) = ∅. It follows that the

associated n-simplex in |US| has zero preimages under f. Therefore f has degree zero.

2. (A Degree One Map) Deﬁne g : R → R by sending each x ∈ R to the greatest

even integer which is less than or equal to x. Cover R with open balls of radius 1

centered at all integers; call this U1. Then U4, the cover consisting of open balls of radius 4 centered at each multiple of 4, has Lebesque number 2, so S = 2 suﬃces.

44 Let u0, u1 be the balls of radius 4 around 0 and 4 respectively. Then if vi is the ball of radius 1 centered at i,

−1 g (u0) = {v−1, v0, v1, v2, v3},

−1 g (u1) = {v3, v4, v5, v6, v7}.

To make g into a well-deﬁned simplicial map g, declare g(v3) = u1. Then the

preimage of the simplex [u0, u1] under g is {[v2, v3]} (positively oriented), and deg g = 1.

2 2 3. (A Degree Zero Map) Deﬁne f : R → R by f(x, y) = f(x, |y|). The image of f

is the ﬂasque space R × R+, so we expect f to have degree zero. Choose a regular 2 cover U1 of R consisting of open squares of edge length 1 centered at {(k, n)k∈Z}

for all even integers n, and {(k + 1/2, m)k∈Z} for all odd integers m. Since f is

distance-decreasing, we may let R = 1; U1 has Lebesgue number at least 1/2, so

S = 2 is suﬃcient. Let u0, u1, u2 be the squares in U2 centered at (0, 0), (2, 0), and

(1, 2). If v(i,j) is the square of radius 1 centered at (i, j),

−1 f (u0) = {v(−1,0), v(0,0), v(1,0), v(−1/2,1), v(1/2,1), v(−1/2,−1), v(1/2,−1)}

−1 f (u1) = {v(1,0), v(2,0), v(3,0), v(3/2,1), v(5/2,1), v(3/2,−1), v(5/2,−1)}

−1 f (u2) = {v(0,±2), v(1,±2), v(2,±2), v(1/2,±1), v(3/2,±1), v(1/2,±3), v(3/2,±3)}.

Make the simplicial map f well-deﬁned by declaring

f(v(1,0)) = f(v(1/2,±1)) = u0, f(v(3/2,±1)) = u1.

45 The preimage of [u0, u1, u2] consists of

[v(1/2,1), v(3/2,1), v(1,2)], [v(1/2,1), v(3/2,1), v(0,2)], [v(1/2,1), v(3/2,1), v(2,2)]

(positive orientation) and

[v(1/2,−1), v(3/2,−1), v(1,−2)], [v(1/2,−1), v(3/2,−1), v(0,−2)], [v(1/2,−1), v(3/2,−1), v(2,−2)]

(negative orientation). Therefore deg f = 3 − 3 = 0.

3.4 The Coarse Brouwer Fixed Point Theorem

(Many thanks to John Roe for his insights which allowed the material in this section to be put in the proper context)

There most likely exist a number of “fixed point theorems” for coarse maps which are in some way analogous to the classical Brouwer theorem. One such result can be found in a paper of Tomohiro Fukaya [4], and deals with orbits of points and coronas of compactifications. The theorem which follows here is an attempt to modify the degree- theoretic proof of the Brouwer fixed point theorem to a coarse framework. We begin by defining a coarse fixed point property. This property shall be expressed in terms of open cones; recall the definition of such objects (cf. Section 6.2 in [10]):

Deﬁnition 3.4.1. Let X be a compact metrizable space. The open cone on X, denoted

OX, is the space obtained from [0, 1) × X by collapsing {0} × X to a point. The open cone is given a metric coarse structure as follows. We may suppose that X is a compact subset of the unit sphere in a real Hilbert space E; let φ be a homeomorphism of [0, 1) onto [0, ∞). Then, the map

OX → E, (t, x) 7→ φ(t)x,

46 identiﬁes OX with a subset of E and so deﬁnes a metric on the open cone of X.

Deﬁnition 3.4.2. Let X be a compact metric space. A coarse map f from the open cone

OX to itself has the coarse ﬁxed point property (CFPP) if there is a constant R and a

sequence x1, x2,... of points in OX such that:

1. For every bounded subset B of OX, there exists N such that xn 6∈ B for all n ≥ N

(denote this by xn → ∞);

2. For each n, xn and f(xn) are within distance R of a common ray (dependent on

xn).

(One may ask how this property relates to an “open cone version” of the classical

ﬁxed point property for continuous maps. This question is addressed in the following

subsection.)

The goal of this section is to prove the following coarse analogue of the Brouwer

ﬁxed point theorem:

n n Theorem 3.4.3 (Coarse Fixed Point Theorem). Let f : R × R+ → R × R+ be a coarse map. Then f has the CFPP.

n n n n−1 Observe that R × R+ = OD and R = OS .

Proof. We shall prove this theorem using coarse degree theory, beginning with a series

of lemmas:

n n Lemma 3.4.4. Any reﬂection map r : R → R has degree −1.

Proof. By applying a coarse homotopy, we may assume that r reﬂects points across the

hyperplane X = {(0, x2, . . . , xn)}, that is, r sends (x1, x2, . . . , xn) to (−x1, x2, . . . , xn).

n Take a locally ﬁnite triangulation ∆ of R such that the interior of each n-simplex is n disjoint from X. We may express R as a union of spaces D1 = {(x1, x2, . . . , xn)|x1 ≤ 0},

47 D2 = {(x1, x2, . . . , xn)|x1 ≥ 0}. Letting

X X ∆1 := σ, ∆2 := σ,

σ∈∆∩D1 σ∈∆∩D2

n the chain z = ∆1 − ∆2 is a cycle representing the generator of HXn(R ) (identifying

each n-chain [a0, . . . , an] with the point (a0, . . . , an)). Since r induces the chain map

sending ∆1 to ∆2 and vice versa, this induced map sends z to −z. Thus r has degree −1.

Note that we may also use the combinatorial degree formula discussed previously to prove the above result.

n n Lemma 3.4.5. The antipodal map a : R → R sending x = (x1, x2, . . . , xn) to −x =

n (−x1, −x2,..., −xn) has degree (−1)

Proof. The map a is a composition of n reﬂection maps.

n n Lemma 3.4.6. Suppose a coarse map f : R → R does not have the CFPP. Then f has degree (−1)n.

Proof. We wish to construct a coarse homotopy between the antipodal map a and f.

Consider the linear homotopy

Ht(x) := (1 − t)f(x) − tx

along the line segment between each a(x) and f(x). We claim that Ht is a coarse

0 0 0 n homotopy. Choose S > 0 and let x = (x1, . . . , xn), x = (x1, . . . , xn) be points in R 0 0 such that d(x, x ) ≤ S; thus, each |xi −xi| is at most S. Let f(x1, . . . , xn) = (y1, . . . , yn), 0 0 0 0 f(x1, . . . , xn) = (y1, . . . , yn). By the coarseness of f, there exists T > 0 such that each

48 0 |yi − yi| is at most T . Then for each i,

0 0 0 0 |((t − 1)xi + tyi) − ((t − 1)xi + tyi)| = |(t − 1)(xi − xi) + t(yi − yi)|

0 0 ≤ |(t − 1)||xi − xi| + |t||yi − yi| ≤ S + T for all t ∈ [0, 1]. It follows that the collection {Ht} is uniformly bornologous.

Uniform pseudocontinuity is straightforward to check; since t 7→ Ht(x) is continuous for each x, we may take the controlled set F in the deﬁnition of pseudocontinuity to be a radius 1 neighborhood of the diagonal, and the condition will be satisﬁed.

n To show that {Ht} is uniformly proper, let B be a bounded subset of R . We may assume, replacing B with a larger set if necessary, that B is a ball of radius S centered

n the origin for some S ≥ 0. Then, a point x ∈ R is an element of

[ −1 VB := ht (B) t∈[0,1] if and only if the line segment Lx connecting −x and f(x) intersects B. We claim that there exists a constant C > 0 such that, if x ∈ VB, then d(f(x), Ox) ≤ C · S. Since f is coarse, there exists A > 0 such that d(f(x), 0) ≤ d(f(x), f(0)) + d(f(0), 0) ≤

A · d(x, 0) + d(f(0), 0). Therefore, there exists a constant K such that d(f(x), 0) ≤

K · max{d(x, 0), 1}. Let C = 2(1 + K) and suppose x ∈ VB. There are two possible cases for this to occur:

1. Suppose d(x, 0) ≤ 2S. Then d(f(x), 0) ≤ 2SK ≤ CS; since 0 ∈ Ox, the claim is

proven in this case.

2. Suppose d(x, 0) ≥ 2S. Since x ∈ VB, there exists a t ∈ [0, 1] such that the point

p = Ht(x) = (1 − t)f(x) − tx lies in VB. We shall form two similar triangles

n in R as in Figure 3.1. The ﬁrst (smaller) triangle has as its vertices the points −x, 0, and p. Then, there exists a point q such that the (larger) triangle with

49 Figure 3.1: Construction of similar triangles.

vertices −x, f(x), and q is similar to the ﬁrst triangle; i.e. the line segment

from f(x) to q is parallel to the segment from p to 0. Note that, by the triangle

inequality, d(−x, p) ≥ d(x, 0)/2, while d(−x, f(x)) ≤ (K + 1)d(x, 0) by deﬁnition

of K. Therefore, by similarity,

d(−x, f(x)) d(f(x), q) = · d(p, 0) ≤ 2(K + 1)S ≤ CS. d(−x, p)

The claim is proven. It follows that for x ∈ VB, x and f(x) lie within distance CR of a common ray. Since f is assumed to not have the CFPP, the set of all x such that x and

f(x) lie within distance CR of a common ray is bounded. Therefore, VB is bounded,

proving uniform boundedness of Ht. Thus Ht is indeed a coarse homotopy from a degree (−1)n map and f.

n The proof of the coarse ﬁxed point theorem now follows. From a coarse map f : R × n n+1 n+1 R+ → R ×R+ we construct a coarse map g : R → R by letting g(x1, . . . , xn+1) =

50 f(x1, . . . , xn+1) if xn+1 ≥ 0, g(x1, . . . , xn+1) = f(x1,..., −xn+1) if xn+1 ≤ 0. The image

n of g is a subset of R × R+ and so g has degree 0. By the last lemma, g has the CFPP n+1 n on R ; by the way we have deﬁned g, f must have the CFPP on R × R+ as well.

3.4.1 The Weak Fixed Point Property

We now wish to briefly discuss how the CFPP in Definition 3.4.2 relates to two different

“coarse fixed point” properties for maps on open cones, both of which may be more immediately reminiscent of the classical fixed point property. The first, stronger, fixed point property arises from the idea that the open cone of a point is a ray and that bounded sets in coarse spaces are coarsely equivalent to points.

Definition 3.4.7. Let X be a compact metrizable space. A coarse map f : OX → OX has the strong coarse fixed point property if there exists a point ζ ∈ OX (called the coarse fixed point of f), a constant R > 0, and a sequence of points xn → ∞ in OX such that, for all n, both xn and fn are within distance R of Oζ.

It is obvious that if a map f has the strong CFPP, it has the CFPP in the context of Deﬁnition 3.4.2. From the above deﬁnition we may also derive a weaker version of the strong CFPP:

Deﬁnition 3.4.8. A coarse map f : OX → OX has the weak coarse ﬁxed point property

(WCFPP) if there exists ζ ∈ X such that, for any neighborhood U of ζ, there is a

sequence of points xn → ∞ in OX such that xn and f(xn) lie in OU for every n.

Remark 3.4.9. Equivalently, we may say that for each n, the rays containing xn and

f(xn) are within arbitrarily small “angular distance” of Oζ, in the following sense: embed X into a sphere Sr; then, say that the angular distance between two rays Ox,

Oy is the distance between x and y in Sr.

We claim that the WCFPP is, in fact, weaker than the CFPP, so that we have

Strong CFPP =⇒ CFPP =⇒ Weak CFPP :

51 Lemma 3.4.10. If f satisfies the coarse fixed point property, then it satisfies the weak coarse fixed point property.

Proof. Suppose f has the CFPP. Then, there is some R > 0 and a sequence of points xn → ∞ in OX such that, for each n, both xn and f(xn) lie within distance R of Oζn

for some ζn ∈ X. Since X is a compact metric space, by passing to a subsequence we may assume that ζn converge to a point ζ in X.

If dX is the angular distance between two rays in OX, and if Rn is the ray containing

xn, we have that d(xn, Oζn) R dX (Rn, Oζn) ≤ ≤ , d(xn, 0) d(xn, 0)

so dX (Rn, O) → 0 as n → ∞. Also, by construction, dX (Oζn, Oζ) → 0 as n → ∞.

Therefore, dX (Rn, Oζ) → 0; that is, if U is a neighborhood of ζ, xn ∈ OU for suﬃciently

large n. Similarly, f(xn) ∈ OU for suﬃciently large n.

Since X is compact, there exists a countable base U1 ⊃ U2 ⊃ · · · for the neighbor-

hoods of ζ. For each k, choose nk such that xn, f(xn) lie in OUk for n ≥ nk. Then, the sequence (xkn ) has the properties needed for the WCFPP.

52 Chapter 4

Geometric Modules

In this chapter, we shall introduce the notion of so-called “controlled” and “cocontrolled” modules; which are expressible in terms of direct and inverse limits of modules with certain geometric ﬁnite generation properties. We shall also show that the coarse chain and cochain modules as deﬁned in the introductory chapter can be expressed as such modules. Furthermore, we will relate the coarse cochain complex of a product space to a “geometric tensor product” of the chain complexes of the individual spaces in the product.

4.1 The Controlled and Bounded Coarse Categories

Deﬁnition 4.1.1. Let X be a coarse space. Then, deﬁne the controlled coarse category

of X, denoted C(X), to be the category whose objects are the controlled subsets E of

X × X and whose morphisms are the inclusion maps E,→ F for E,F ∈ C(X) with

E ⊂ F .

Additionally, let B(X) denote the bounded coarse category of X, the subcategory of

C(X) whose objects consist of all bounded subsets of X × X.

53 4.2 Geometric Modules

Deﬁnition 4.2.1. Let X be a coarse space and R be a unital ring. Then, a free geometric

(X,R)-module (or, more simply, a geometric module in the case that X and R are

understood) M is a pair of functors M = (iM , pM ) from C(X) to the category of R-

modules which satisﬁes the following axioms:

1. iM is covariant and pM is contravariant;

M M 2. For all E ∈ C(X), i (E) = p (E) (we shall denote this module ME);

3. M∅ is the zero module;

4. When B ∈ B(X), MB is a free module of ﬁnite rank;

M 5. (Consistency Axiom) Given E,F,G ∈ C(X) with E,F ⊂ G, and if iAA0 : MA → 0 M MA0 is the morphism induced by A ⊂ A (similarly, pA0A : MA0 → MA), then the diagram

M iEG - ME MG

M M pE,E∩F pGF ? iM ? E∩F,F- ME∩F MF

commutes.

M M Remark 4.2.2. The family {ME, iEF , pFE} forms an inverse-direct system as in Deﬁni- tion 1.5 of [3].

Our main examples of geometric modules arise from coarse cochain groups:

Example 4.2.3. Let X be a boundedly ﬁnite coarse space and R be a unital ring. If

E ⊂ X × X is a controlled set, let

(q+1) E := {(x0, . . . , xq)|(xi, xj) ∈ E for all i, j}

54 and

q (q+1) IE := {ψ ∈ CX (X)|ψ = 0 on E }.

M M q M Then, define M = (i , p ) as follows; let ME = CX (X)/IE and pFE be the quotient M homomorphism from MF to ME (observe that IF ⊂ IE when E ⊂ F ). To define iFE, q (q+1) let E ⊂ F and ψ ∈ ME. Then there exists ψ ∈ CX (X) such that ψ = 0 outside E and such that all the representatives of ψ in CXq(X) agree with ψ on E(q+1). Define

M iEF (ψ) to be the class of ψ in MF . Clearly, (iM , pM ) satisﬁes the ﬁrst three geometric module axioms. Additionally, if

B is bounded, MB is generated as a free R-module by the equivalence classes of the characteristic functions of points in B(q+1). Since B, and hence B(q+1), is a ﬁnite set,

ﬁnite generation follows.

To prove consistency, let E,F,G be as in the statement of the axiom and let ψ ∈ ME

M M with a canonical representative ψ as given in the deﬁnition of i above. Then iEG(ψ) is 0 M M 0 the class of ψ in MG (call it ψ ), and (pGF ◦ iFG)(ψ) is the class of ψ in MF . Therefore, M M q (q+1) (q+1) (pGF ◦ iEG)(ψ) is represented by φ ∈ CX (X) such that φ = ψ on E ∩ F M 00 and φ = 0 otherwise. On the other hand, pE,E∩F (ψ) is represented by ψ such that 00 (q+1) 00 M M ψ = ψ on (E ∩ F ) and ψ = 0 otherwise. Then (iE∩F,F ◦ pE,E∩F )(ψ) is the class 0 q 0 00 (q+1) in MF represented by φ ∈ CX (X) such that φ = ψ = ψ on (E ∩ F ) and φ = 0 otherwise.

(q+1) Finally, note that (x0, . . . , xq+1) ∈ (E ∩ F ) if and only if (xi, xj) ∈ E ∩ F

(q+1) (q+1) for all i, j, and that (x0, . . . , xq+1) ∈ E ∩ F if and only if (xi, xj) ∈ E and

(q+1) (q+1) (q+1) M M (xi, xj) ∈ F for all i, j. Therefore, (E∩F ) = E ∩F and so (pGF ◦iFG)(ψ) = M M (iE∩F,F ◦ pE,E∩F )(ψ). We shall refer to the geometric module M = (iM , pM ) deﬁned above as the geometric

module associated to CXq(X). Later in this section, we will construct an R-module

isomorphic to CXq(X) from M by ﬁrst observing that M is a so-called cocontrolled

geometric module and using that fact to form its so-called total module.

55 Remark 4.2.4. There exists an alternative formulation of the geometric module associ-

q q ated to CX (X); ﬁrst, deﬁne NE to be the submodule of CX (X) consisting of those

(q+1) N N functions whose supports lie in E . Then let iEF be the inclusion and pFE be the (q+1) map which sends ψ ∈ NF to the function which agrees with ψ on E and is zero otherwise. Then the geometric modules (iM , pM ) (as deﬁned in the previous example)

N N and (i , p ) are isomorphic in the sense that the module isomorphism φE : ME → NE

M N which sends a class in ME to its representative in NE satisﬁes φF ◦ iEF = iEF ◦ φE and M N φE ◦ pFE = pFE ◦ φF .

We now return to general discussion of geometric modules.

M Lemma 4.2.5. If E ⊂ F are controlled sets and M is a geometric module, then pFE ◦ M M M iEF is the identity homomorphism on ME (so iEF is a split injection and pFE is a split surjection).

Proof. By the Consistency Axiom, we have commutativity of the diagram

M iEF - ME MF

M M pEE pFE

? M ? iEE- ME ME.

M M M M Since iEE = pEE is the identity map on ME, it follows that pFE ◦ iEF = idME .

Lemma 4.2.6. If E ∈ C(X) can be decomposed as a disjoint union of controlled sets

M M E = F tG, then i and p give an inclusion of MF ⊕MG into ME as a split summand, via the isomorphisms

M ∼ M M M ∼ M M iFE(MF ) = (iFE ◦ pEF )(ME), iGE(MG) = (iGE ◦ pEG)(ME)

M M M M and decomposition ME = (iFE ◦ pEF )(ME) ⊕ (iGE ◦ pEG)(ME) ⊕ M.

56 M M Proof. By the previous lemma, iFE and iGE give inclusions of MF and MG into direct ∼ summands of ME. Therefore, there exists a module M such that ME = MF ⊕ M

M and iFE is the inclusion map MF ,→ MF ⊕ M. By the consistency axiom, there is a commutative diagram

M iFE- MF MF ⊕ M

M M pE,E∩F pEG ? iM ? E∩F,G - M∅ = 0 MG

M M M so it follows that iGE maps MG into M. Therefore, iFE ⊕ iGE) gives an inclusion from M M M M MF ⊕ MG into a direct summand of ME, and (pEF ⊕ pEG) ◦ (iFE ⊕ iGE) is the identity on MF ⊕ MG.

4.2.1 Morphisms of Geometric Modules

We shall deﬁne two notions of morphisms of geometric modules; we shall start by giving the more restrictive deﬁnition and generalize it later.

Deﬁnition 4.2.7. Suppose M = (iM , pM ), N = (iN , pN ) are geometric X-modules.

Then, a strict morphism from M to N is a family of R-module homomorphisms

M N {φE : i (E) → i (E)}

for all E ∈ C(X) such that, for all E ⊂ F in C(X),

N M N M (iEF ◦ φE) = (φF ◦ iEF ) and (pFE ◦ φF ) = (φE ◦ pFE).

In other words, a strict geometric module morphism is a family of R-module homomorphisms where each is simultaneously a natural transformation from iM to iN and from pM to pN .

57 Lemma 4.2.8. If M = (iM , pM ) and N = (iN , pN ) are the geometric modules associated to CXq(X) and CXq+1(X) respectively, and ∂ : CXq(X) → CXq+1(X) is the coarse

coboundary map, then there exists a strict morphism

{∂E : ME → NE}

q such that, if ψ ∈ CX (X) represents ψ ∈ ME (in the sense that ψ agrees with all coset

(q+1) q+1 representatives of ψ on E ), then ∂ψ ∈ CX (X) represents ∂Eψ ∈ NE.

Proof. For E ⊂ X ×X controlled, deﬁne ∂E : ME → NE as follows: for ψ ∈ ME, choose

q a representative ψ ∈ CX (X). Then, if ∂ψ is the class of ∂ψ in NF , deﬁne

∂Eψ := ∂ψ.

q (q+1) Observe that ∂E is well-deﬁned: if φ ∈ CX (X) is a map whose restriction to E is

(q+2) zero, consider, for (x0, . . . , xq+1) ∈ E ,

q+1 X i ∂φ(x0, . . . , xq+1) = (−1) φ(x0,..., xˆi, . . . , xq+1). i=0

(q+1) Since each (x0,..., xˆi, . . . , xq+1) lies in E , ∂φ(x0, . . . , xq+1) = 0. Thus if two maps agree on E(q+1), their coboundaries will agree on E(q+1).

We now wish to show that the homomorphisms ∂E have the desired naturality

M M properties with respect to the iEF and pFE.

First, let E ⊂ F be controlled sets and let ψ ∈ ME. Then, ψ is represented by

q (q+1) M ψ ∈ CX (X) such that Supp(ψ) ⊂ E and iEF (ψ) is the class of ψ in MF . Then, M (∂F ◦ iEF )(ψ) is the class of ∂ψ in NF . On the other hand, ∂E(ψ) is the class of ∂ψ q (q+1) in NE, and this element is represented by φ ∈ CX (X) such that Supp(φ) ⊂ E .

N M N Then, (iEF ◦ ∂E)(ψ) is the class of φ in NF . However, (∂F ◦ iEF )(ψ) and (iEF ◦ ∂E)(ψ) are both represented by maps which agree with ∂ψ on E(q+2), therefore the equivalence

58 classes are equal.

Similarly, let ψ ∈ MF be represented by ψ. Then ∂F (ψ) is the class of ∂ψ in NF ,

N M and (pFE∂F )(ψ) is the class of ∂ψ in NE. Also, pFE(ψ) is the class of ψ in MF , and M N (∂E ◦pFE)(ψ) is the class of ∂ψ in NE. Therefore, the equivalence classes of (pFE∂F )(ψ) M and (∂E ◦ pFE)(ψ) are equal. N M N M Since (iEF ◦ ∂E) = (∂F ◦ iEF ) and (pFE∂F ) = (∂E ◦ pFE), it follows that the family

{∂E} induced by ∂ is a geometric module morphism.

4.3 Controlled and Cocontrolled Modules

M Observe that if M is a geometric module, and E ∈ C(X), the collection {(MB, iBC )} taken over all bounded subsets of E is a direct system, directed by inclusion. Therefore,

we may discuss the R-module

lim M . −→ B B⊂E

M M Also, for all bounded B ⊂ E, the map iBE : MB → ME satisﬁes the property iCE ◦ M M iBC = iBE. By the universal property of direct limits, there exists a unique R-module homomorphism

iM : lim M → M E −→ B E B⊂E such that, if α is the canonical map from M to lim M , then iM ◦ α = iM . B B −→ B E B BE

M Lemma 4.3.1. The map iE is injective.

Proof. If m is an element of lim M , then there exists a bounded subset B of E and −→ B M M m ∈ MB such that αB(m) = m. Then, iE (m) is deﬁned to be iBE(m) ∈ ME. Since M M the maps iBE are injective, so is iE .

M Deﬁnition 4.3.2. A geometric module M is said to be cocontrolled if iE as deﬁned above is an isomorphism for all E.

59 M Analogously, the collection {(MB, pCB)} over all bounded subsets B of E form an inverse system, and there exists a surjection

pM : M → lim M E E ←− B B⊂E

arising from the universal property of inverse limits. We say that a geometric module

M is controlled if pE is an isomorphism for all E. The motivating example for the above deﬁnitions is:

Lemma 4.3.3. The geometric module associated to CXq(X) is cocontrolled. (Analo-

gously, the geometric module associated to CXq(X) is controlled.)

M Proof. It suﬃces to show that iE is a surjection. Let E ⊂ X × X and let ψ ∈ ME. Then, ψ is represented by ψ ∈ CXq(X) such that Supp(ψ) ⊂ E(q+1). Since E(q+1) is a

controlled subset of Xq+1, there exists a bounded B0 ⊂ Xq+1 such that Supp(ψ) ⊂ B0.

q+1 Letting πi : X → X, 0 ≤ i ≤ q be the coordinate projection maps, the set

0 B := {(πi(x), πj(x))|x ∈ B , 0 ≤ i, j ≤ q}, is a bounded subset of E. Then B0 ⊂ B(q+1) ⊂ E(q+1), so Supp(ψ) ⊂ B(q+1). Therefore,

M the element of MB represented by ψ is sent to ψ via iE .

4.4 Generalized Morphisms

The generalized definition of a geometric module morphism uses the notion of cofinal subsets of controlled or bounded sets. Recall how such a subset is defined:

Deﬁnition 4.4.1. A subset C of a directed set I (with preorder ≤) is coﬁnal if, for each i ∈ I, there exists c ∈ C with i ≤ c.

j Recall (cf. Exercise 5.22 in [22]) that if X is a directed, {Mi, φi } a direct (inverse)

60 system indexed by I and C is a coﬁnal subset of X, that the direct (inverse) limit of the Mi over I is isomorphic to the direct (inverse) limit over C. We shall now introduce a less restrictive version of Deﬁnition 4.2.7:

Deﬁnition 4.4.2. Let X,Y be coarse spaces, f : X → Y be a coarse map, and

M = (iM , pM ),N = (iN , pN ) be geometric X- and Y - modules respectively. A covariant morphism from M to N over f, denoted (f, {φE}) is a family of R-module homomorphisms

φE : ME → N(f×f)(E), over all E ∈ C(X), with the property that for all E ∈ C(X), deﬁning the subcategory

BE of {B ∈ B(X) | B ⊂ E} to be

−1 BE := {E ∩ (f × f) (B) | B ∈ B(Y )},

then, for all B ⊂ C in BE, the diagrams

M M i - i - MB MC MB ME

φB φC and φB φE

? N ? ? N ? i- i- N(f×f)(B) N(f×f)(C) N(f×f)(B) N(f×f)(E) and

M M p - p - MC MB ME MB

φC φB and φE φB

? N ? ? N ? p- p- N(f×f)(C) N(f×f)(B) N(f×f)(E) N(f×f)(B) commute.

Deﬁnition 4.4.3. A covariant morphism (f, {φE}) from M to N is said to be of controlled

61 type if, deﬁning a coﬁnal subcategory C0 of C(X) by

C0 = {F ∩ (f × f)−1(E) | E ∈ C(Y ),F ∈ C(X)}, for all E ⊂ F in C0, the diagram

M i - ME MF

φE φF

? N ? i- N(f×f)(E) N(f×f)(F ) commutes.

A covariant morphism is said to be of cocontrolled type if, with C0 as deﬁned above, for all E ⊂ F in C0, the diagram

M p - MF ME

φF φE

? N ? p- N(f×f)(F ) N(f×f)(E) commutes.

Remark 4.4.4. In the case that f is the identity map from X to itself, the subcategory

0 BE is equal to {B ∈ B(X) | B ⊂ E} for each E, and C is equal to C(X). Therefore, any morphism over the identity map has the naturality properties for a strict mor-

phism on bounded sets; if that morphism is also of controlled and cocontrolled type

simultaneously, it is a strict morphism.

Deﬁnition 4.4.5. Let X,Y,M,N, and f be as in the previous deﬁnition. A con-

0 travariant morphism from N to M over f, denoted (f, {φE}), is a family of R-module

62 homomorphisms

0 φE : N(f×f)(E) → ME,

over all E ∈ C(X), with the property that for all E ∈ C(X), deﬁning the subcategory

BE of {B ∈ B(X) | B ⊂ E} to be

−1 BE := {E ∩ (f × f) (B) | B ∈ B(Y )},

then, for all B ⊂ C in BE, the diagrams

N N i- i- N(f×f)(B) N(f×f)(C) N(f×f)(B) N(f×f)(E)

0 0 0 0 φB φC and φB φE

? M ? ? M ? i - i - MB MC MB ME and

N N p- p- N(f×f)(C) N(f×f)(B) N(f×f)(E) N(f×f)(B)

0 0 0 0 φC φB and φE φB

? M ? ? M ? p - p - MC MB ME MB commute.

Deﬁnition 4.4.6. A contravariant morphism (f, {φE}) from M to N is said to be of controlled type if, for the subcategory

C0 = {F ∩ (f × f)−1(E) | E ∈ C(Y ),F ∈ C(X)}

63 of C(X), and for all E ⊂ F in C0, the diagram

N i- N(f×f)(E) N(f×f)(F )

0 0 φB φC

? M ? i - ME MF commutes. A contravariant morphism is said to be of cocontrolled type if for all E ⊂ F in C0 as deﬁned above, the diagram

N p- N(f×f)(F ) N(f×f)(E)

0 0 φC φB

? M ? p - MF ME commutes.

Remark 4.4.7. Observe that BE as defined above is a cofinal subcategory of {B ∈ B | B ⊂ E}; indeed, if B is a bounded subset of E, then B ⊂ E ∩ (f × f)−1(f × f)(B), and E ∩ (f × f)−1(f × f)(B) is bounded because f is proper. Similarly, C0 is a cofinal subcategory of C(X).

Example 4.4.8. If X and Y are boundedly ﬁnite coarse spaces, any coarse map f : X →

Y induces a covariant morphism on the geometric modules M and N associated to

CXq(X) and CXq(Y ). For E ∈ C(X), we deﬁne ME to be the set of all formal linear combinations X αx(x0, . . . , xq), αx ∈ R (q+1) x=(x0,...,xq)∈E

(and deﬁne NF analogously). Then, for all E ∈ C(X), let

−1 BE := {E ∩ (f × f) (f × f)(B) | B ∈ B(X)}.

64 For E ∈ C(X), deﬁne φE by

! X X φE αx(x0, . . . , xq) := αx(f(x0), . . . , f(xq)). x x

(q+1) (q+1) Observe ﬁrst that, for (x0, . . . , xq) ∈ E ,(f(x0), . . . , f(xq)) ∈ (f ×f)(E) , so the image of each φE indeed lies in N(f×f)(E).

To show that (f, {φE}) is a morphism, let B ⊂ C be members of BE. Then, for

X m = αx(x0, . . . , xq) ∈ MC , x∈C(q+1)

M X pCB(m) = αx(x0, . . . , xq), x∈B(q+1) and

M X (φB ◦ pCB)(m) = αx(f(x0), . . . , f(xq)). x∈B(q+1) On the other hand, X φC (m) = αx(f(x0), . . . , f(xq)) x∈C(q+1) and

N X (p(f×f)(C),(f×f)(B) ◦ φC )(m) = αx(f(x0), . . . , f(xq)). (q+1) (f(x0),...,f(xq))∈(f×f)(B)

(q+1) (q+1) By construction of BE,(x0, . . . , xq) ∈ B if and only if (x0, . . . , xq) ∈ C and

(q+1) M N (f(x0), . . . , f(xq)) ∈ (f × f)(B) . Therefore, φB ◦ p = p ◦ φC . X For all E ⊂ F in C and m = αx(x0, . . . , xq) ∈ ME, x∈E(q+1)

M X N (φF ◦ iEF )(m) = αx(f(x0), . . . , f(xq)) = (i(f×f)(E),(f×f)(F ) ◦ φE)(m). x∈E(q+1)

Therefore, (f, {φE}) is indeed a morphism, and is in fact of controlled type.

65 Example 4.4.9. Analogously, we may construct a contravariant morphism from the geometric module N 0 associated to CXq(Y ) to the module M 0 associated to CXq(Y ) as

0 q (q+1) follows. If ψ ∈ N(f×f)(E) is represented by ψ ∈ CX (Y ) such that Supp ψ ⊂ E , ∗ q then deﬁne φE(ψ) to be the element of ME represented by f ψ ∈ CX (X). An argu-

ment similar to the previous one shows that (f, {φE}) is a morphism of cocontrolled type.

Remark 4.4.10. The introduction of the coﬁnal sets BE are necessary in the above example in order for the morphism induced by f to satisfy the compatibility criteria in

M N the deﬁnition. For arbitrary E ⊂ F in C(X), φE ◦ p and p ◦ φF are not in general

equal. For example, let M = N be the module associated to CX1(Z) and let f : Z → Z be the map which sends z to the largest even integer less than or equal to z. Then,

letting B = {(0, 0)}, C = {(0, 0), (1, 1)} and m = (0, 0) + (1, 1) ∈ MC ,

M N (φB ◦ pCB)(m) = (0, 0), while (p(f×f)(C),(f×f)(B) ◦ φC )(m) = 2(0, 0).

M N Here, the failure of φE ◦ p to equal p ◦ φE in general arises from the fact that the set {x ∈ F | (f × f)(x) ∈ (f × f)(E)} is not necessarily equal to E.

Lemma 4.4.11. If M,N are cocontrolled geometric modules, if (f, {φE}) is a geometric module morphism from M to N, and if

iM : lim M → M , iN : lim M → N E −→ B E (f×f)(E) −→ C (f×f)(E) B⊂E C⊂(f×f)(E)

are the isomorphisms from Lemma 4.3.1, then, for all E ∈ C(E),

φ ◦ iM = iN ◦ lim φ . E E (f×f)(E) −→ B B⊂E

66 Similarly, if M and N are controlled modules,

lim φ ◦ pM = pN ◦ φ . ←− B E (f×f)(E) E B⊂E

Proof. We shall prove this for case where M and N are cocontrolled; the controlled argument is analogous. Let E ∈ C(X) and deﬁne

M 0 := lim M ,N 0 := lim N , E −→ B (f×f)(E) −→ (f×f)(B) B⊂E B⊂E

0 0 the limits taken over bounded B. If αB : MB → ME, γ(f×f)(B) : N(f×f)(B) → N(f×f)(E) 0 0 are the canonical maps, then there exists a unique φ : ME → N(f×f)(E) such that φ◦α = γ ◦φ for all B ∈ B ; in fact, φ = lim φ . Now, recall that iM ◦α = iM B (f×f)(B) B E −→ B E B BE N N and i(f×f)(E)◦γ(f×f)(B) = i(f×f)(B),(f×f)(E). By the compatibility condition with respect M to iBE for morphisms,

M M N N φE ◦ iE ◦ αB = φE ◦ iBE = i(f×f)(B),(f×f)(E) ◦ φB = i(f×f)(E) ◦ γ(f×f)(B) ◦ φB.

Therefore,

lim φ = φ = (iN )−1 ◦ φ ◦ iM . −→ B (f×f)(E) E E

4.4.1 Composition of Morphisms

Suppose that L, M, and N are geometric modules over coarse spaces X, Y , Z, and

(f, {φE}): L → M,(g, {ψF }): M → N are covariant geometric module morphisms. We wish to show that these morphisms may be “composed” in such a way as to obtain a morphism from L to N.

Deﬁnition 4.4.12. Let L, M and N be geometric modules with morphisms (f, {φE}),

67 (g, {ψF }) as in the previous paragraph. For E ∈ C(X), deﬁne

(ψ ∗ φ)E : LE → N(g×g)(f×f)(E)

to be ψ(f×f)(E) ◦ φE.

Lemma 4.4.13. The pair (g ◦ f, {(ψ ∗ φ)E}), called the composition of (f, {φE}) and

(g, {ψF }) is a geometric module morphism from L to N.

Proof. Let E ⊂ C(X) and consider the subcategory

0 −1 −1 BE := {E ∩ (f × f) ◦ (g × g) (B) | B ∈ B(Z)},

0 0 0 of {B ∈ B(X) | B ⊂ E}. For all B ⊂ C in BE, there exist bounded subsets B ⊂ C in C(Z) such that

B = E ∩ (f × f)−1 ◦ (g × g)−1(B0),C = E ∩ (f × f)−1 ◦ (g × g)−1(C0).

Then, since (g × g)−1(B0) is bounded in Y × Y , B is an element of

−1 BE = {E ∩ (f × f) (D) | D ∈ B(Y )}

(and an analogous argument shows the same for C) and so the diagrams

L L i - i - LB LC LB LE

φB φC and φB φE

? M ? ? M ? i- i- M(f×f)(B) M(f×f)(C) M(f×f)(B) M(f×f)(E) from Deﬁnition 4.4.2 commute, as do the analogous diagrams with p’s in place of i’s.

68 Similarly,

(f × f)(B) = (f × f)(E) ∩ (g × g)−1(B0)

(and similarly for C), so the diagrams

M M i - i - M(f×f)(B) M(f×f)(C) M(f×f)(B) M(f×f)(E)

ψ(f×f)(B) ψ(f×f)(C) and ψ(f×f)(B) ψ(f×f)(E)

? N ? ? N ? i- i- N(g×g)(f×f)(B) N(g×g)(f×f)(C) N(g×g)(f×f)(B) N(g×g)(f×f)(E) commute, as do the analogous diagrams with p. It follows that the diagrams

L L i - i - LB LC LB LE

(ψ ∗ φ)B (ψ ∗ φ)C and (ψ ∗ φ)B (ψ ∗ φ)E

? N ? ? N ? i- i- N(g×g)(f×f)(B) N(g×g)(f×f)(C) N(g×g)(f×f)(B) N(g×g)(f×f)(E)

(and the analogous diagrams with p) commute, proving the claim.

Since the construction of the coﬁnal subcategories C0 of C(X) in Deﬁnition 4.4.3 was done in an similar way to the contruction of the BE, a straightforward analogue to the previous argument gives:

Lemma 4.4.14. If (f, {φE}): L → M and (g, {ψF }): M → N are morphisms of

(co)controlled type, then (g ◦ f, {(ψ ∗ φ)E}) is of (co)controlled type as well.

Remark 4.4.15. As a result of the discussion in the section, we may form a category whose objects are geometric modules and morphisms are geometric module morphisms (the above deﬁnition of composition of geometric module morphisms in con- junction with standard properties of R-module homomorphisms assures us that geometric module morphisms are indeed morphisms in the category). We may also form

69 speciﬁc subcategories of the “geometric module category” – the categories whose objects are (co)controlled modules and morphisms are geometric module morphisms of

(co)controlled type.

4.5 Partial Modules and Closures

Deﬁnition 4.5.1. Let B0 be either the category B(X) or a coﬁnal subcategory. Then a

partial geometric module based on B0 is a pair of functors M = (iM , pM ) from B0 to the

category of R-modules which satisﬁes the axioms for a geometric module.

Theorem 4.5.2. Given a partial geometric module M, there exist unique (up to geo-

metric module isomorphism) cocontrolled and controlled modules extending it. We shall

call these modules the cocontrolled and controlled closures of M respectively.

Proof. Assume for the time being that B0 = B(X). We shall only prove the cocontrolled

version, as the controlled case is analogous. We shall deﬁne a pair of functors N =

(iN , pN ), show that these functors give a geometric module, then show that the module

is cocontrolled. First, deﬁne

N := lim M , E −→ B B⊂E

M the direct limit with respect to the direct system {MB, iBC }. Now suppose that E ⊂ F are controlled sets. For each bounded subset B of E, B is also a subset of F . Therefore,

since NF is the direct limit with respect to bounded subsets of F , there is a canonical

F F F M map αB : MB → NF with the property that αB = αC ◦ iBC for bounded subsets B ⊂ C of E. By the universal property of direct limits, since

N = lim M E −→ B B⊂E

there is a unique homomorphism from NE to NF causing commutativity of a diagram.

N Deﬁne iEF to be this homomorphism.

70 Conversely, for all bounded subsets B ⊂ C of F , the map ψB : MB → NE deﬁned

E M by ψB := αB∩E ◦ pB,B∩E satisﬁes

E M E M M ψB := αB∩E ◦ pB,B∩E = (αC∩E ◦ iB∩E,C∩E) ◦ pB,B∩E (property of map to direct limit)

E M M = αC∩E ◦ pC,C∩E ◦ iBC (consistency axiom)

M = ψC ◦ iBC .

By the universal property of the direct limit, there therefore exists a unique homomor-

N phism from NF to NE causing diagram commutativity; deﬁne pFE to be this homomorphism.

Again, to ensure that N = (iN , pN ) is indeed a geometric module, the consistency

axiom is the only non-trivial one to check. Let E,F,G be controlled sets with E,F ⊂ G

E and let m ∈ NE. Then, for some bounded B ⊂ E, there exists m ∈ MB with αB = m; M G also, iEG(m) is αB(m) ∈ NG. Similarly, there exists a bounded subset C of G and 0 N G 0 N N F M 0 m ∈ MC such that iEG(m) = αC (m ), and (pGF ◦ iEG)(m) = (αC∩F ◦ pC,C∩F )(m ). N E∩F M N On the other hand, pE,E∩F (m) = (αB∩(E∩F ) ◦ pB,B∩(E∩F ))(m). Then pE,E∩F (m) = 00 N N F 00 m ∈ MD for some bounded subset D of E ∩ F and (iE∩F,F ◦ pE,E∩F )(m) = αD(m ). By choosing a bounded subset A of G suﬃciently large (A = B ∪ (C ∪ D) will

N suﬃce), iEG sends m, represented by m ∈ MB, to the element of NG represented M N N by iBA(m) ∈ MA. Similarly, pGF sends iEG(m) to the element of NF represented M M N by (pA,A∩F ◦ iBA)(m). On the other hand, pE,E∩F sends m to the element of NE∩F M N N represented by pB,B∩F (m) ∈ MB∩F , and iE∩F,F sends pE,E∩F (m) to the element of NF M M represented by (iB∩F,A∩F ◦ pB,B∩F )(m). Since B ∩ (A ∩ F ) = B ∩ F , the consistency axiom gives

M M M M (iB∩F,A∩F ◦ pB,B∩F )(m) = (pA,A∩F ◦ iBA)(m).

N N N N N N Therefore, (iE∩F,F ◦ pE,E∩F )(m) = (pGF ◦ iEG)(m). This shows that (i , p ) is a

71 geometric module.

N N To show that (i , p ) is cocontrolled, note that for all bounded B, NB = MB. Then,

lim N = lim M = N , −→ B −→ B E B⊂E B⊂E and ψ : lim N → N is the identity map. −→ B E B⊂E In the case that B0 is a proper coﬁnal subcategory of B(X), the above proof proceeds largely unaltered, the only change being that any instance of a bounded set B which may not necessarily lie in B0 (for example, C ∩ E for E controlled and C ∈ B0) must be

replaced with some B ∈ B0 containing B.

Obviously the objects NE are unique up to isomorphism by deﬁnition of a cocon-

N trolled module. Also the iEF are unique; as we have seen, an element m ∈ NE can be represented by m ∈ MB. Choosing C ⊂ F suﬃciently large that B ⊂ C and that

N M iEF m is represented by an element of MC , the element iBC (m) is that representative. N M Therefore the morphisms iEF depend only on the (ﬁxed) morphisms iBC . N Proof of uniqueness of the pFE is analogous. An element n ∈ NF is represented by N n ∈ MB; taking B ⊂ F suﬃciently large, we may assume that pFE(n) is represented by M N an element of MB∩E. Then, that representative is pB,B∩E(n), so p depends only on M the pCB.

Lemma 4.5.3. If M, N are geometric modules, and M 0, N 0 are the (co)controlled

closures of the partial geometric modules formed by restricting M and N to bounded

sets, then a geometric module morphism from M to N induces a geometric module

morphism from M 0 to N 0.

Proof. Let M 0 := lim M ∼ lim M . By the compatibility condition on φ with E −→ B = −→ B B B⊂E B∈BE respect to iBC , for each E there exists a homomorphism

φ0 := lim φ : M 0 → N 0 . E −→ B E (f×f)(E) B∈BE

72 Since (f, {φE}) is a morphism, for all E ∈ C(X) and B ⊂ C in BE (as given in Deﬁnition

M N 4.4.2) such that φC ◦ iBC = if×f(B),f×f(C) ◦ φB (and similarly for p). Since φB = 0 0 0 φB for bounded B, it follows that the analogous compatibilities hold for φB and φC .

Additionally, by the deﬁnition of the direct limit and the compatibility of φB, φC with

M 0 respect to i for B,C ∈ BE, and because iBE for B ∈ BE is the canonical map to the 0 M 0 N 0 0 direct limit, φC ◦ iBE = if×f(B),f×f(C) ◦ φB as well (and similarly for p). It follows that 0 (f, {φE}) is a morphism. The inverse limit argument is analogous.

4.6 Total Modules

Deﬁnition 4.6.1. Suppose M = (iM , pM ) is a cocontrolled module. Then, deﬁne the total module of M to be the R-module

lim M ∼ lim lim M . ←− E = ←− −→ B E∈C(X) E∈C(X) B⊂E

Analogously, if (i, p) is a controlled module, its total module is deﬁned to be

lim M ∼ lim lim M . −→ E = −→ ←− B E∈C(X) E∈C(X) B⊂E

Lemma 4.6.2. The module CXq(X) is isomorphic to the total module of its associated cocontrolled geometric module M. (Similarly, CXq(X) is isomorphic to the total module of its associated controlled geometric module).

q Proof. Let T denote the aforementioned total module. The quotient maps qE : CX (X) →

M ME from Example 4.2.3 satisfy pFE ◦qF = qE for all E ⊂ F . Therefore, if γE : T → ME are the canonical projection maps from the inverse limit, there exists a (unique) homo-

q morphism p : CX (X) → T such that γE ◦ p = qE. By an analogue of the proof of Lemma 4.3.3, p is a surjection. So it remains to prove that p is injective as well.

73 Let f lie in the kernel of p; then, for all E, qE(f) = (γE ◦ p)(f) = 0. For x ∈ Xq+1, there exists a controlled E ⊂ Xq+1 such that x ∈ E(q+1) (for example, let E =

{(πi(x), πj(x))} where the maps πi are coordinate projections). Then, since qE(f) = 0,

q all representatives g of qE(f) in CX (X) must satisfy g(x) = 0. It follows that f(x) = 0 for all x ∈ Xq+1; therefore f is the zero map and so p is injective.

Lemma 4.6.3. If M and N are controlled geometric modules, a covariant morphism of controlled type from M to N induces a module homomorphism from the total module of M to the total module of N. In this way, the formation of total modules deﬁnes a functor from the category of controlled geometric modules and covariant morphisms of controlled type to the category of R-modules.

Similarly, if M and N are cocontrolled geometric modules, a contravariant morphism

of cocontrolled type from N to M induces a module homomorphism from the total module

of N to the total module of M.

Proof. Again, we shall prove only the covariant case; as the contravariant argument is

analogous. Similar to the proof of Lemma 4.5.3, compatibility of morphisms of controlled

M type with respect to iEF ensures the existence of a homomorphism

lim φ : lim M → lim N −→ E −→ E −→ (f×f)(E) E∈C(X) E∈C(X) E∈C(X)

In the case that M and N are both controlled, the universal property of direct limits

ensures the existence of a unique homomorphism

ψ : lim N → lim N , −→ (f×f)(E) −→ F E∈C(X) F ∈C(Y )

since the canonical maps

α : N → lim N (f×f)(E) (f×f)(E) −→ F F ∈C(Y )

74 N are obviously compatible with the iEF . Therefore, we have a ring homomorphism

φ := ψ ◦ lim φ : lim M → lim N −→ E −→ E −→ F E∈C(X) E∈C(X) F ∈C(Y ) between total modules.

We claim that the mapping C0 from the category of controlled modules to the category of R-modules deﬁned by associating to controlled module M its total module

Tot(M) and to each morphism (f, {φE}) the R-module homomorphism φ is a functor. This follows from the fact that the “composition” operation ∗ as deﬁned in Deﬁnition

4.4.12 commutes with direct limits. To see this, let L, M, and N be controlled modules

over X, Y , Z and (f, {φE}): L → M,(f, {φE}): M → N be morphisms of cocontrolled

type. For x ∈ Tot(L), there exists E ∈ C(X) and x ∈ LE such that the image of x in

Tot(L) is x. Then φ(x) ∈ Tot(M) is the image of φE(x) ∈ M(f×f)(E) and ψ ◦ φ(x) ∈

Tot(N) is the image of ψ(f×f)(E) ◦ φE(x) ∈ N(g×g)(f×f)(E). On the other hand, ψ ∗ φ(x)

is the image of ψ(f×f)(E) ◦ φE(x), so the homomorphisms agree on Tot(L).

Remark 4.6.4. If M and N are cocontrolled modules and (f, {φE}) is a covariant cocontrolled morphism from M to N, an analogue of the previous argument shows that

the inverse limit homomorphism

lim φ : lim M → lim N ←− E ←− E ←− (f×f)(E) E∈C(X) E∈C(X) E∈C(X)

is a module homomorphism, but not necessarily a homomorphism between the total

modules of M and N.

Example 4.6.5. Let M,N be the modules associated to CXq(X), CXq(Y ) and deﬁne

(f, {φE}) for a coarse map f as in Example 4.4.8. Then {φE} induces a homomorphism lim φ on the total module lim M (the direct limits taken over all controlled E) as −→ E −→ E

75 follows: for m ∈ lim M , there exists E ∈ C(X) and −→ E

X m = αx(x0, . . . , xq) x∈E(q+1)

and the image of m in lim M is m. Then, −→ E

X (lim φ )(m) := φ (m) = α (f(x ), . . . , f(x )). −→ E E x 0 q x∈E(q+1)

That is to say, lim φ = f : CX (X) → CX (Y ). −→ E ∗ q q

4.7 Cocontrolled and Controlled Tensor Products

Suppose that X and Y are coarse spaces and M = (i, p) and M 0 = (i0, p0) are geometric

X- and Y - modules respectively. Then, one may form a partial geometric (X × Y )-

module N = (i ⊗ i0, p ⊗ p0) by deﬁning, for all bounded subsets B ⊂ B0 of X × X,

C ⊂ C0 of Y × Y :

0 • NB×C := MB ⊗ MC ;

0 0 • (i ⊗ i )B×C,B0×C0 := iBB0 ⊗ iCC0 ;

0 0 • (p ⊗ p )B0×C0,B×C := pB0B ⊗ pC0C .

Thus (i ⊗ i0, p ⊗ p0) is deﬁned only on bounded subsets of X × Y of the form B × C,

where B and C are bounded subsets of X and Y respectively.

Lemma 4.7.1. The pair of functors (i ⊗ i0, p ⊗ p0) is a partial geometric module over the subcategory of B(X × Y ) whose objects are all the bounded sets of the form B × C;

B ∈ B(X),C ∈ B(Y ).

Proof. Since i, i0 are covariant and p, p0 are contravariant, i ⊗ i0 is covariant and p ⊗ p0 is contravariant; also, since i(B) = p(B) and i0(C) = p0(C) for all B ∈ B(X),C ∈ B(Y ),

76 (i ⊗ i0)(B × C) = (p ⊗ p0)(B × C). To verify the next two axioms for a geometric module, note that the tensor product of the zero module with itself is the zero module and the tensor product of two ﬁnitely generated free modules is ﬁnitely generated and free. Finally, to verify the consistency axiom, let B ⊂ C ⊂ D be bounded subsets of

X × X and B0 ⊂ C0 ⊂ D0 be bounded subsets of Y × Y . Then

0 0 0 0 (p ⊗ p )D×D0,C×C0 ◦ (i ⊗ i )B×B0,C×C0 = (pDC ⊗ pD0C0 ) ◦ (iBC ⊗ iB0C0 ) =

0 0 = (pDC ◦ iBC ) ⊗ (pD0C0 ◦ iB0C0 ), and

0 0 (i ⊗ i )(B×B0)∩(C×C0),C×C0 ◦ (p ⊗ p )B×B0,(B×B0)∩(C×C0) =

0 0 = (i ⊗ i )(B∩C)×(B0∩C0),C×C0 ◦ (p ⊗ p )B×B0,(B∩C)×(B0∩C0) =

0 = (iB∩C,C ◦ pB,B∩C ) ⊗ (iB0∩C0,C0 ◦ pB0,B0∩C0 ).

0 0 From the consistency axioms on (i, p) and (i , p ), we know pDC ◦ iBC = iB∩C,C ◦ pB,B∩C

0 0 0 and pD0C0 ◦ iB0C0 = iB0∩C0,C0 ◦ pB0,B0∩C0 , so the result follows.

Deﬁnition 4.7.2. Given a geometric X-module (i, p) and a geometric Y -module (i0, p0),

0 0 their cocontrolled tensor product, denoted (i ⊗cc i , p ⊗cc p ), a geometric X × Y -module, is the cocontrolled closure of (i ⊗ i0, p ⊗ p0).

0 0 Similarly, the controlled tensor product (i ⊗c i , p ⊗c p ) is the controlled closure of (i ⊗ i0, p ⊗ p0).

4.8 Tensor Products

Now, let X and Y be coarse spaces and consider the geometric modules associated

p q p to CX (X) and CX (Y ). For these geometric modules, denote by CX (X)E and

q CX (Y )F the R-modules which correspond to the controlled sets E ⊂ X×X, F ⊂ Y ×Y

77 p q respectively. Then, the cocontrolled tensor product CX (X) ⊗cc CX (Y ) is isomorphic to the R-module

lim lim CXp(X) ⊗ CXq(Y ) . ←− −→ B C E∈C(X),F ∈C(Y ) B⊂E,C⊂F B∈B(X),C∈B(Y )

n X p q Additionally, the modules C := CX (X) ⊗cc CX (Y ) form a chain complex, p+q=n p q with coboundary map induced by the coboundaries on CX (X)B ⊗CX (Y )C by taking limits.

Recall the Eilenberg-Zilber theorem from algebraic topology (i.e. [16], Theorem

59.2), which states that, if X and Y are topological spaces, and C∗(X) a singular

cochain complex over X, there exists a chain homotopy equivalence between the cochain

complexes C∗(X × Y ) and (C∗(X) ⊗ C∗(Y ))∗ deﬁned by

M (C∗(X) ⊗ C∗(Y ))n := Cp(X) ⊗ Cq(Y ). p+q=n

In coarse algebraic topology, it is not necessarily the case that the coarse cochain com-

plex of a product space is chain homotopic to such an algebraic tensor product. However,

we shall prove in a later section that:

Theorem 4.8.1. There exists a chain homotopy equivalence between the cochain com-

n X p q plexes CX (X × Y ) and CX (X) ⊗cc CX (Y ). p+q=n We shall proceed by ﬁrst showing that there exists an explicit cochain description

∗ ∗ of the tensor product CX (X) ⊗cc CX (Y ), as follows:

Deﬁnition 4.8.2. Let X and Y be coarse spaces. Deﬁne CXp(X)⊗ˆ CXq(Y ) to be the

module of equivalence classes of (possibly inﬁnite) formal sums

X σi ⊗ τi, i

78 p q where each σi ⊗ τi is an elementary tensor in CX (X) ⊗ CX (Y ), and such that for each pair of controlled subsets E ⊂ Xp+1,F ⊂ Y q+1, only ﬁnitely many

σi|E ⊗ τi|F are nonzero. The equivalence relation is given by declaring any two such decompositions

X X 0 0 σi ⊗ τi, σj ⊗ τj i j

p+1 q+1 to be equivalent if and only if, for all (x0, . . . , xp, y0, . . . , yq) ∈ X × Y ,

X X 0 0 σi(x0, . . . , xp)τi(y0, . . . , yq) = σj(x0, . . . , xp)τj(y0, . . . , yq). i j

Remark 4.8.3. The sum

X σi(x0, . . . , xp)τi(y0, . . . , yq) i

described above indeed exists for any pair of points x = (x0, . . . , xp), y = (y0, . . . , yq) by the aforementioned controlled ﬁniteness property of sums, as {x} and {y} are controlled

subsets of Xp+1 and Y q+1 respectively.

We then have a cochain complex CX∗(X)⊗ˆ CX∗(Y )∗, where

n M CX∗(X)⊗ˆ CX∗(Y ) = Cp(X)⊗ˆ Cq(Y ), p+q=n

and the coboundary map is given by extending the usual tensor product coboundary to

formal sums.

Remark 4.8.4. The extended coboundary map indeed preserves the “controlled ﬁnite-

79 ness” property of sums in the cochain complex. If

X p q ψ = σi ⊗ τi ∈ CX (X)⊗ˆ CX (Y ), i then

X p ∂ψ = ∂X σi ⊗ τi + (−1) σi ⊗ ∂Y τi. i

p+2 q+1 Let E ⊂ X , F ⊂ Y be controlled. By deﬁnition, only ﬁnitely many τi have support which lie in F . For each j = 0, . . . , p + 2,

σi(x0,..., xˆj, . . . xp+2) 6= 0

for some (x0, . . . , xp+2) ∈ E if and only if σi is nonzero on some element of the set

{(x0,..., xˆj, . . . xp+2)|(x0, . . . , xp+2) ∈ E}, which is a controlled subset of Xp+1. Therefore, there are only ﬁnitely many i for which

σi(x0,..., xˆj, . . . xp+2) 6= 0 for some element of E. Since this is true for all j, there are only ﬁnitely many i for which p+2 X j ∂X σi(x0, . . . , xp+2) = (−1) σi(x0,..., xˆj, . . . xp+2) 6= 0; j=0 P P thus the sum i ∂X σi ⊗τi is controlledly ﬁnite. The proof for i σi ⊗∂Y τi is analogous.

Furthermore,

Theorem 4.8.5. For all p, q, there is a cochain isomorphism from the cocontrolled

p q p q tensor product CX (X) ⊗cc CX (Y ) to CX (X)⊗ˆ CX (Y ).

p+1 q+1 p q Proof. Let B ⊂ X , C ⊂ Y be bounded and CX (X)B, CX (Y )C be the sub-

80 modules of CXp(X), CXq(Y ) consisting of those maps whose supports lie in B and C respectively. Any element

n X p q σi ⊗ τi ∈ CX (X)B ⊗ CX (Y )C i=1 can be decomposed as an equivalent element

n X X σi(x)χ{x} ⊗ τi(y)χ{y}, i=1 x∈B,y∈C where χA is the characteristic function of A (note that the inner sum is ﬁnite by boundedness of B and C). This element, in turn, is equivalent to

n X X σi(x)τi(y)χ{x} ⊗ χ{y}. i=1 x∈B,y∈C

P P 0 0 p q It follows that two elements σi ⊗τi, σj ⊗τj of CX (X)B ⊗CX (Y )C are equivalent if and only if

X X 0 0 σi(x)τi(y) = σj(x)τj(y) for all x ∈ B, y ∈ C.

Given controlled subsets E ⊂ Xq+1, F ⊂ Y q+1,

lim CXp(X) ⊗ CXq(Y ) −→ B C B⊂E,C⊂F B∈B(X),C∈B(Y )

P is therefore isomorphic to the module ME,F of equivalence classes of elements σi(x)τi(y), such that there exist bounded subsets B ⊂ E and C ⊂ F such that the supports of

each σi and τi are subsets of B and C respectively, the isomorphism given by the above decomposition.

The module lim lim CXp(X) ⊗ CXq(Y ) , where the inverse limit is ←−E,F −→B⊂E,C⊂F B C

81 taken over all controlled E ⊂ Xp+1, F ⊂ Y q+1, is therefore isomorphic to the module of

q+1 q+1 sequences {(ψE,F ) | E ⊂ X ,F ⊂ Y controlled} of elements ψE,F ∈ ME,F , where each ψE,F is represented by some

X 0 φE,F ⊗ φE,F

0 0 0 0 and if E ⊂ E, F ⊂ F , the restriction of ψE,F to E × F is equivalent to ψE0,F 0 . The map sending each such (ψE,F ) to the element represented by the sum

X σi ⊗ τi such that, for x ∈ Xp+1, y ∈ Y p+1 and controlled sets E,F with x ∈ E, y ∈ F ,

X X 0 σi(x) ⊗ τi(y) = φE,F (x) ⊗ φE,F (y), is an isomorphism from lim lim CXp(X) ⊗ CXq(Y ) to the module of ←−E,F −→B⊂E,C⊂F B C equivalence classes of sums X σi ⊗ τi that are ﬁnite on controlled sets. Therefore, by the explicit deﬁnition of elements in

CXp(X)⊗ˆ CXq(Y ), it follows that

CXp(X)⊗ˆ CXq(Y ) ∼ lim lim CXp(X) ⊗ CXq(Y ) , = ←− −→ B C E,F B⊂E,C⊂F where the inverse limit is taken over all controlled E ⊂ Xp+1, F ⊂ Y q+1. The set

of controlled subsets of Xp+1 of the form E(p+1) is coﬁnal in the set of all controlled

subsets of Xp+1 (similarly for Y ). To see this, let F be a controlled subset of Xp+1 and

82 p+1 let πi, 0 ≤ i ≤ p + 1, be the canonical projections from X to X. Then

E = {(πi(x), πj(x))|x ∈ E, 0 ≤ i, j ≤ p} is a controlled subset of X × X and F ⊂ E(p+1). Similarly, the set of bounded subsets of each E(p+1) of the form B(p+1) is coﬁnal in the set of all bounded subsets of E(p+1).

Therefore, the modules

CXp(X)⊗ˆ CXq(Y ) ∼ lim lim CXp(X) ⊗ CXq(Y ) = ←− −→ B C E,F B⊂E,C⊂F and

CXp(X) ⊗ CXq(Y ) ∼ lim lim CXp(X) ⊗ CXq(Y ) cc = ←− −→ B C E,F B⊂E,C⊂F are isomorphic.

∗ ∗ We shall use the preceding explicit description of CX (X)⊗cc CX (Y ) when demonstrating the existence of a chain homotopy equivalence between it and CX∗(X × Y ), in the proof of Theorem 4.8.1.

4.8.1 The Cross Product Map

The ﬁrst (and more straightforward) map in the cochain equivalence between CXn(X ×

X p q Y ) and CX (X) ⊗cc CX (Y ) asserted in Theorem 4.8.1 is the cross product. Let p+q=n σ ⊗ τ ∈ CXp(X)⊗ˆ CXq(Y ) be an elementary tensor; then the cross product σ × τ ∈

CXp+q(X × Y ) is given by

(σ × τ) ((x0, y0),..., (xp+q, yp+q)) = σ(x0, . . . , xp)τ(yp, . . . , yp+q).

83 p q We extend the cross product to CX (X) ⊗cc CX (Y ) in the obvious way, i.e.

! X X × σi ⊗ τi = σi × τi. i i

p q p+q The extended cross product indeed maps CX (X)⊗cc CX (Y ) to CX (X ×Y ); given

σ ∈ CXp(X), τ ∈ CXq(Y ) and E ⊂ (X × Y )p+q+1 controlled, the projections of the set

p q+1 S = {(x0, . . . , xp, yp, . . . , yp + q)|∃(a0, . . . , ap−1) ∈ Y , (bp+1, . . . , bp+q) ∈ X with

((x0, a0),..., (xp−1, ap−1), (xp, yp),..., (bp+1, yp+1),..., (bp+q, yp+q)) ∈ E}

to Xp+1 and Y q+1 are controlled as well. Therefore the set

Supp(σ × τ) ∩ S

is of the form

B × B0,

where B ⊂ Xp+1, B0 ⊂ Y q+1 are bounded. It follows that

Supp(σ × τ) ∩ E

is bounded in (X × Y )p+q+1.

p Remark 4.8.6. The cross product map respects the equivalence relation on CX (X) ⊗cc CXq(Y ); indeed, if

X X 0 0 p q σi ⊗ τi, σj ⊗ τj ∈ CX (X)⊗ˆ CX (Y ) i j

84 p+1 q+1 are equivalent, then, for all (x0, . . . , xp, y0, . . . , yq) ∈ X × Y ,

X X 0 0 σi(x0, . . . , xp)τi(y0, . . . , yq) = σj(x0, . . . , xp)τj(y0, . . . , yq). i j

It follows that

X X (σi × τi) ((x0, y0),..., (xp+q, yp+q)) = σi(x0, . . . , xp)τi(yp, . . . , yp+q) = i i

X 0 0 X 0 0 = σj(x0, . . . , xp)τj(yp, . . . , yp+q) = (σj × τj) ((x0, y0),..., (xp+q, yp+q)) . j j

p q Now, consider an arbitrary element of CX (X) ⊗cc CX (Y ) represented by

X σi ⊗ τi

p+q+1 and a controlled set E ⊂ (X × Y ) , and let E1,E2 be the restrictions of E to the

ﬁrst p + 1 X-coordinates and last q + 1 Y -coordinates respectively. Then, E1 and E2 are controlled, and only ﬁnitely many

σi|E1 ⊗ τi|E2

are nonzero by deﬁnition of the cocontrolled tensor product. Therefore,

X Supp σi × τi ∩ E

is the union of ﬁnitely many bounded sets and is therefore bounded. That is to say,

X p+q σi × τi ∈ CX (X × Y ).

Additionally, the cross product is indeed a cochain map; that is, ×∂ = ∂×. It

85 p q suﬃces to show this for an elementary tensor σ ⊗ τ ∈ CX (X) ⊗cc CX (Y ) and extend by linearity. We have that

p (∂σ × τ + (−1) σ × ∂τ) ((x0, y0),..., (xp+q+1, yp+q+1)) =

p = ∂σ(x0, . . . , xp+1)τ(yp+1, . . . , yp+q+1) + (−1) σ(x0, . . . , xp)∂τ(yp, . . . , yp+q+1)

p+1 X i = (−1) σ(x0,..., xˆi, . . . , xp+1)τ(yp+1, . . . , yp+q+1)+ i=0

q+1 X p+j (−1) σ(x0, . . . , xp)τ(yp,..., ypˆ+j, . . . , yp+q+1) j=0

p X i = (−1) σ(x0,..., xˆi, . . . , xp+1)τ(yp+1, . . . , yp+q+1)+ i=0

p+1 +(−1) σ(x0, . . . , xp)τ(yp+1, . . . , yp+q+1)+

q+1 X p+j + (−1) σ(x0, . . . , xp)τ(yp,..., ypˆ+j, . . . , yp+q+1)+ j=1

p p+1 +(−1) (−1) σ(x0, . . . , xp)τ(yp+1, . . . , yp+q+1)

p X i = (−1) σ(x0,..., xˆi, . . . , xp+1)τ(yp+1, . . . , yp+q+1)+ i=0

q+1 X p+j + (−1) σ(x0, . . . , xp)τ(yp,..., ypˆ+j, . . . , yp+q+1). j=1

Conversely,

∂(σ × τ) ((x0, y0) ..., (xp+q+1, yp+q+1)) =

p+q+1 X k (−1) (σ × τ) (x0, y0),..., (x\k, yk) ..., (xp+q+1, yp+q+1) j=0

p X k = (−1) σ(x0,..., xˆk, . . . , xp+1)τ(yp+1, . . . , yp+q+1)+ k=0

86 p+q+1 X k + (−1) σ(x0, . . . , xp)τ(yp,..., yˆk, . . . , yp+q+1) k=p+1

p X i = (−1) σ(x0,..., xˆi, . . . , xp+1)τ(yp+1, . . . , yp+q+1)+ i=0

q+1 X p+j + (−1) σ(x0, . . . , xp)τ(yp,..., ypˆ+j, . . . , yp+q+1). j=1

Given n ∈ N, taking the sum of the cross products on each individual

p q CX (X) ⊗cc CX (Y ), p + q = n results in the cross product map

M p q n × : CX (X) ⊗cc CX (Y ) → CX (X × Y ). p+q=n

Clearly, this cross product commutes with the coboundary map by linearity.

4.8.2 Decomposing elements of CXn(X × Y )

In order to deﬁne a map from CXn(X × Y ) to

M p q CX (X) ⊗cc CX (Y ), p+q=n

n n n we shall ﬁrst express elements in CX (X × Y ) as elements of CX (X) ⊗cc CX (Y ). An element ψ ∈ CXn(X × Y ) is a map from (X × Y )n+1 to R with cocontrolled support. We wish to decompose ψ into a sum

X n n σi ⊗ τi ∈ CX (X) ⊗cc CX (Y ) i

87 such that, for all ((x0, y0),..., (xn, yn)),

X σi(x0, . . . , xn)τi(y0, . . . , yn) = ψ((x0, y0),..., (xn, yn)). i

To that end, let χA denote the characteristic function of the set A and consider the following decomposition of ψ:

X 0 (n) 0 (n) ψ((x, y0), (x , y1),..., (x , yn)) ⊗ χ{(y0,...,yn)}(y, y , . . . , y ). n+1 (y0,...,yn)∈Y

0 (n) 0 (n) n Observe that each σ(x, x , . . . , x ) := ψ((x, y0), (x , y1),..., (x , yn)) lies in CX (X), and the characteristic function of a point in Y n+1 clearly lies in CXn(Y ). Furthermore,

this sum is controlledly ﬁnite: suppose we restrict our points ((x, y),..., (x(n), y(n)) to

n+1 n+1 a controlled subset of (X × Y ) . Then EX , the projection of E to X is controlled,

as is EY , and we obtain

0 X 0 (n) 0 (n) ψ := ψ((x, y0), (x , y1),..., (x , yn))|EX ⊗ χ{(y0,...,yn)}(y, y , . . . , y ). (y0,...,yn)∈EY

0 (n) Only ﬁnitely many of the terms ψ((x, y0), (x , y1),..., (x , yn)) are nonzero by the cocontrolled support condition on ψ; therefore the sum is ﬁnite on E. The sum of

0 n n tensors ψ represents a decomposition of ψ in CX (X)⊗cc CX (Y ). This decomposition

n n is unique up to equivalence in CX (X) ⊗cc CX (Y ), as any such decomposition

X σi ⊗ τi i

of ψ will, by deﬁnition, satisfy

X σi(x0, . . . , xn)τi(y0, . . . , yn) = ψ((x0, y0),..., (xn, yn)). i

88 Now, given non-negative integers p, q such that p + q = n, we wish to obtain an

p q P element of CX (X) ⊗cc CX (Y ) from the previous decomposition σi ⊗ τi. This will be done via the shuﬄe map. The next subsection provides a deﬁnition of this map.

4.8.3 The Shuﬄe Map

The following deﬁnitions can be found in many standard algebraic topology texts, for instance, [2] page 184.

Deﬁnition 4.8.7. If p, q are non-negative integers, a (p, q)-shuﬄe (µ, ν) is a pair of disjoint sets of integers

µ = {µ1, . . . , µp}, ν = {ν1, . . . , νq} such that

1 ≤ µ1 < µ2 < . . . < µp ≤ p + q, 1 ≤ ν1 < ν2 < . . . < νq ≤ p + q;

additionally, deﬁne µ0 = ν0 = 0, µp+1 = νq+1 = p + q + 1. Let sgn(µ, ν) denote the sign of the permutation (µ1, µ2 . . . , µp, ν1, ν2, . . . , νq) of (1, 2, . . . , p + q).

Then, given a coarse space X, a (p, q)-shuﬄe (µ, ν) and a (p + 1)−tuple of points

p+1 0 0 (x0, . . . , xp) in X , deﬁne a (p+q+1)−tuple of points γµ(x0, . . . , xp) = (x0, . . . , xp+q+1) ∈ Xp+q+1 by

0 xi = xj if µj ≤ i < µj+1

Analogously, given a (q + 1) tuple of points (y0, . . . , yq), deﬁne γν by γν(y0, . . . , yq) =

0 0 (y0, . . . , yp+q+1), where 0 yi = yj if νj ≤ i < νj+1.

Deﬁnition 4.8.8 (The (p, q)-shuﬄe map). Given maps σ ∈ CXn(X), τ ∈ CXn(Y ), and

89 p, q non-negative integers such that p + q = n, deﬁne

X 0 0 p q Sh (σ ⊗ τ) = σi ⊗ τi ∈ CX (X) ⊗cc CX (Y ) (p,q) by

X 0 0 σi(x0, . . . , xp) ⊗ τi (y0, . . . , yq) :=

X sgn(µ, ν)σ (γµ(x0, . . . , xp, y0, . . . , yq)) ⊗ τ (γν(x0, . . . , xp, y0, . . . , yq)) , (µ,ν)

where the right-hand sum is taken over all (p, q)-shuﬄes (µ, ν).

Definition 4.8.9 (The shuffle map). For σ, τ given as in the previous definition, define

M Sh(σ ⊗ τ) ∈ CXp(X) ⊗ CXq(Y ) p+q=n

by n X Sh(σ ⊗ τ) := Sh (σ ⊗ τ). (i,n−i) i=0

p q We then deﬁne the shuﬄe map on CX (X)⊗cc CX (Y ) by extending Sh to formal sums of elementary tensors in the obvious way.

Observe that for

X n n σj ⊗ τj ∈ CX (X) ⊗cc CX (Y ), j

the element   X Sh  σj ⊗ τj j

L p q p+1 q+1 indeed lies in p+q=n CX (X)⊗ccCX (Y ). Given controlled sets E ⊂ X ,F ⊂ Y , the set

{γµ(x0, . . . , xp) × γν(y0, . . . , yq)| (x0, . . . , xp, y0, . . . , yq) ∈ E × F }

90 is controlled in (X × Y )n+1 for each (p, q)-shuﬄe (µ, ν). Therefore, the set

  X Supp  σjγµ ⊗ τjγν ∩ (E × F ) j

is bounded in Xp+1 × Y q+1. Since there are ﬁnitely many (p, q) shuﬄes for a given p, q,

it follows that Sh maps to the desired direct sum of tensor products.

We may now deﬁne the Alexander-Whitney map, which provides half of the desired

cochain homotopy equivalence:

Deﬁnition 4.8.10. Let ψ ∈ CXn(X × Y ) with decomposition represented by

0 X n n ψ = σi ⊗ τi ∈ CX (X) ⊗cc CX (Y ) i

as given in the previous subsection. Then, the Alexander-Whitney map

n M p q A : CX (X × Y ) → CX (X) ⊗cc CX (Y ) p+q=n

is given by

A(ψ) := Sh(ψ0).

It is a combinatorial exercise (for example, refer to the arguments in Section 2.3 of

[17]) to show that A is a chain map.

4.8.4 Construction of Homotopy Equivalences

Proposition 4.8.11. The maps

n M p q A : CX (X × Y ) → CX (X) ⊗cc CX (Y ) p+q=n

91 Figure 4.1: Path corresponding to the shuﬄe µ = {1, 4, 5, 6}, ν = {2, 3, 7}. and

M p q n × : CX (X) ⊗cc CX (Y ) → CX (X × Y ) p+q=n are cochain homotopy inverses.

We shall prove this proposition by constructing explicit chain homotopies between

× ◦ A and the identity map on CXn(X × Y ) (similarly for A ◦ ×). First, we provide an alternative deﬁnition of (p, q) shuﬄes which will be used in the following proof.

Remark 4.8.12. Let p, q be non-negative integers and Λ(p,q) be a lattice with vertices 0 {(i, j) | 0 ≤ i ≤ p, 0 ≤ j ≤ q}. Given a (p, q)-shuffle (µ, ν), define µi, 0 ≤ i ≤ p + q, by 0 0 0 µi = µj if i ≥ j. Similarly, define νi, 0 ≤ i ≤ p + q, by νi = νj if i ≥ j. Then, (µ, ν) may be characterized by a path

0 0 0 0 0 0 γ = {(µ0, ν0), (µ1, ν1),..., (µp+q, νp+q)}

on Λ(p,q) as in Figure 4.1; that is to say, γ is a sequence of p + q + 1 points starting at (0, 0) and such that each point is obtained from the one before it by moving either one

step upwards or one step to the right on Λ(p,q) Conversely, if

92 γ = {(µ0, ν0), (µ1, ν1),..., (µp+q, νp+q)}

is a path of length n on Λ(p,q) such that µ0 = ν0 = 0, µp+q, νp+q = p + q, and either

µi+1 = µi + 1 or νi+1 = νi + 1 for each i, then γ corresponds to a (p, q)-shuﬄe (µ, ν). Therefore (p, q)-shuﬄes are in one-to-one correspondence with paths γ as characterized above. Additionally, if (µ, ν) corresponds with γ, and if |γ| is the number of squares lying under the path γ in Λ(p,q), then

sgn(µ, ν) = (−1)|γ|.

n th Let ψ ∈ CX (X × Y ); observe that, if γ is a p, q-shuﬄe, γi,x represents the i coordinate of γu(x), and sp is the map sending yi to yi+p, the map (×A)ψ may be written

(×A)ψ ((x0, y0),..., (xn, yn)) =

X X |γ| (−1) ψ ((γ0,x, spγ0,y), (γ1,x, spγ1,y),..., (γn,x, spγn,y)) , (4.1) p+q=n (p,q)−shuﬄes γ i.e. as a sum of shuﬄes which have been shifted upward p units in the y−direction in

+ the lattice. Let Λn be the subgraph of Λ(n,n) with vertices

{(xi, yj) : 0 ≤ i ≤ j ≤ n}.

+ Remark 4.8.13. From this point forwards, a path on Λn will refer to a list of n vertices

{(xi1 , yj1 ), (xi2 , yj2 ),..., (xin , yjn )}

with im ≤ im+1 and jm ≤ jm+1 for all m.

Then, the collection of these vertically shifted shuﬄes for all p, q is the set of all

+ length n increasing paths δ on Λn such that δ begins at (x0, yk) and ends at (xl, yn) for

93 some k, l.

The chain homotopy between ×A and the identity will be constructed in terms of these paths.

The following calculations take place in the group Gn generated by the set of signed

+ 0 paths on Λn with formal addition as the operation. Given a shuﬄe γ in (4.1), let γ + n be the path on Λn associated to γ. Then we may identify with ×A : CX (X × Y ) → CXn(X × Y ) the element

X |γ| 0 Γn := (−1) γ ∈ Gn, γ

where the sum is taken over all shuﬄes γ appearing in (4.1).

Remark 4.8.14. As was noted previously, there exists an identiﬁcation between a path

0 n n γ ∈ Gn and its associated map γ :(X × Y ) → (X × Y ) , and so we shall continue to use the two interchangeably.

We now deﬁne maps h, π on G∗ as follows:

Deﬁnition 4.8.15. Given a path γ ∈ Gn, for some n, let

• hγ ∈ Gn be the path {(x0, y0), γ};

• πγ ∈ Gn+1 be the path obtained by sending each vertex (xi, yj) in γ to (xi+1, yj+1).

Remark 4.8.16. 1. Note that, if γ is the empty path with no vertices, we say that

hγ = {(x0, y0)}.

n 2. Given γ ∈ Gn−1 and ψ ∈ CX (X × Y ), the maps ψ(hγ) and ψ(πγ) are elements of CXn(X × Y ) as well.

∗ Remark 4.8.17. If ∂ is the coboundary map on CX (X × Y ), an element γ ∈ Gn has

n−1 associated elements γ∂, ∂γ ∈ Gn, given as follows. For φ ∈ CX (X × Y ), there exist

γ0, γ1, . . . , γn ∈ Gn such that

n X i (∂φγ ((x0, y0),..., (xn, yn)) = (−1) φ ◦ γi ((x0, y0),..., (xn, yn)) ; i=0

94 then, n X i γ∂ = (−1) γi. i=0 Similarly, given ψ ∈ CXn(X × Y ), write

n+1 X j 0 ∂ (ψγ) ((x0, y0),..., (xn+1, yn+1)) = (−1) ψ ◦ γj ((x0, y0),..., (xn+1, yn+1)) ; j=0

then, n+1 X i 0 ∂γ = (−1) γi. i=0 Observe that the elements γ∂ and ∂γ are derived from γ by given by removing vertices

in γ and expanding the lattice, respectively. The fact that ×A is a chain map, translated

to the language of paths, results in the identity

Γn∂ = ∂Γn−1.

n n−1 Deﬁnition 4.8.18. Let Pn : CX (X × Y ) → CX (X × Y ) be the map associated to

n+1 the element (−1) Πn ∈ Gn, where

n−1 X i i Πn = (−1) h(πh) Γn−i−1. i=0

Proposition 4.8.19. The maps P∗ provide a chain homotopy between ×A and the identity on CX∗(X × Y ); that is, for all n, ψ ∈ CXn(X × Y ),

∂Pnψ + Pn+1∂ψ = (×A)ψ − ψ.

n Proof. Let In be the path associated to the identity on CX (X × Y ), i.e.,

In = {(x0, y0), (x1, y1),..., (xn, yn)}.

95 It suﬃces to show that

n ∂Πn + Πn+1∂ = Γn + (−1) In.

By deﬁnition,

n−1 !  n  X i 1 X j 1 ∂Πn + Πn+1∂ = ∂ (−1) h(πh) Γn−i−1 +  (−1) h(πh) Γn−j ∂. i=0 j=0

We shall employ the following properties of the path maps π, h and ∂:

Lemma 4.8.20. For any n and Γ ∈ Gn,

1. ∂ (hΓ) = πhΓ − hπΓ + h (∂Γ);

2. (hΓ) ∂ = Γ − h (Γ∂);

3. ∂ (πhΓ) = πhπΓ − πh (∂Γ), so

i−1 X ∂ (πh)iΓ = (πh)i (∂Γ) + (−1)k(πh)kπ(πh)i−kΓ; k=0

4. (πhΓ) ∂ = πΓ − πh (Γ∂), so

i−1 X (πh)iΓ ∂ = (πh)i (Γ∂) + (−1)k(πh)kπ(πh)i−k−1Γ. k=0

The proof of this technical lemma is a combinatorial exercise.

By the properties given above,

n−1 ! X i i ∂Πn = ∂ (−1) h(πh) Γn−i−1 i=0

n−1 n−1 n−1 X i i X i i X 1 i = (−1) πh(πh) Γn−i−1 − (−1) hπ(πh) Γn−i−1 + (−1) h∂ (πh) Γn−i−1 i=0 i=0 i=0

96 n−1 n−1 X i i X i i = (−1) πh(πh) Γn−i−1 − (−1) hπ(πh) Γn−i−1+ i=0 i=0

n−1 " i−1 # X 1 i X k k i−k + (−1) h (πh) (∂Γn−i−1) + (−1) (πh) π(πh) Γn−i−1 i=0 k=0 while  n  X j 1 Πn+1∂ =  (−1) h(πh) Γn−j ∂ j=0

n n X j j X j j = (−1) (πh) Γn−j − (−1) h (πh) Γn−j ∂ j=0 j=0 which, after a change of index, equals

n−1 X i+1 i+1 (−1) (πh) Γn−i−1− i=−1

n " i−1 # X i i X k k i−k−1 (−1) h (πh) (Γn−i∂) + (−1) (πh) π(πh) Γn−i . i=0 k=0

Clearly, adding together the ﬁrst sums from both expressions above gives Γn; after a straightforward calculation, one can show that the majority of the remaining terms

cancel when adding, leaving

n n n n n (−1) h(πh) Γ0∂ = (−1) (hπ) {(x0, y0)} = (−1) In.

Therefore

n ∂Πn + Πn+1∂ = Γn + (−1) In.

Remark 4.8.21. Alternatively, we could deﬁne Πn by using the sum of paths in each Γj

97 which are not the “vertical” path

{(x0, y0), (x0, y1),..., (x0, yn)}

instead of Γj itself. The proof then proceeds almost exactly as before, with the exception of some additional bookkeeping and a slight redeﬁnition of the map h, but allows us to

avoid using the empty path Γ0∂.

All that remains is to deﬁne a chain homotopy Q between A× and the identity on

M p q CX (X) ⊗cc CX (Y ). p+q=n

First, given σ ∈ CXp(X), p > 0, let σ0 ∈ CXp−1(X) be the function

0 p σ (x0, . . . , xp−1) = (−1) σ(x0, . . . , xp−1, xp−1).

p q p−1 q Then we deﬁne Q(p,q) : CX (X) ⊗cc CX (Y ) → CX (X) ⊗cc CX (Y ), p > 0 to act on elementary tensors as follows:

Deﬁnition 4.8.22. Let σ ∈ CXp(X), τ ∈ CXq(Y ); then

0 0 Q(p,q)(σ ⊗ τ) := A(σ × τ) − (σ ⊗ τ).

p We may then extend Q(p,q) linearly to a map (also called Q(p,q)) on all of CX (X)⊗cc CXq(Y ), since if X σi ⊗ τi

restricts to a ﬁnite sum on controlled subsets of Xp+1 × Y q+1, then

X 0 σi ⊗ τi

98 restricts to a ﬁnite sum on controlled subsets of Xp × Y q+1 as well.

The case p = 0 is handled similarly. For τ ∈ CXq(Y ), deﬁne τ ∈ CXq−1(Y ) by

q τ(y0, . . . , yq − 1) = (−1) τ(y1, y1, . . . , yq).

Then, Q(0,q) is deﬁned by

Q(0,q)(σ ⊗ τ) := A(σ × τ) − (σ ⊗ τ).

For a given n, the map

M p q M k l Qn : CX (X) ⊗cc CX (Y ) → CX (X) ⊗cc CX (Y ) p+q=n k+l=n−1

is then the sum of the individual Q(p,q).

Proposition 4.8.23. The Q∗ give the desired homotopy equivalence Q.

Proof. It suﬃces to prove the proposition for an elementary tensor and extend by lin-

earity. Let σ ∈ CXp(X), τ ∈ CXq(Y ) (assume p > 0). First, observe that

p 0 X p+i 0 ∂σ (x0, . . . , xp) = (−1) σ (x0,..., xˆi, . . . , xp) i=0

p−1 X p+i 2p = (−1) σ(x0,..., xˆi, xp, xp) + (−1) σ(x0, . . . , xp−1, xp−1), i=0 while

0 p+1 (∂σ) (x0, . . . , xp) = (−1) ∂σ(x0, . . . , xp, xp)

p−1 X p+i+1 = (−1) σ(x0,..., xˆi, xp, xp). i=0

99 Therefore, if we let σ00 be the element of CXp(X) given by

00 σ (x0, . . . , xp) = σ(x0, . . . , xp−1, xp−1),

it follows that ∂σ0 + (∂σ)0 = σ00.

It follows that

∂Q(σ ⊗ τ) + Q∂(σ ⊗ τ) =

= ∂A(σ0 × τ) + Q(∂σ ⊗ τ) + (−1)pQ(σ ⊗ ∂τ) − ∂(σ0 ⊗ τ)

= A(∂σ0 × τ) + (−1)p−1A(σ0 × ∂τ) + A (∂σ)0 × τ + (−1)pA(σ0 × ∂τ)−

∂σ0 ⊗ τ − (−1)p−1(σ0 ⊗ ∂τ) − (∂σ)0 ⊗ τ − (−1)p(σ0 ⊗ ∂τ)

= A(∂σ0 × τ) + A (∂σ)0 × τ − ∂σ0 ⊗ τ − (∂σ)0 ⊗ τ

= A(σ00 × τ) − (σ00 ⊗ τ).

Now, observe that A(σ00 ×τ) and A(σ ×τ) diﬀer only on the (p, q)-shuﬄe which contains

xp; that is, on

γ = {(x0, y0), (x1, y0),..., (xp, y0), (xp, y1),..., (xp, yq)}.

Under the γ-component of A, σ × τ maps to σ ⊗ τ, while σ00 × τ maps to σ00 ⊗ τ.

Therefore,

A(σ00 × τ) − σ00 × τ = A(σ × τ) − (σ × τ), proving the result. The proof for p = 0 is analogous.

100 Chapter 5

Coarse Universal Coeﬃcient and K¨unnethTheorems

5.1 A Coarse Universal Coeﬃcient Theorem

n For any ring R, there exists a natural pairing h·, ·i between CX (X; R) and CXn(X; R) as follows. An element of ψ ∈ CXn(X; R) is a map ψ : Xn+1 → R with cocontrolled

support, while σ ∈ CXn(X; R) is a formal linear combination

X σ = ri · xi i

n+1 where each ri ∈ R, xi ∈ E for some controlled E ⊂ X , and the sum is ﬁnite on bounded sets. Then, X hψ, σi := ri · ψ(xi). i The goal of this section is to prove the following result for a general PID R and coarse

space X:

101 Theorem 5.1.1 (Coarse Universal Coeﬃcient Theorem). If

n δ : HX (X; R) → Hom(HXn(X; R),R)

∗ is the homomorphism deﬁned by the pairing above, and if DE denotes the submodule of CX∗(X; R) associated to the controlled set E (as deﬁned in Example 4.2.3), the

following is true:

n ∗ 1. If each H (DE) is a projective module, then

ker δ ∼ lim 1Hn−1(D∗ ); = ←− E

n ∗ 2. If each H (DE) is ﬁnitely generated and projective, then δ is surjective.

Remark 5.1.2. To simplify the proof, we shall assume that the coarse structure of X is

countably generated (that is, X is metrizable). This assumption allows us to consider

the inverse systems of modules in the following proof to be inverse sequences. The proof

should be readily modiﬁable to general coarse spaces, but care must be taken to show

that the higher derived functors of the inverse limit vanish; see the discussion in Remark

2.3.3.

Proof. We shall ﬁrst construct δ via a composition of homomorphisms, show that it has

the attributes claimed in the proposition, then show that δ agrees with the obvious map

n from HX (X; R) to Hom(HXn(X),R). Let C, D be the geometric modules associated

∗ to CX∗(X; R), CX (X; R). Recall that for each controlled E ⊂ X × X there exist

associated submodules CE, DE, which are ﬁnitely generated if E is bounded, and such that

C ∼ lim C ,D ∼ lim D , E = −→ B E = ←− B B⊂E B⊂E where the limits are taken over all bounded subsets B of E. Observe also that for

102 n ∼ bounded B, DB = Hom((Cn)B,R), as (Cn)B is ﬁnitely generated as an R-module by (n+1) n the points of B and DB is generated by the characteristic functions of points in (n+1) B . Also, by the fact that for any direct system {Mi} of R-modules,

! Hom lim M ,R ∼ lim Hom(M ,R) −→ i = ←− i i i

n (cf. [21], Proposition 7.96), the modules (Cn)E are dual to DE. Standard results in homological algebra (i.e. [13], Proposition 2.8) give a universal coeﬃcient short exact sequence

n−1 n 0 → Ext(H (DE),R) → Hn(CE) → Hom(H (DE),R) → 0. (5.1)

∗ The fact that the submodules of CX (X),CX∗(X) associated to bounded sets are ﬁnitely generated is important here, as the construction of the above short exact se-

quence involves taking duals.

We therefore have a collection of short exact sequences for each controlled set E.

Additionally,

C ∼ lim C ,D ∼ lim D , = ←− E = ←− E E⊂X×X E⊂X×X with the limits taken over all controlled subsets E of X × X. Since the direct limit

functor is exact, we may take the direct limit over all controlled E ⊂ X ×X of the short

exact sequence (5.1) and obtain

0 → lim Ext(Hn−1(D ),R) → lim H (C ) → lim Hom(Hn(D ),R) → 0. −→ E −→ n E −→ E E∈C(X) E E

Since homology commutes with direct limits, the middle term is isomorphic to HXn(X). Now, given an R-module M, let M ∗ denote the dual module Hom(M,R). Recall that

the functor M 7→ M ∗ is contravariant and left-exact, and that its ﬁrst right derived

103 functor is Ext(M,R). Therefore, dualizing the above short exact sequence, we obtain

(5.2)

!∗ α 0 - lim (Hn(D ))∗ - (HX (X))∗ - −→ E n E !∗ ! - lim Ext(Hn−1(D ),R) - Ext lim (Hn(D ))∗ - ··· . −→ E −→ E E E Using the fact that !∗ lim M ∼ lim M ∗, −→ i = ←− i i i note that the ﬁrst non-zero term in the above long exact sequence,

!∗ lim (Hn(D ))∗ , −→ E E

is isomorphic to lim (Hn(D ))∗∗. Additionally, for each E there exists a canonical map ←− E n βE from each H (DE) to its double dual; applying the inverse limit functor to the

collection {βE} gives a map

β = lim β : lim Hn(D ) → lim (Hn(D ))∗∗ . ←− E ←− E ←− E E∈C(X) E E

Then, composing β with the surjection

γ : HXn(X) → lim Hn(D ) ←− E E

from the Milnor lim-1 sequence

γ 0 - lim 1 Hn−1(D ) - HXn(X) - lim Hn(D ) - 0, ←− E ←− E E

we obtain a homomorphism from HXn(X) to lim (Hn(D ))∗∗. Finally, composing with ←− E

104 the map α in sequence (5.2) gives the desired module homomorphism

n δ = α ◦ β ◦ γ : HX (X) → Hom(HXn(X),R).

n Observe that if each H (DE) is a projective module, then each βE is an injection; since the inverse limit functor is left exact, β is injective as well. In this case, ker δ = ker γ =

lim 1Hn−1(D ) as desired. Also in this case, α is an isomorphism, so coker δ = coker β. ←− E n When each H (DE) is ﬁnitely generated and projective, each βE is an isomorphism, as is α; therefore, δ is surjective.

Finally, we need to show that δ as deﬁned in the proof is the explicit homomorphism

claimed in the hypothesis. Let [ψ] be an element of HXn(X) represented by a cocycle

ψ. The image of ψ in each DE, denoted ψE, is an equivalence class of maps in DE which agree with ψ on E(q+1) (as deﬁned in Example 4.2.3). Then γ([ψ]) maps to the sequence

{[ψ ]} over all controlled E ⊂ X ×X. The collection {[ψ ]} maps to lim (Hn(D ))∗∗ E E −→ E n ∗ as follows: for each E, define a map θE :(H (DE)) → R by letting θE(mE) := mE(ψE). These maps fit together to define a map θ : lim ((Hn(D ))∗ → R. Since α as defined −→ E above is the dual of (the inverse limit of) a classical universal coefficient map, it acts

on θ as follows: For an element [σ] ∈ HXn(X) represented by

X σ = ri · xi ∈ CXn(X),

xi∈F

P P α(θ)[σ] = i ri · ψF (xi) = i ri · ψ(xi). Therefore, δ agrees with the homomorphism induced by the pairing of coarse cochains with coarse chains.

5.2 A Coarse K¨unnethTheorem

In this section, we wish to establish a coarse analogue of the K¨unneththeorem for cohomology.

105 Recall the classical statement of the theorem (cf. [16], Theorem 60.3):

Theorem 5.2.1. Let C, D be cochain complexes of modules over a PID such that

Hp(D) is ﬁnitely generated in all dimensions; then, there is a short exact sequence

M M 0 → Hp(C) ⊗ Hq(D) → Hn(C ⊗ D) → Tor(Hp+1(C),Hq(D)) → 0. p+q=n p+q=n

To develop a coarse version of this theorem, the requirement that one of the cochain

complexes in the tensor product must have ﬁnitely generated cohomology will be re-

placed with the following condition.

Deﬁnition 5.2.2. A coarse space X is said to be tamely coarse if there exists a subcate-

gory C0 of the controlled subsets of X×X which is coﬁnal in the controlled category C(X)

∗ of X such that, if CE is the quotient complex of CX (X) associated to the controlled

0 set E, then for each E,F ∈ C with E ⊂ F , the standard quotient maps pEF : CF → CE induce isomorphisms on coarse cohomology.

n A fundamental example of a tamely coarse space is R .

2 Example 5.2.3. The space X = {n | n ∈ N}, equipped with its metric coarse structure as

a subset of R, is not tamely coarse. In this space, every neighborhood E of the diagonal (1) is the union of the diagonal with ﬁnitely many points x1, . . . , xk. Also, E = X, and a

0 cochain ψ ∈ CE is a cocycle if and only if ψ is constant on {x1, . . . , xk}. Every coﬁnal subcategory of controlled sets as described in the deﬁnition of tame coarseness must

contain neighborhoods E,F of the diagonal such that E is a proper subset of F . In this

0 0 case, H (CF ) is a proper subgroup of H (CE); therefore the induced homomorphism

0 0 from H (CF ) to H (CE) can not be an isomorphism.

Lemma 5.2.4. Uniformly contractible spaces are tamely coarse.

Proof. See Theorem 5.28 in [18], which states that, if X is uniformly contractible and

∗ E is a neighborhood of the diagonal in X × X, the restriction map pE from CX (X)

106 0 to CE induces an isomorphism on cohomology. Letting C be the coﬁnal subgategory

0 consisting of all neighborhoods of the diagonal, we see that for E ⊂ F in C , both pE and pF induce isomorphisms. As pE = pEF ◦ pF and pE, pF and pEF are chain maps, it follows that pEF induces a cohomology isomorphism as well.

The goal of this section is to prove the following:

Theorem 5.2.5. [The Coarse K¨unnethTheorem] Let R be a PID, X a coarse space and Y a tamely coarse space such that the modules HXq(Y ) are ﬁnitely presented. If

∗ ∗ C := CX (X; R), D := CX (Y ; R) and CE, DF denote the subcomplexes of C and D associated to controlled sets E and F , then the cross product map

M p q n ∼ n κ : HX (X) ⊗ HX (Y ) → HX (C ⊗cc D) = HX (X × Y ) p+q=n

has the following properties:

• κ is a surjection if, for all p, q and controlled E ⊂ X × X,

lim Tor(Hp(C ),Hq(D)) = lim 1 Tor(Hp(C ),Hq(D)) = 0; ←− E ←− E E∈C(X)

• κ is an isomorphism if, in addition to the previous condition, lim 1 Hp(C ) = 0 ←− E for all p.

Again, for simplicity’s sake, we shall assume that the coarse structures on X and

Y are countably generated. (Refer to the remarks following the proof of the coarse

Universal Coeﬃcient theorem.)

Proof. We begin the proof by stating some facts about the inverse limit and lim-1

functors.

Lemma 5.2.6. Under the hypotheses of Theorem 5.2.5, the following are true:

107 1. Tor(lim Hp(C ),Hq(D)) ∼ lim Tor(Hp(C ),Hq(D)); ←− E = ←− E

2. (lim 1 Hp(C )) ⊗ Hq(D) ∼ lim 1(Hp(C ) ⊗ Hq(D)). ←− E = ←− E

Proof. 1. See result (1.6) in [24]

2. Since Hq(D) is a ﬁnitely presented module, we know from [5] that the map

! Y p q Y p q α : H (CE) ⊗ H (D) → (H (CE) ⊗ H (D)) E E

P 0 P 0 sending (hE) ⊗ h to ( hE ⊗ h ) is an isomorphism. Let

Y p Y p 0 Y p q Y p q d : H (CE) → H (CE), d : (H (CE)⊗H (D)) → (H (CE)⊗H (D)) E E E E

be the maps whose kernels are lim Hp(C ) and lim(Hp(C ) ⊗ Hq(D)) and whose ←− E ←− E cokernels are lim 1 Hp(C ) and lim 1(Hp(C ) ⊗ Hq(D)) respectively. That is, if ←− E ←− E p p pi : H (CEi ) → H (CEi−1 ) denotes the maps comprising the inverse sequence, d

is deﬁned by d(h0, h1,...) := (h0 − p1(h1), h1 − p2(h2),...).

From the commutative diagram

! ! Y p q d ⊗-1 Y p q H (CE) ⊗ H (D) H (CE) ⊗ H (D) E E α α

? 0 ? Y p q d - Y p q (H (CE) ⊗ H (D)) (H (CE) ⊗ H (D)) , E E

and the fact that α is an isomorphism, observe that the images of d ⊗ 1 and d0 are

isomorphic. Since tensoring with Hq(D) is right exact, taking the tensor product

- - Y p d - - 0 ker d H (CE) im d 0 E

108 with Hq(D) results in

! - q - Y p q d ⊗-1 q - ··· ker d ⊗ H (D) H (CE) ⊗ H (D) im d ⊗ H (D) 0. E

Therefore the cokernel of d ⊗ 1, which is isomorphic to coker d, is isomorphic to

(lim 1 Hp(C )) ⊗ Hq(D). ←− E

Since Y is tamely coarse, there exists a subcategory C0 of controlled subsets of Y ×Y

as speciﬁed in Deﬁnition 5.2.2. For each controlled E ⊂ X × X and F ⊂ F 0 ∈ C0, if we

p p let HE := H (CE), there is a commutative diagram

- M p q - n - M p+1 q - 0 HE ⊗ H (DF 0 ) H (CE ⊗ DF 0 ) Tor(H )E,H (DF 0 )) 0

id ⊗ pFF 0 id ⊗ pFF 0 Tor(id, pFF 0 ) ? ? ? - M p q - n - M p+1 q - 0 HE ⊗ H (DF ) H (CE ⊗ DF ) Tor(HE ,H (DF )) 0,

the direct sums taken over all p + q = n.

Here, the rows are the K¨unneth theorem sequences, and the vertical maps are deﬁned

by the following: if θ is a cochain map, θ denotes the map on cohomology induced by

θ. Since each pFF 0 is an isomorphism, the maps id ⊗ pFF 0 and Tor(id, pFF 0 ) are as well.

By the short ﬁve lemma, id ⊗ pFF 0 is also an isomorphism. It follows that lim 1Hn(C ⊗ D ) = 0 for all n, and by the Milnor lim-1 sequence, ←− E F F

Hn(lim C ⊗ D ) ∼ lim Hn(C ⊗ D ). ←− E F = ←− E F F F

n n 0 0 Furthermore, lim H (C ⊗ D ) is isomorphic to H (C ⊗ D 0 ) for any ﬁxed F ∈ C , ←−F E F E F

109 and a commutative diagram argument analogous to the one above shows that

n n H (C ⊗ D 0 ) ∼ H (C ⊗ lim D ). E F = E ←− F F

Therefore,

Hn(lim C ⊗ D ) ∼ Hn(C ⊗ D) and so Hn(C ⊗ D) ∼ Hn(lim(C ⊗ D)). ←− E F = E cc = ←− E F E

We shall deﬁne the coarse K¨unnethmap κ via a series of intermediate steps. First, for controlled E, let γ : lim Hp(C ) → Hp(C ) be the canonical projection map. E ←−E E E q Clearly, if 1 denotes the identity map on H (D), then γE ⊗ 1 is a homomorphism from lim Hp(C ) ⊗ Hq(D) to Hp(C ) ⊗ Hq(D); also, the γ ⊗ 1 are compatible with the ←−E E E E p q maps in the inverse system {H (CE) ⊗ H (D)}. Therefore, by the universal property of inverse limits, there exists a unique map

! τ : lim Hp(C ) ⊗ Hq(D) → lim (Hp(C ) ⊗ Hq(D)) . p,q ←− E ←− E E E

Taking the direct sum of these τp,q over all p + q = n we obtain a homomorphism

! M M M τ := τ : lim Hp(C ) ⊗ Hq(D) → lim (Hp(C ) ⊗ Hq(D)) . p,q ←− E ←− E p+q=n p+q=n E p+q=n E

Now, take KE to be the classical K¨unneththeorem map

M p q n KE : H (CE) ⊗ H (D) → H (CE ⊗ D), p+q=n

K0 to be the composition of lim 1 K with the isomorphism ←− E

(lim 1Hp(C )) ⊗ Hq(D) ∼ lim 1(Hp(C ) ⊗ Hq(D)) ←− E = ←− E E E

110 as discussed in Part 2 of 5.2.6, and deﬁne K to be the composition limK ◦ τ. Then, if ←− E E p p we let HE := H (CE), we have a diagram (direct sums taken over all p + q = n):

M α0⊗1 M α⊗1 M ··· - (lim 1Hp−1) ⊗ Hq(D) - Hp(C) ⊗ Hq(D) - (lim Hp ) ⊗ Hq(D) - 0 ←− E ←− E E K0 κ K ? β0 ? β ? 0 - lim 1Hn−1(C ⊗ D) - Hn(C ⊗ D) - lim Hn(C ⊗ D) - 0. ←− E cc ←− E E

(5.3)

p Here, the top row comes from tensoring the Milnor lim-1 sequence of {H (CE)} with Hq(D) and taking the direct sum over all p + q = n, and the bottom row is the lim-1

n sequence of {H (CE ⊗ D)}.

p Proposition 5.2.7. If h ∈ H (C) is represented by a sequence {cE} ∈ C, where cE ∈

0 p 0 CE for each E, and h ∈ H (D) is represented by c ∈ D, then the cross product κ as deﬁned on the elementary tensor h ⊗ h0 by

0 0 κ(h ⊗ h ) := [{cE ⊗ c }]

makes the above diagram commute.

Proof. We shall ﬁrst show commutativity for the right side of the diagram. Let (α⊗1) :

L Hp(C) ⊗ Hq(D) → L(lim Hp(C )) ⊗ Hq(D), β : Hn(C ⊗ D) → lim Hn(C ⊗ D) ←− E cc ←− E be the maps in the diagram. It suﬃces to show commutativity for an elementary tensor

0 p q e = [{cE}] ⊗ [c ] ∈ H (C) ⊗ H (D). First, by the deﬁnition above,

0 κ(e) = [{cE ⊗ c }],

111 and

(β ◦ κ)(e) = {[c ⊗ c0]} ∈ lim Hn(C ⊗ D). E ←− E

0 Similarly, (α ⊗ 1)(e) = {[cE]} ⊗ [c ], and

0 0 (K ◦ (α ⊗ 1))(e) = {[cE] × [c ]} = {[cE ⊗ c ]}.

Therefore, β ◦ κ = K ◦ (α ⊗ 1).

Demonstrating commutativity of the left side of the diagram is somewhat more technical. First, let

ψ = [h] ⊗ h0 ∈ (lim 1Hp−1(C )) ⊗ Hq(D) ←− E E

be represented by ! 0 0 Y p−1 q ψ = h ⊗ h ∈ H (CE) ⊗ H (D). E

By Milnor’s proof of the existence of the lim 1 sequence ([15]), the image of [h] in Hp(C), ←− α0([h]), is equal to the image of h under a Mayer-Vietoris coboundary map as follows:

p−1 let h = {[cE]}, where each [cE] ∈ H (CE) is represented by the cocycle cE. Choosing a basis [ Ei; Ei ⊂ Ej for all i < j i∈N

for the controlled subsets of X × X, pull {cE} back to

Y Y φ = ({c | i odd }, {−c | j even }) ∈ Cp−1 ⊕ Cp−1. Ei Ej Ei Ej i odd j even

Then, there exists {φ0 } ∈ Cp such that the image of {φ0 } in Q Cp ⊕Q Cp (i and j as E E Ei Ej 0 0 0 before) is equal to the coboundary of φ, ∂φ. Then, α ([h]) = [{φE}], and so (α ⊗1)(ψ) = 0 0 0 0 0 0 0 [{φE}] ⊗ h . If h is represented by the cocycle c , then κ ◦ (α ⊗ 1)(ψ) = [{φE ⊗ c }]. On

112 the other hand, K0(ψ) is represented by

0 Y n−1 {[cE ⊗ c ]} ∈ H (CE ⊗ D), E which pulls back to

!   0 0 Y Y ω = ({cEi ⊗ 2c | i odd }, {−cEj ⊗ c | j even }) ∈ CEi ⊗ D ⊕  CEj ⊗ D . i odd j even

Recall the deﬁnition of the coboundary map on a tensor cochain complex C∗ := C∗ ⊗D∗; if c ∈ Cp and c0 ∈ Dq are cochains, then

0 0 p 0 ∂C(c ⊗ c ) := ∂C (c) ⊗ c + (−1) c ⊗ ∂D(c ).

However, since c0 is a cocycle, the summands of ∂ω containing (−1)p−1∂2c0 and (−1)p−1∂c0 vanish; i.e. only the ∂{cEi } and ∂{cEj } terms survive. As a result (by an argument 0 0 0 analogous to the preceding paragraph’s), it follows that (β ⊗ 1) ◦ K (φ) = [{φE ⊗ c }] as well.

We shall need the following fact when computing the kernel and cokernel of κ:

Lemma 5.2.8. Under the hypotheses of Theorem 5.2.5, the map τ is surjective.

Proof. Deﬁne d and d0 as in the proof of Part 2 of Lemma 5.2.6; i.e. the homomorphisms

Y p Y p 0 Y p q Y p q d : H (CE) → H (CE), d : (H (CE) ⊗ H (D)) → (H (CE) ⊗ H (D)) E E E E

whose kernels are lim Hp(C ) and lim(Hp(C )⊗Hq(D)) and cokernels are lim 1 Hp(C ) ←− E ←− E ←− E

113 and lim 1(Hp(C ) ⊗ Hq(D)) respectively. There exists a commutative diagram ←− E

! Y d ⊗ 1 - (lim Hp(C )) ⊗ Hq(D) - Hp(C ) ⊗ Hq(D) - im d ⊗ Hq(D) - 0 ←− E E E 0 00 τp,q τ τ ? ? ? Y d0 0 - lim (Hp(C ) ⊗ Hq(D)) - (Hp(C ) ⊗ Hq(D)) - im d0 - 0. ←− E E E

0 00 The maps τ and τ are isomorphisms; therefore τp,q is surjective by the Snake Lemma. L The result follows from the fact that τ = τp,q.

Applying the Snake Lemma to Diagram (5.3), there exists an exact sequence

ker K0 → ker κ → ker K → coker K0 → coker κ → coker K → 0. (5.4)

We claim that:

Lemma 5.2.9. Under the hypotheses of Theorem 5.2.5:

1. ker K =∼ ker τ;

2. coker K ∼ coker lim K ; = ←− E

3. coker K0 ∼ lim 1 coker K . = ←− E

Proof. Note that applying the inverse limit functor to the classical K¨unnethshort exact

sequence

- M p q K-E n T-E M p+1 q - 0 H (CE)) ⊗ H (D) H (CE ⊗ D) Tor(H (CE), ⊗H (D)) 0 p+q=n p+q=n

results in a long exact sequence

(5.5)

114 M lim KE lim TE 0 - lim(Hp(C ) ⊗ Hq(D)) ←− - lim Hn(C ⊗ D) ←− - ←− E ←− E p+q=n 1 M δ M lim KE lim Tor(Hp+1(C ),Hq(D)) - lim 1(Hp(C ) ⊗ Hq(D)) ←− - ←− E ←− E p+q=n p+q=n 1 lim TE M lim 1 Hn(C ⊗ D) ←− - lim 1 Tor(Hp+1(C ),Hq(D)) - 0. ←− E ←− E p+q=n 1. By an earlier discussion, K is the composition lim K ◦τ, and lim K is an injection ←− E ←− E since eack K is injective and lim is left exact. E ←−

2. Since K = lim K ◦ τ, and τ is surjective, their cokernels are isomorphic. ←− E

3. K0 is the composition of lim 1 K with an isomorphism; therefore their cokernels ←− E are isomorphic.

Remark 5.2.10. The previous lemma and the following remarks shall be used to prove that certain modules in the exact sequences we have constructed vanish. Because of this, it is suﬃcient to just show the existence of isomorphisms; constructing the module isomorphisms explicitly is not necessary.

Additionally, by breaking the long exact sequence (5.5) into short exact sequences, we conclude that

M • coker lim K is isomorphic to im lim T , a submodule of lim Tor(Hp+1(C ),Hq(D)); ←− E ←− E ←− E p+q=n M • coker lim 1 K ∼ lim 1 Tor(Hp+1(C ),Hq(D)); ←− E = ←− E p+q=n

• ker lim 1 K ∼ L lim Tor(Hp(C ),Hq(D)) / ker δ, and ker δ ∼ lim Hn−1(C ⊗ ←− E = ←− E = ←− E D)/ L lim Hp−1(C ) ⊗ Hq(D); ←− E

• ker lim 1 K = 0 when all lim 1 Hp(C ) = 0. ←− E ←− E

115 To calculate ker K =∼ ker τ, consider ﬁrst the short exact sequence

i Y d 0 - lim Hp(C ) - Hp(C ) - im d - 0, ←− E E E E

where d is again deﬁned as in the proof of Part 2 of 5.2.6.

Tensoring with Hq(D) results in a long exact sequence

(5.6)

Tor(i, 1) Y Tor(d, 1) 0 - Tor(lim Hp(C ),Hq(D)) - Tor( Hp(C ),Hq(D)) - Tor(im d, Hq(D)) ←− E E

i ⊗ 1 Y d ⊗ 1 - (lim Hp(C )) ⊗ Hq(D) - ( Hp(C )) ⊗ Hq(D) - im d ⊗ Hq(D) - 0. ←− E E

Applying the Snake Lemma to the commutative diagram

i ⊗ 1 Y d ⊗ 1 ··· - (lim Hp(C )) ⊗ Hq(D) - ( Hp(C )) ⊗ Hq(D) - im d ⊗ Hq(D) - 0 ←− E E

τ =∼ ? ? ? Y 0 - lim(Hp(C ) ⊗ Hq(D)) - (Hp(C ) ⊗ Hq(D)) - im d0 - 0 ←− E E

we see that the kernel of τ : (lim Hp(C )) ⊗ Hq(D) → lim(Hp(C ) ⊗ Hq(D)) is isomor- ←− E ←− E phic to ker(i ⊗ 1). By breaking up the top row, which is the sequence (5.6), into short

exact sequences, we see that ker(i ⊗ 1) is in turn isomorphic to

Y p q ∼ Tor( H (CE),H (D)) / im Tor(d, 1) = coker Tor(d, 1).

On the right-hand side of the isomorphism, Tor(d, 1) is treated as a map from

Y p q Tor H (CE),H (D)

116 to itself, and so

coker Tor(d, 1) ∼ lim 1 Tor(Hp(C ),Hq(D)) = ←− E

The module ker(i ⊗ 1) is thus isomorphic to lim 1 Tor(Hp(C ),Hq(D)). Therefore, ←− E

M ker K ∼ ker τ ∼ lim 1 Tor(Hp(C ),Hq(D)). = = ←− E p+q=n

In conclusion, recalling the snake lemma sequence (5.4):

p q • Suppose each Tor(H (CE),H (D)) = 0. Then coker K, which is isomorphic to a M submodule of lim Tor(Hp+1(C ),Hq(D)), vanishes. Also, coker K0, which ←− E p+q=n M is isomorphic to lim 1 Tor(Hp(C ),Hq(D)) vanishes as well. Therefore κ is ←− E p+q=n surjective.

• If the previous condition holds, and if each lim 1 Hp(C ) = 0, then ker K0 ∼ ←− E = ker lim 1 K ∼ 0, and lim Hp(C ) ∼ Hp(C). Thus ←− E = ←− E =

M M M lim(Hp(C )) ⊗ Hq(D) ∼ Hp(C) ⊗ Hq(D) ∼ lim(Hp(C ) ⊗ Hq(D)), ←− E = = ←− E

so ker τ = 0. It follows that κ is injective as well.

5.3 Examples

m q ∼ 1. Let Y = R . Then Y is tamely coarse and HX (Y ) = Z for q = m and 0 otherwise. Since HXq(Y ) is torsion free, all Tor terms vanish, and the K¨unneth

formula gives an isomorphism

p ∼ p+m m HX (X) = HX (X × R ).

2. For a positive integer s, deﬁne Xs to be the open cone of the s-fold dunce cap

117 (recall the deﬁnition of this object from [16], page 41, for instance). Then Xs, being the open cone of a compact, locally contracible metrizable space, is uniformly contractible (see Example 5.27 in [18]) and therefore tamely coarse and

3 ∼ p HX (Xs) = Z/sZ,HX (Xs) = 0 otherwise

(since the ordinary reduced cohomology of Xs is Z/sZ in dimension 2 and zero otherwise). Let X = Xs, Y = Xt for positive integers s, t; then the only nonzero K¨unnethhomomorphism maps HX3(X) ⊗ HX3(Y ) to HX6(X × Y ). If d = 3 3 ∼ gcd(s, t), then HX (X) ⊗ HX (Y ) = Z/dZ.

In this case, the lim-1 terms vanish, so ker κ =∼ ker K and coker κ =∼ coker K. Addi- tionally, Tor(H4(X),H3(Y )) = 0, so coker K = 0, and lim 1 Tor(H3(X),H3(Y )) = ←− 0, so ker K = 0. It follows that

6 ∼ HX (Xs × Xt) = Z/dZ

M Also, the K¨unneth map from 0 =∼ HXp(X) ⊗ HXq(Y ) to HXq(X × Y ) has p+q=5 cokernel equal to

3 3 ∼ ∼ Tor(HX (X),HX (Y )) = Tor(Z/sZ, Z/tZ) = Z/dZ.

5 ∼ Thus HX (Xs × Xt) = Z/dZ as well.

118 Chapter 6

Coarse CW-Complexes and Fibrations

6.1 Coarse CW-Complexes

We begin this chapter by introducing a type of coarse space analogous to CW-complexes in topology.

Deﬁnition 6.1.1. Let R+ denote the ray [0, ∞). For n ≥ 1, a coarse n-cell (or coarse n n−1 n-disk), denoted D , is the space R ×R+ equipped with the Euclidean metric coarse structure. A coarse 0-cell, D0, is a point.

n n A coarse n-sphere, denoted S , is the space R .

The basics of the above ideas were ﬁrst addressed in this dissertation at the beginning

of Chapter 3.

n Each coarse n-disk D = {(x1, ··· , xn−1, t) | t ≥ 0}, contains a coarse (n−1)-sphere

as the subspace {(x1, ··· , xn−1, 0)}. We shall refer to this (n−1)-sphere as the boundary of Dn, denoted ∂Dn.

Deﬁnition 6.1.2. A coarsely connected ﬁnite coarse CW-complex is a coarse space X

constructed by the following procedure:

119 1. The 0-skeleton X0 of X is a single point.

2. Inductively, form the p-skeleton Xp from Xp−1 by attaching ﬁnitely many coarse

p p−1 p p-cells Dα via coarse maps fα : S = ∂Dα → Xp−1. In other words, Xp is p the quotient space of the coarsely disjoint union Xp−1 tα Dα with respect to the p identiﬁcations x ∼ f(x) for x ∈ ∂Dα.

3. We shall assume that all CW-complexes in this chapter are ﬁnite-dimensional;

that is to say, there exists some ﬁnite n for which X = Xn (we call this n the dimension of X).

Remark 6.1.3. From the construction of coarse CW-complexes given above, and the discussion of gluing coarse spaces in Section 1.4.1, observe that the image of a coarse

1-cell R+ in the 1-skeleton of a CW-complex is coarsely equivalent to R+ itself. In other words, the image of R+ in the 1-skeleton is flasque. In higher dimensions, the image of a coarse n-cell in the n-skeleton is not in general flasque. For example, consider a coarse CW-complex X consisting of one 0-cell, one 1-cell, and one 2-cell D glued with respect to f : ∂D = R → R+ defined by f(x) = |x|. It can be shown that X is coarsely 2 equivalent to R ; this construction is analogous to forming a classical 2-sphere from a 2-disk by identifying all points along the boundary of the disk.

Remark 6.1.4. As remarked prior to Definition 1.2.1, we shall work only in the context of coarsely connected spaces for simplicity’s sake. If we wish to relax this restriction, we may generalize the above definition so that the 0-skeleton is a coarsely disjoint union of points (cf. Definition 1.2.2).

As is the case for classical CW-complexes, we can compute the coarse cohomology of a coarse CW-complex, in part, by examining the coarse cohomology of its p-skeleta relative to its (p − 1)-skeleta. (This fact will be further examined in the following section.) In particular:

120 p Lemma 6.1.5. If X is a coarse CW-complex, then HX (Xn,Xn−1) is free abelian on the p-cells of X when n = p and is zero otherwise.

We prove this by ﬁrst by proving a similar result:

Lemma 6.1.6. Let D be a coarse n-cell, ∂D its boundary, and Y be a path metric

space. If f : ∂D → Y is a coarse map and Y ∪f D denotes the quotient space of Y with D attached, then   Z if p = n, p ∼ HX (Y ∪f D,Y ) =  0 otherwise.

Proof. Observe via a long exact sequence calculation that

  Z if p = n, HXp(Dn,Sn−1) =∼  0 otherwise.

n n Let D = R × R+ denote a coarse n-cell and S = R = ∂D denote the boundary of D.

Also, let y0 := f(0). Gluing D to Y via f results in the quotient space Y ∪f D; denote p ∼ by D and S the images of D and S in Y ∪f D. We wish to prove that HX (D,S) =

p 0 HX (D, S) for all p; then, by excising Y , the complement of D, from Y ∪f D, we shall conclude that

p ∼ p 0 0 ∼ p ∼ p HX (Y ∪f D,Y ) = HX (Y ∪f D) \ Y ,Y \ Y ) = HX (D, S) = HX (D,S).

Let dY be the metric on Y and consider the following subspaces of D:

n n B := {(x, t) ∈ R × R+|t ≤ dY (f(x), y0)},C := {(x, t) ∈ R × R+|t ≥ dY (f(x), y0)}.

We wish to show the following:

Lemma 6.1.7. 1. The space C is coarsely equivalent to C, the image of C in Y ∪f D;

121 2. The spaces B and S are coarse homotopy equivalent.

Proof. Before proving these results, we shall establish a metric on D which induces its

0 0 coarse structure (cf. Remark 1.4.6). For x, x ∈ D, deﬁne dD(x, x ) to be the inﬁmum of the lengths of all paths from x to x0, that is to say,

0 0 dD(x, x ) = inf {d(x, x1) + d(x1, x2) + ··· + d(xm−1, xm) + d(xm, x )} x1,...,xm∈D

with  d (s, t) if s, t ∈ D \ S,  D  ∼  d(s, t) = dY (s, t) if s, t ∈ S,    inf {dD(s, y)} if s ∈ D \ S, t ∈ S. y∈f −1({t})

−1 1. First, by removing the bounded set f ({y0}), we may assume that C and S are disjoint. Then, the quotient map q : C → C is a bijection; in fact, q(c) = c for all

c ∈ C. It therefore suﬃces to prove that q and q−1 are coarse maps.

Since the quotient map from D to D is distance-decreasing and proper on D \ S,

it follows that q is a coarse map.

−1 0 0 0 0 To show coarseness of q , let (x, t), (x , t ) ∈ C with dD((x, t), (x , t )) ≤ R. By the way C has been deﬁned, any path from (x, t) to (x0, t0) which passes through

S must have length greater than or equal to t + t0, so

0 0 0 2R ≥ 2(t + t ) ≥ (t + t ) + dY (f(x, 0), y0) + dY (f(x , 0), y0).

0 Since dY (f(y0), f(x, 0)) ≤ 2R (similarly for (x , 0)), and f is proper, there exists

S such that dD((x, 0), (0, 0)) ≤ S. Therefore,

−1 −1 0 0 0 0 0 0 dD(q ((x, t)), q ((x , t ))) = dD((x, t), (x , t )) ≤ |t − t | + dD((x, 0), (x , 0))

122 0 0 ≤ t + t + dD((x, 0), (0, 0)) + dD((x , 0), (0, 0)) ≤ 2R + 2S.

Additionally, the map q−1 is proper, as a set

{(x1, . . . , xn, t)} ⊂ C

is bounded if and only if the collection {xi, t} is bounded in R.

2. We begin by showing that:

Lemma 6.1.8. The spaces B and S are coarse homotopy equivalent.

Proof. Deﬁne the maps α : B → S, i : S → B by α((x, t)) = (x, 0) and

i(x, 0) = (x, 0). The map i is obviously coarse, and α is proper since f is proper.

0 0 0 Additionally, α is bornologous, as dD((x, 0), (x , 0)) ≤ dD((x, t), (x , t )) for any (x, t), (x0, t0). The composition α ◦ i is the identity map on S, and there exists a

coarse homotopy hs from i ◦ α to the identity on B via

hs((x, t)) = (x, s · t).

Composing the coarse homotopy hs with the quotient map gives a coarse homotopy between α ◦ i, the image of α ◦ i under the quotient map, and the identity on B.

Since i ◦ α is the identity on S, it follows that B and S are coarse homotopy

equivalent.

Deﬁnition 6.1.9. Let X, Y be metric spaces, A ⊂ X and f : A → Y be a coarse map.

For a ﬁxed point y ∈ f(A) and a ∈ A, let Ba ⊂ X be the set Ba := {x ∈ X | dX (a, x) ≤

123 dY (f(a), y)}. Then, the image of

[ B := Ba a∈A in X ∪f Y is called a neighborhood of the image of A in X ∪f Y under f.

For example, the set B, with B as described in the proof of Lemma 6.1.6 is a neighborhood of S under f.

We now return to the proof of Lemma 6.1.6. Recall that we deﬁned subsets B, C of the coarse (n + 1)-disk D by

n n B := {(x, t) ∈ R × R+ | t ≤ dY (f(x), y0)},C := {(x, t) ∈ R × R+ | t ≥ dY (f(x), y0)}.

A straightforward calculation shows that B and C form a coarsely excisive pair; note

that a radius R neighborhood of B is of the form {(x, t) ∈ D | t ≤ dY (f(x), y0) + R}

and a radius R neighborhood of C is of the form {(x, t) ∈ D | t ≥ dY (f(x), y0) − R}. Therefore the intersection of these two neighborhoods is a radius R neighborhood of p ∼ B ∩C = {(x, t) ∈ D | t = dY (f(x), y0)}. From Lemma 6.1.8, we have that HX (D,S) = HXp(D,B); additionally, since B and C are coarsely excisive, letting F = B ∩ C, we

have, as a consequence of Proposition 2.3.1,

HXp(D,B) = HXp(B ∪ C,B) =∼ HXp(C,F ).

Similarly, from result (2) of Lemma 6.1, HXp(D, S) =∼ HXp(D, B), and B and C are coarsely excisive. Since B ∩ C = F , it follows from Proposition 2.3.1 that

Xp(D, B) =∼ HXp(C, F ),

124 Also, by result (1) of Lemma ,

HXp(C,F ) =∼ HXp(C, F ).

Therefore, HXp(D, S) =∼ HXp(D,S). Again, by Prop. 2.3.1 for the coarsely excisive

pair (D,Y ) in Y ∪f D,

p ∼ p p HX (Y ∪f D,Y ) = HX (D,Y ∩ D) = HX (D, S),

p ∼ p so we have that HX (Y ∪f D,Y ) = HX (D,S), proving Lemma 6.1.6.

The proof of Lemma 6.1.5 now follows by induction.

6.2 Coarse Cellular Cochain Complexes

Deﬁnition 6.2.1. Let X be a coarse CW-complex. Then, the nth cellular cochain group

n n of X, denoted C (X), is the relative coarse cohomology group HX (Xn,Xn−1).

n These cochain groups form a cochain complex with coboundary map ∂ : HX (Xn,Xn−1) →

n+1 HX (Xn+1,Xn) coming from the long exact sequence of the triple (Xn+1,Xn,Xn−1) (as in Remark 2.2.2). We claim that, for any coarse CW-complex X,

Proposition 6.2.2. The cohomology groups Hn(C∗(X)) and HXn(X) are isomorphic.

0 ∗ ∼ Proof. Clearly when dim X = 0, i.e. when X is a single point, H (C (X)) = Z and Hn(C∗(X)) = 0 otherwise. Suppose now that X has dimension greater than zero. We

have already shown that Cn(X) is free abelian, generated by the n-cells of X; also, since n n ∼ n HX (Xp,Xp−1) = 0 for p 6= n, it follows that HX (Xn−1,Xn−2) = HX (Xn+1,Xn) =

125 0. Therefore there is a diagram with an exact row and an exact column

? n−1 αn−-1 n - n - HX (Xn−1,Xn−2) HX (Xn+1,Xn−1) HX (Xn+1,Xn−2) 0

βn ? n HX (Xn,Xn−1)

∂n ? n+1 HX (Xn+1,Xn) where the horizontal row comes from the long exact sequence of the triple (Xn+1,Xn,Xn−1)

and the vertical column comes from the long exact sequence of the triple (Xn+1,Xn−1,Xn−2). Results from classical algebraic topology (for example, the remarks preceding [8],

Theorem 3.5) give us that βn ◦ αn−1 = ∂n−1. Therefore,

n n ∗ ker ∂n ∼ HX (Xn+1,Xn−1) ∼ n H (C (X)) = = = HX (Xn+1,Xn−2). im ∂n−1 im αn−1

We now wish to prove that

Lemma 6.2.3.   p HX (X) if p < n, p ∼ HX (Xn) =  0 if p > n.

Proof of Lemma. The proof shall be analogous to the proof of [8], Lemma 2.34. Suppose

ﬁrst that p < n. From the long exact relative cohomology sequence of (Xn+1,Xn), we

126 obtain

- p - p - p - p+1 - HX (Xn+1,Xn) HX (Xn+1) HX (Xn) HX (Xn+1,Xn) ;

p p+1 p ∼ since p < n, HX (Xn+1,Xn) and HX (Xn+1,Xn) are 0. Therefore, HX (Xn+1) =

p HX (Xn). Similarly, from the the long exact relative sequence of (Xn+2,Xn+1), we obtain

- p - p - p - p+1 - HX (Xn+2,Xn+1) HX (Xn+2) HX (Xn+1) HX (Xn+2,Xn+1) ,

p and, as was the case with the previous sequence, the middle terms HX (Xn+2) and p p ∼ HX (Xn+1) are isomorphic. Proceeding by induction, we conclude that HX (Xn) = p ∼ p ∼ HX (Xn+1) = HX (Xn+2) = ··· . Since X is ﬁnite-dimensional, it follows that p ∼ p HX (Xn) = HX (X).

Now suppose that p > n. Repeating the previous argument for the long exact

p p p sequences of HX (Xn,Xn−1),HX (Xn−1,Xn−2),...,HX (X1,X0) gives us that

p ∼ p ∼ ∼ p ∼ HX (Xn) = HX (Xn−1) = ··· = HX (X0) = 0,

since p > n implies that p > 0 and X0 is a bounded coarse space.

∗ As a consequence of Lemma 6.2.3 and the long exact sequence of HX (Xn+1,Xn−2), n ∼ n n it follows that HX (Xn+1,Xn−2) = HX (Xn+1), which is in turn isomorphic to HX (X).

n ∗ n Therefore, H (C (X)), which is isomorphic to HX (Xn+1,Xn−2), is isomorphic to HXn(X).

127 6.3 Coarse Fibrations

We shall now introduce the notion of a “coarse ﬁbration” on coarse CW-complexes.

First, if E, B, and X are general spaces, and π : E → B and f : X → B are maps, recall (from [8], page 406, for example) that the pullback f ∗(E) is deﬁned to be the set

f ∗(E) := {(x, e) ∈ X × E | f(x) = π(e)}.

In the case that X and E are coarse spaces, we may make f ∗(E) a coarse space by giving it the structure of a subspace of X × E.

Definition 6.3.1. Let E, B, and X be coarse spaces and π : E → B be a bornologous (but not necessarily coarse) map. If ht : X × [0, 1] → B is a coarse homotopy (as defined in ˜ ∗ Definition 1.4.2), a coarse lifting of ht via π is a coarse homotopy ht : h0(E)×[0, 1] → E ∗ such that, if pX : h0(E) × [0, 1] → X × [0, 1] is the canonical projection, the diagram

˜ ∗ ht - h0(E) × [0, 1] E

pX π

? h ? X × [0, 1] t - B commutes. If every coarse homotopy ht : X × [0, 1] → B has a coarse lifting via π, we say that π has the coarse homotopy lifting property (CHLP) with respect to X.

In order to establish the notion of coarse fibrations, we must first recall the definition

of a characteristic map for n-cells in a CW-complex (cf. [8], page 519):

Deﬁnition 6.3.2. Let X be a (coarse) CW-complex, D a (coarse) n-cell, and f : D → Xn be the (coarse) map which glues D to the (n − 1)-skeleton of X. The characteristic map

Φ for D is the composition f- D Xn ,→ X.

128 If X is a coarse CW-complex, Φ is obviously a coarse map by construction.

Deﬁnition 6.3.3. Let E be a coarse space and B an n-dimensional coarse CW-complex.

A (strong) coarse ﬁbration from E to B is an essentially surjective (as in Prop. 3.1.1), bornologous map π : E → B such that:

• If dim B ≥ 2, π has the coarse homotopy lifting property with respect to all coarse

spaces X, and;

• If Φα : R+ → B is the characteristic map for a coarse 1-cell of B, then the pullback ˜ ∗ space R := Φα(E) is flasque (and hence its coarse cohomology groups are all zero), with a “flasqueness map” f (as defined by Defn. 1.5.4) on R˜ given by lifting a

ﬂasqueness map on Φα(R+).

The second condition is necessary because flasqueness as defined in this dissertation is not a coarse homotopy property. It is, however, possible to define flasque spaces as having a type of “coarse null-homotopy” property (cf. [14]), in which case, the second criterion for coarse fibrations can be incorporated into the first.

Example 6.3.4. The most basic examples of coarse ﬁbrations are when E is the product space F ×B and π : F ×B → B is the projection map. In this case, any coarse homotopy ˜ ∗ ht : X × [0, 1] → B can be pulled back to a homotopy ht : h0(E) × [0, 1] → E by letting h˜t(x, (f, b)) = (f, ht(x)).

The motivating examples in this section will be semidirect products of groups:

Deﬁnition 6.3.5. Let N and H be groups and let φ : H → Aut(N) be a group homomorphism. The semidirect product of N by H with respect to φ, denoted G = N oφ H, is the set G = N × H imbued with the group operation

(n, h)(n0, h0) = (n · φ(h)n0, h · h0)

Equivalently, semidirect products may also be deﬁned in terms of group extensions:

129 Deﬁnition 6.3.6. If β α 0 - N - G - H - 0

is a split exact sequence of groups, then G is said to be a semidirect product of N by

The two deﬁnitions of semidirect product are related in the following way: Given a

short exact sequence as in the second deﬁnition, and denoting by γ : H → G the splitting

homomorphism (i.e. α◦γ = idH ), a map φ as in the ﬁrst deﬁnition of semidirect product can be constructed by

φ(h)n := β−1 γ(h)β(n)γ(h−1) .

We shall show that the coarse cohomology of certain semidirect products of groups

(viewed as coarse spaces) is isomorphic to the coarse cohomology of the corresponding

product spaces.

Remark 6.3.7. Before proceeding, we should note that not every semidirect product of

groups has coarse cohomology isomorphic to the direct product. For example, let F be

the free group on 2 generators and Z ,→ F be an injection which maps the generator of

Z to one of the generators of F . Then, we have a group extension

0 - Z - F - F/Z - 0.

It is known ([19], Proposition 3.45) that HX1(F ) is inﬁnite-dimensional, and HXp(F )

is zero otherwise. However, F/Z is an unbounded coarse space, so it follows from the 1 coarse K¨unneththeorem that HX (Z × F/Z) = 0.

Example 6.3.8. Let G be the fundamental group of the Klein bottle, G = ha, b | ab =

b−1ai with the coarse structure induced by the word metric d(g, h) = |g−1h|. The group

G is a semidirect product of Z with Z. Give B = R the obvious coarse CW-complex

structure, with a 0-cell at the origin and 1-cells for the rays R≤0 and R≥0. Deﬁne

130 α β α β π : G → R as follows: if g ∈ G has the reduced word representation a 1 b 1 ··· a n b n , then π(g) = α1 + ··· + αn. As π maps onto Z, it is essentially surjective to R. Also, the ﬁber F is the subgroup generated by b with the subspace coarse structure inherited

from G; F is coarsely equivalent to Z. Since the base space R has dimension 1, we only need to check the second property of Deﬁnition 6.3.3 to see that π is a coarse ﬁbration.

n The space A = im(π) is the union of two ﬂasque spaces A− := {a | n ≤ 0} and

n m n A+ := {a | n ≥ 0} with preimages G+ and G−. Here, G+ = {b a | n ≥ 0} (similarly

m n m n+1 for G−). Then G+ (similarly G−) is ﬂasque with respect to the map f(b a ) = b a . Since d(g, g · a) = 1, f is close to the identity.

(−1)k, and

n Finally, f will clearly move G+ outside any bounded set B for sufficiently large n, since f increases word length for members of G+. Therefore G+ (similarly, G−) is flasque and so π is a coarse fibration.

There also exists a weaker notion of coarse ﬁbration, which is suﬃcient for our purposes:

Deﬁnition 6.3.9. Let E be a coarse space and B an n-dimensional coarse CW-complex.

A weak coarse ﬁbration from E to B is a bornologous, essentially surjective map π :

E → B such that, for n ≥ 2:

• If dim B ≥ 2 and if Dn is a coarse n-cell in B with boundary Sn−1 a union of of

n−1 n−1 (n − 1)-cells D− ∪ D+ , there exist coarse deformation retractions (as deﬁned n n−1 n−1 in Defn. 1.4.4) gt, ht of D onto D− and D+ respectively such that, if Φ is

131 the characteristic map for Dn,

n n Φ ◦ gt : D × [0, 1] → B, Φ ◦ ht : D × [0, 1] → B

lift via π;

• If dim B ≥ 2, Dn and Sn−1 are as in the previous part, and f is the gluing map

n for D , then for every coarse deformation retraction ht : N × [0, 1] → N of a neighborhood N ⊂ B of Sn−1 under f (as deﬁned in Lemma 6.1) onto Sn−1,

Φ ◦ ht : N × [0, 1] → B

has a coarse lifting via π;

∗ • If Φα : R+ → B1 is the characteristic map for a coarse 1-cell of B, then Φα(E) is ∗ flasque, a flasqueness map on Φα(E) given by lifting a flasqueness map on Φα(R+).

Example 6.3.10. For a basic example of the ﬁrst part of the above deﬁnition, give

2 1 1 R a coarse CW-structure consisting of a 0-cell at the origin, 1-cells D+ and D− at the 2 2 positive and negative x-axes, and 2-cells D+ and D− at the upper and lower half-planes. 3 2 Consider the map π : R → R given by π(x, y, z) = π(x, y). An example of a “liftable” 2 1 2 coarse deformation retraction ht from D+ to D+ is given by expressing (x, y) ∈ D+ in polar coordinates (r, θ) and letting

ht(r, θ) = (r, max{θ − πt, 0});

in other words, ht moves each point at a constant radial speed clockwise along a semi- ˜ ∗ 3 3 circle in the upper half-plane. Then ht has a coarse lifting ht : h0(R ) × [0, 1] → R given by

h˜t((r, θ), (r, θ, z)) = (r, max{θ − πt, 0}, z).

132 Remark 6.3.11. The distinction between (strong) coarse fibrations and weak coarse fibrations is analogous to the distinction between Hurewicz and Serre fibrations in classical topology. Recall that a Hurewicz fibration is a surjective map which has the homotopy lifting property with respect to all spaces, whereas a Serre fibration has the homotopy lifting property with respect to disks.

Example 6.3.12. Let HR denote the real Heisenberg group

n 1 x z o H = 0 1 y | x, y, z ∈ R R 0 0 1

−1 2 with coarse structure from the metric d(M,N) = |MN |. Then the map π : HR → R 1 x z which sends 0 1 y to (x, y) is a weak coarse ﬁbration. 0 0 1 2 The coarse space R has a coarse CW-complex structure consisting of a 0-cell (the 1 1 origin), two 1-cells (the positive and negative x-axes D+ and D−), and two 2-cells 2 2 (the upper and lower half-planes D+ and D−). Also, the gluing maps in this case are the identity maps on cells. In this case, it suﬃces to show that the following coarse

2 1 2 deformation retract Ht from D+ onto D+ lifts. Let A1 be the subspace of D+ bounded 1 by D− and the ray R1 := {−x, x |x ≥ 0}, A2 be the subspace bounded by R1 and the 1 ray R2 := {x, x |x ≥ 0}, and A3 be the subspace bounded by R2 and D+. Also, deﬁne 2 ft : D+ → A2 ∪ A3 by ft(x, y) = (x, (1 − t)y + t(−x)) if (x, y) ∈ A1 and the identity otherwise, gt : A2 ∪ A3 → A3 by gt(x, y) = ((1 − t)x + ty, y) if (x, y) ∈ A2 and the

1 identity otherwise, and ht : A3 → D+ by ht(x, y) = (x, (1 − t)y). Then, deﬁne Ht by

  f3t if 0 ≤ t < 1/3,   Ht := g3t−1 if 1/3 ≤ t < 2/3,    f3t−2 if 2/3 ≤ t < 1.

133 There exist coarse liftings of ft, gt, and ht deﬁned by

     1 x z 1 x (1 − t)z − tx2/2      ˜      ft (x, y), 0 1 y = 0 1 (1 − t)y + t(−x)           0 0 1 0 0 1

if (x, y) ∈ A1, and the identity otherwise;

     1 x z 1 (1 − t)x + ty (1 − t)z + ty2/2           g˜t (x, y), 0 1 y = 0 1 y            0 0 1 0 0 1 if (x, y) ∈ A2, and the identity otherwise;

  1 x (1 − t)z   ˜   ht ((x, y), ) = 0 1 (1 − t)y .     0 0 1

Therefore Ht has a coarse lifting H˜t.

We shall only prove that f˜t is a coarse homotopy, as the proof forg ˜t and h˜t are analogous. We shall ﬁrst prove that f˜t is uniformly proper. Since ft is obviously uniformly proper in the places where it is the identity, we only need to check uniform properness of ft restricted to A1. Note that if the set of images

   1 x (1 − t)z − tx2/2      B = 0 1 (1 − t)y + t(−x)      0 0 1  is bounded, then the set of all x in the matrices in B is bounded as well. Similarly, as the set of all (1 − t)y + t(−x) is bounded, and the set of all t(−x) is bounded, it follows that the set of (1 − t)y is bounded. Therefore, the set of y is bounded as well. Finally,

134 as the set of all (1 − t)z + tx2/2 is bounded, the set of all z must be bounded as well.

It follows that [ f −1(B) t∈[0,1] is bounded. The map ft is in fact continuous, so pseudocontinuity follows. Finally, we need to show that f˜t is uniformly bornologous. First, suppose

      1 x z 1 x0 z0           0 0   A = (x, y), 0 1 y ,B = (x , y ), 0 1 y0 ,             0 0 1 0 0 1

     −1 1 x00 z00 1 x (1 − t)z − tx2/2 1 x0 (1 − t)z0 − tx02/2             0 1 y00 := 0 1 (1 − t)y + t(−x) 0 1 (1 − t)y0 + t(−x0) .             0 0 1 0 0 1 0 0 1

Then, by another matrix calculation, x00 = x − x0 and y00 = t(x0 − x) + (1 − t)(y − y0),

so |x00| ≤ R and |y00| ≤ 2R. Finally,

z00 = −tx2/2 + z − tz + txx0 − tx02/2 − z0 + tz0 − xy0 + txy0 + x0y0 − tx0y0 =

= −tx2/2 + txx0 − tx02/2 + (1 − t)(z − z0) + (1 − t)y0(x − x0)

≤ | − tx2/2 + txx0 − x02/2| + (1 − t)|z − z0 + y0(x − x0)| =

√ √ 2 = t| x/ 2 − x0/ 2 | + (1 − t)|z − z0 + y0(x − x0)|

≤ R2/2 + R

135 2 for all t ∈ [0, 1]. Therefore, d(f˜t(A), f˜t(B)) ≤ R /2 + R for all t.

1 1 Similarly, the coarse deformation retract of the f-neighborhood A1∪A3 onto D−∪D+ can be lifted to a coarse deformation retract. Finally, the pullbacks

1 n 1 x z o 1 n 1 x z o D˜ := 0 1 0 , x ≤ 0 , D˜ := 0 1 0 , x ≥ 0 − 0 0 1 + 0 0 1

˜ ˜ 1 ˜ 1 ˜ 1 ˜ 1 are ﬂasque, the ﬂasqueness maps f : D− → D− andg ˜ : D+ → D+ given by

1 x z 1 x−1 z f˜ 0 1 0 = 0 1 0 0 0 1 0 0 1 and similarly forg ˜.

Proposition 6.3.13. If B is a coarsely connected coarse CW-complex and π : E → B

−1 is a (weak) coarse ﬁbration, then the ﬁbers Fb := π ({b}) are all coarse homotopy equivalent.

Proof. The proposition is obviously true if dim B = 0. We shall ﬁrst assume that dim

B ≥ 2 and π is a strong coarse ﬁbration. Let b, b0 ∈ B. Then, there exists a coarse homotopy ht : [0, 1] → B deﬁned by

  b if 0 ≤ t < 1, ht =  0 b if t = 1.

∗ Note that, in this case, h0(E) = Fb. By the CHLP, ht pulls back to a coarse homotopy ˜ ˜ ˜ ht : Fb × [0, 1] → E such that h1(x) ∈ Fb0 for all x ∈ Fb. Then ht provides half of the desired coarse homotopy equivalence, with the pullback of the inverse homotopy

 b0 if t = 0, 0  ht =  b if 0 < t ≤ 1

136 providing the other half.

When dim B = 1, the notions of weak and strong coarse ﬁbrations are equivalent.

In this case, a map g from b to b0 is given by a composition of maps, each of which is either a flasqueness map on a 1-cell or the inverse of a flasqueness map, so g is coarse homotopy equivalent to the identity on {b}. Since each of these flasqueness maps lifts to a flasqueness map, g also lifts to a mapg ˜ : Fb → Fb0 which is coarse homotopy equivalent to the identity map on Fb. It follows that Fb and Fb0 are coarse homotopy equivalent.

The ﬁnal case is when dim B = n ≥ 2 and π is a weak coarse ﬁbration. By taking

“liftable” coarse deformation retracts Dn → Dn−1 as speciﬁed in the deﬁnition of weak

0 0 ﬁbration, we can coarsely homotope the points b and b to points bn−1, bn−1 ∈ Bn−1. Iterating this process for decreasing dimension, b and b0 can be coarsely homotoped via

0 a series of liftable deformation retracts to points b1, b1 ∈ B1. Lifting these deformation −1 retracts, we see that F and F 0 are coarse homotopy equivalent to F ,F 0 ⊂ π (B ) b b b1 b1 1 respectively; ﬁnally, by the previous paragraph’s argument, F F 0 are coarse homotopy b1 b1 equivalent.

6.4 The Main Theorem

The motivation behind studying coarse fibrations is that, analogously to classical fibrations, the coarse cohomology of the total space can be determined from the cohomology of the base space and fiber via spectral sequences:

Theorem 6.4.1. If B is a coarse CW-complex, E is a coarse space, and π : E → B is a coarse fibration with fiber F , then there exists a first-quadrant spectral sequence

p,q {Er , dr} with

p,n−p n n (a) The stable terms E∞ are isomorphic to the successive quotients Fp /Fp+1 in a n n n n ﬁltration 0 ⊂ Fn ⊂ · · · F0 = HX (E) of HX (E);

137 2 ∼ p q (b) Ep,q = HX (B; HX (F )).

Proof. The argument is analogous to the proof of Theorem 1.3 in [7]. The outline of the proof is as follows:

−1 1. If Bn is the n-skeleton of B, then let Xn := π (Bn). We shall construct an

p q p+q explicit isomorphism Ψ from HX (Bp,Bp−1) ⊗ HX (F ) to HX (Xp,Xp−1) as the composition of three isomorphisms constructed as intermediate steps.

2. Construct the ﬁrst-quadrant spectral sequence explicitly via an exact couple. Part

(a) of Theorem 6.4.1 shall then follow from known facts about ﬁrst-quadrant

spectral sequences.

p+q p+q+1 3. Argue that, if d1 : HX (Xp,Xp−1) → HX (Xp+1,Xp) is the ﬁrst diﬀeren-

p p+1 tial deﬁning the spectral sequence, and if ∂ : HX (Bp,Bp−1) → HX (Bp+1,Bp) is the cellular coboundary map, then the diagram

p q ∂ ⊗-1 p+1 q HX (Bp,Bp−1) ⊗ HX (F ) HX (Bp+1,Bp) ⊗ HX (F )

Ψ Ψ ? ? p+q d1 - p+q+1 HX (Xp,Xp−1) HX (Xp+1,Xp)

commutes. This fact will then imply part (b) of Theorem 6.4.1.

6.4.1 Construction of the Isomorphism

In this section, we construct the map Ψ indicated in part (1) of the outline.

−1 Lemma 6.4.2. If Bn is the n-skeleton of B, then let Xn := π (Bn), there exists an isomorphism

p q p+q Ψ: HX (Bp,Bp−1) ⊗ HX (F ) → HX (Xp,Xp−1).

138 Proof. We will construct Ψ from the commutative diagram

p q ∼ q HX (Bp,Bp−1) ⊗ HX (F ) = ⊕αHX (F )

p Ψ ⊕α =∼ ? ? ˜ ∗ p+q Φ- M p+q ˜ p ˜p−1 HX (Xp,Xp−1) ∼ HX (Dα, Sα ). = α

p p−1 p−1 p−1 Let D denote R × R+ and S its boundary R For each characteristic map p ˜ p ∗ ˜p−1 ˜ p Φα : Dα → Bp for a p-cell of Bp, let Dα := Φα(Xp) and Sα be the part of Dα over p−1 p p−1 Sα . We then have a map Φ:˜ tα(D˜α,Sα ) → (Xp,Xp−1) Using the construction from

Lemma 6.1, there exists an f-neighborhood N of Bp−1 in Bp which maps onto Bp−1 via p a coarse homotopy as follows. For a p-cell Dα which has an image in Bp via the gluing map fα, let p Nα = {(x, t) ∈ Dα | t ≤ d(f(x), 0)}.

S p Then, N is the image of α Nα in B . −1 By the CHLP, it follows that the inclusion Xp−1 ⊂ π (N) is a coarse homotopy equivalence. By excision, we conclude that Φ˜ induces the isomorphism Φ˜ ∗ in the dia-

p gram. The top isomorphism comes from the fact that HX (Bp,Bp−1) is a direct sum of Z’s, one for each p-cell of B. p p p To construct the α making up the ﬁnal isomorphism, let D˜ = D˜α and consider HXp+q(D˜ p, S˜p−1). If p ≥ 2, then Sp−1 can be expressed as the union of coarse (p − 1)-

p−1 p−1 p−1 p cells, S = D+ ∪ D− with intersection a coarse (p − 2)-sphere. Since D is coarse p−1 homotopy equivalent to D− , the equivalence given by a liftable coarse deformation ˜ p ˜ p−1 retract, the CHLP gives a coarse homotopy of D onto D− . Therefore, the inclusion ˜ p−1 ˜ p ∗ map j : D− ,→ D induces an isomorphism j on coarse cohomology, so

∗ ˜ p ˜ p−1 HX (D , D− ) = 0

139 by the discussion of relative coarse cohomology in Section 2.2. It follows that the

coboundary map

p+q−1 ˜p−1 ˜ p−1 p+q ˜ p ˜p−1 ∂ : HX (S , D− ) → HX (D , S ).

˜ p ˜p−1 ˜ p−1 in the long exact cohomology sequence of the triple (D , S , D− ),

- p+q−1 ˜ p ˜ p−1 - p+q−1 ˜p−1 ˜ p−1 ∂- p+q ˜ p ˜p−1 - ··· HX (D , D− ) HX (S , D− ) HX (D , S ) ···

is an isomorphism. ˜ p−1 ˜ p−1 ˜ p−1 ˜ p−1 ˜p−1 Since the pair (D− , D+ ) is coarsely excisive, with D− ∪ D+ = S and ˜ p−1 ˜ p−1 ˜p−2 D− ∩ D+ = S , we have by excision (Prop. 2.3.1) an isomorphism

∗ p+q−1 ˜p−1 ˜ p−1 p+q−1 ˜ p−1 ˜p−2 i : HX (S , D− ) → HX (D+ , S )

induced by inclusion. Composing ∂ with the inverse of i∗ gives an isomorphism

p+q−1 ˜ p−1 ˜p−2 p+q ˜ p ˜p−1 p : HX (D+ , S ) → HX (D , S ).

k+q−1 ˜ k−1 k˜−2 k+q ˜ k ˜k−1 Iterating this process, we obtain isomorphisms k from HX (D+ , S ) to HX (D , S ) for each 2 ≥ k ≥ p. In the case of k = 1, we use the fact that HXn(D˜ 1) = 0 for all n

(as stated in the deﬁnition of a coarse ﬁbration) and so there exists an isomorphism

q 0 q+1 1 0 1 : HX (S˜ ) → HX (D˜ , S˜ )

q q 0 from the long exact relative cohomology sequence, and HX (F ) = HX (S˜ ); let 0 : HXq(F ) → HXq(S˜0) be the identity isomorphism. Composing these isomorphisms

140 gives the isomorphism

p q p+q p p−1 := p ◦ p−1 ◦ · · · 0 : HX (F ) → HX (D˜ , S˜ );

p taking the direct sum over all p-cells results in the map ⊕αα in the diagram.

6.4.2 Construction of the Spectral Sequence

Now, we shall construct the spectral sequence stated in the theorem. The discussion in this section is directly analogous to that in [7], Chapter 1; as a result, some of the more technical details may be omitted. As in the classical case, the spectral se-

p+q,p p+q p+q,p quence will arise from an exact couple (A, E) with A1 = HX (Xp) and E1 = p+q p+q p+q+1 HX (Xp,Xp−1). The first differential d1 : HX (Xp,Xp−1) → HX (Xp+1,Xp) is defined by the map

p+q α- p+q β- p+q+1 HX (Xp,Xp−1) HX (Xp) HX (Xp,Xp−1) from the staircase diagram of long exact sequences

? - p+q - p+q - p+q+1 - HX (Xp+1,Xp) HX (Xp+1) HX (Xp+2,Xp+1)

? - p+q α- p+q β- p+q+1 - HX (Xp,Xp−1) HX (Xp) HX (Xp+1,Xp)

? - p+q - p+q - p+q+1 - HX (Xp−1,Xp−2) HX (Xp−1) HX (Xp,Xp−1) . . . .?

141 This diagram is composed of long exact sequences in relative cohomology placed in a staircase orientation as follows:

- p+q - p+q HX (Xp,Xp−1) HX (Xp)

? p+q - p+q+1 - HX (Xp−1) HX (Xp,Xp−1) .

Recall from the classical construction of the Serre spectral sequence that En+1 is deﬁned

n n to be ker dr/ im dr. The groups Fp filtering HX (X) in part (a) of the theorem are n n n defined by Fp = im(HX (X) → HX (Xp)); since the spectral sequence is a first- n n p,n−p quadrant one, the quotients Fp /Fp+1 converge to E∞ . In order to guarantee that ∗ the spectral sequence converges to HX (X), we require two conditions on the Xp (cf. [7], p.7). The first condition is that Xp = ∅ for p < 0, which is guaranteed by the definition of Xp. The second condition is that the inclusion Xp ,→ X induces an isomorphism on HXn for p sufficiently large with respect to n, which will be guaranteed by the following result:

n Lemma 6.4.3. HX (X,Xp) = 0 for all n ≤ p.

n Proof. Here, we use the fact that HX (B,Bp) = 0 for all p ≤ n and argue by induction on the dimension of B. Obviously the result is true when dim B = 0; in this case, p = 0 is the only possible value, and X = X0. Similarly, if dim B = 1, then the only possible

0 1 values of p are p = 0 and p = 1; clearly HX (X,X1) = HX (X,X1) = 0 since X = X1.

0 As X1 is unbounded, it also follows that HX (X1,X0) = 0. Now assume the result for dim B < m, suppose dim B = m, m ≥ 2 and observe

that for all p ≤ m, the restriction π|Xp : Xp → Bp is a coarse ﬁbration. Observe that 0 for m > 0, X ∪ CXp is unbounded for all p ≤ m, so HX (X,Xp) = 0.

1 We shall now argue the result for HX (X,Xp), p ≥ 1 only, as the proof for gen-

n eral HX (X,Xp), n ≤ p is analogous. From the long exact sequence of the triple

142 (X,Xm−1,Xm−2)

- 0 - 1 - 1 - ··· HX (Xm−1,Xm−2) HX (X,Xm−1) HX (X,Xm−2) ···

0 1 and the fact that HX (Xm−1,Xm−2) = HX (Xm−1,Xm−2) = 0 from the induction 1 ∼ 1 hypothesis, we observe that HX (X,Xm−1) = HX (X,Xm−2). From the long exact

sequence of (X,Xm−2,Xm−3)

- 0 - 1 - 1 - ··· HX (Xm−2,Xm−3) HX (X,Xm−2) HX (X,Xm−3) ···

1 ∼ 1 and the induction hypothesis, we see that HX (X,Xm−2) = HX (X,Xm−3). Iterating this process,

1 ∼ 1 ∼ ∼ 1 ∼ 1 HX (X,X1) = HX (X,X2) = ··· = HX (X,Xm−2) = HX (X,Xm−1).

By Lemma 6.4.2, we note that

1 1 ∼ m 1−m HX (X,Xm−1) = HX (Xm,Xm−1) = HX (Bm,Bm−1) ⊗ HX (F ),

m 1−m 1 and HX (Bm,Bm−1)⊗HX (F ) = 0 since m ≥ 2. Also, obviously HX (X,Xm) = 0

since Xm = X. Alternatively, it appears feasible to modify Bell and Dranishnikov’s “asymptotic

Hurewicz theorem” (Theorem 1 in [1]) to prove the result as well.

By an argument directly analogous to the one found on pages 12-13 of [7], it now follows that the diagram in part (3) of the outline commutes.

143 6.4.3 Examples

3 2 • Let π : R → R be the projection map. Then the fiber F is R and the only 2,1 ∼ term on the E2 page is E2 = Z. This term is therefore unaffected by the higher p,n−p 2,1 ∼ degree differentials and so the only stable term E∞ is E∞ = Z. It follows that 3 3 ∼ n 3 HX (R ) = Z and HX (R ) is zero otherwise.

• The real Heisenberg group HR has the same E2 page as the previous example. n ∼ n 3 Therefore HX (HR) = HX (R ) for all n.

• Similarly, if G is the fundamental group of the Klein bottle, then HXn(G) =∼

n 2 HX (Z ) for all n.

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146 Vita

Steven Hair was born in Shermans Dale, Pennsylvania in 1979. His interest in science and mathematics compelled him to enroll at Virginia Tech, where he obtained a Bache- lor’s degree in mathematics. Subsequently, he remained at Virginia Tech for a Master’s degree, working under Dr. Peter Linnell. Following his Master’s studies, Steven enrolled in the doctoral program at Penn State, completing his studies in 2010.