ON ACYCLICITY PROPERTIES OF COMPLEMENTS OF SUBSETS IN THEHILBERT CUBE

By

ASHWINI K. AMARASINGHE

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2017

c⃝ 2017 Ashwini K. Amarasinghe

To my parents

ACKNOWLEDGMENTS

First, I would like to thank my advisor, Dr. Alexander Dranishnikov, for all his

guidance and for being generous with his time. His deep knowledge of mathematics,

enthusiasm for the subject, and patience were invaluable. I am also very grateful the

members of my committee Dr. Yuli Rudyak, Dr. James Keesling, Dr. Paul Robinson

and Dr. Galina Rylkova for for their advice and support. I thank Margaret Somers,

Stacie Austin, and Connie Doby for all their help during my time at graduate school. I

would also like to express my gratitude to everyone at the mathematics department at

the University of Florida for the warm and collegial atmosphere they provided. Finally, I

thank my wife Udeni for her support and encouragement extended throughout my college

career.

4

TABLE OF CONTENTSpage

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1 Dimension Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Space of Complete Non-negatively Curved Metrics on the Plane. . . . . . . 142.3 Cohomological Dimension of Compact Metric Spaces. . . . . . . . . . . . . 162.4 Z sets, Zn sets and homological Zn sets in Q . . . . . . . . . . . . . . . . . 172.5 Steenrod Homology, Inverse Limits and the First Derived Functor of an

Inverse Sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 CONNECTEDNESS PROPERTIES OF THE COMPLETE NON-NEGATIVELYCURVED METRICS ON THE PLANE . . . . . . . . . . . . . . . . . . . . . . 21

3.1 Infinite Dimensional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 ACYCLICITY OF COMPLEMENTS OF WEAKLY INFINITE-DIMENSIONALSPACES IN HILBERT CUBE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2.1 Alexander Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2.2 Strongly Infinite Dimensional Spaces . . . . . . . . . . . . . . . . . 304.2.3 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . 31

4.3 On Weakly Infinite Dimensional Compacta . . . . . . . . . . . . . . . . . . 334.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.5 Weakly Infinite-dimensional Compacta in Compact Q-manifolds. . . . . . . 384.6 Zero-dimensional Steenrod Acyclicity of Complement of a σ-compact Subset

in the Hilbert Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

ON ACYCLICITY PROPERTIES OF COMPLEMENTS OF SUBSETS IN THEHILBERT CUBE

By

Ashwini K. Amarasinghe

August 2017

Chair: Alexander N. DranishnikovMajor: Mathematics

The first part of this dissertation is on extending a non-separation theorem for the

complement of finite dimensional subspace of the space of complete non-negatively curved

metrics on the plane by Belagradek and Hu. We generalize their result to complements of

weakly infinite dimensional (WID) compacta. An extension for a similar non-separation

theorem by Belagradek and Hu for the moduli space is then obtained for spaces having

Haver’s property-C. It is still an open question whether every weakly infinite dimensional

compact metric space have property-C.

In the second part of this dissertation we answer a question by Banach, Zarichnyi

and Kauty regarding the acyclicity of the complement of closed WID subspaces of

a Hilbert Cube affirmatively. In the process, we define cohomologically strongly

infinite-dimensional (CSID) compacta with respect to any ring. A space that is not

CSID is called cohomologically weakly infinite dimensional (CWID). We prove that the

complement of any CWID compactum in the Hilbert cube is acyclic with respect to any

ring. Consequently, it is shown that the class of strongly infinite-dimensional compacta is

properly contains the class of CSID compacta, and it is further shown that being CWID is

hereditary with respect to closed subsets.

The third part of this dissertation is about extending the above results to σ-compacta

in the Hilbert Cube. We first prove that removing a closed WID subspace does not change

the homology type of a Hilbert cube manifold, and then we prove the main result, that

6

states the complement of a σ-compactum in the Hilbert Cube is Steenrod acyclic in the

dimension 0.

7

CHAPTER 1INTRODUCTION

The concept of topological dimension was first introduced by Brouwer in 1911 in

[Bro11] which was constructed upon Poincare’s observation that the meaning of dimension

is inductive in nature, and the possibility of separating a space by subsets of lower

dimension. In 1913, he proved that there is no homeomorphism between Rn and Rm

for distinct m and n using his definition of dimension [HW41], which was a topological

invariant by definition. Meanwhile Lebesgue had approached in another method to prove

that the dimension of a Euclidean space is topologically invariant by using the concept of

covering dimension [HW41].

The definition of dimension we use throughout this dissertation is due to the theorem

of separators [vM02], which coincides with covering dimension, small and large inductive

dimension [Eng95] in the case of separable metric spaces. We will consider only this type

of spaces throughout this dissertation, unless explicitly noted otherwise. A family of pairs

of disjoint closed sets τ = {(Ai, Bi) : i ∈ Γ} of X is said to be essential if for every

family {Li : i ∈ Γ} where Li is a separator between Ai and Bi, we have∩

i∈Γ Li = ∅

For a non-empty space X, the topological dimension is defined to be the largest number

n such that X has an essential family of n pairs of disjoint closed subsets, but every

family of n + 1 pairs of disjoint closed subsets is inessential. If no such n exists, the

space is called infinite dimensional. A space is called strongly infinite-dimensional if it

has an infinite essential family of pairs of disjoint closed subsets, and it is called weakly

infinite-dimensional if it is not strongly infinite dimensional.

The Chapter 2 of this dissertation starts with the basic definitions of dimension

together with some known results we will use throughout the rest of the text. The

distinctions between different classes of infinite dimensional spaces, namely, strongly

infinite-dimensional spaces, weakly infinite-dimensional spaces, countable dimensional

spaces and spaces with Haver’s property-C will be defined followed by the known results

8

on the relations among the classes. The last section of this chapter includes a discussion

on Cγ topology on the space of functions, and a discussion on Steenrod homology.

In their paper [BH16], Belegradek and Hu proved that the spaces of complete,

non-negatively curved Riemannian metrics on the plane, Rk≥0(R2) cannot be separated

by removing a finite-dimensional subspace. We shall show, by using a theorem on

continuum-wise separation by Mazurkiewicz [Eng95], that the space Rk≥0(R2) cannot

be separated by removing a weakly infinite-dimensional subspace, and if the subspace

under consideration is closed, the complement is path connected. The moduli space

Mk≥0(R2) is defined as the quotient space of Rk

≥0(R2) under the action of Diff(R2) via

pullback. In [BH16], Belagradek and Hu obtained a similar non-separation theorem for the

moduli space under removal of a closed, finite dimensional subspace. In the latter part of

Chapter 3, we prove that this non-separation theorem can be generalized to a space with

property-C, and obtain a necessary condition on the action that would allow one to extend

the non-separation theorem to weakly infinite-dimensional subspaces in Mk≥0(R2). The

results in this chapter were published in 2015 in [Ama17].

Chapters 4 constitutes the second part of this dissertation. In [Kro74], Kroonenberg

proved that the complement of any closed, finite-dimensional subspace of the Hilbert

cube is acyclic. This result was extended to countable dimensional spaces in [BCK11]

by Banakh, Cauty and Karrasev. They defined the concept of a homological Z∞-set,

which is analogous to Z∞-sets of a space. The authors proved that either closed,

countable dimensional subspaces or trt-dimensional subspaces in the Hilbert cube are

homological Z∞ sets and posed the question whether the same is true for closed weakly

infinite-dimensional spaces or spaces with property-C. This question also appeared in

Open Problems in Topology II [Pea07], in a different setting. Namely, whether the

complement of a closed weakly infinite-dimensional subspace in the Hilbert cube is acyclic.

The purpose of the first three sections of Chapter 4 is to prove this in the affirmative by

using an argument based on Alexander duality Theorem.

9

In the process of proving this theorem, we observed that there is a sufficient

cohomological condition for a closed subspace in the Hilbert cube in order for its

complement to have a non-zero cycle. Namely, the existence of a sequence of pure

relative classes αi ∈ Hki(X,Ai), ki > 0, i = 1, 2, . . . , Ai ⊂Cl X such that the cup-product

α1 ∪ αn ∪ · · · ∪ αn = 0 for all n. We call a relative cohomology class α ∈ Hk(X,A) pure

relative if j∗(α) = 0 for the inclusion of pairs homomorphism j∗ : Hk(X,A) → Hk(X).

This means that α = δ(β) for β ∈ Hk−1(A). In section 3 of Chapter 4, we use pure relative

cohomology classes and the existence of a non-zero cup product to define the class of

cohomologically strongly infinite-dimensional compact metric spaces with respect to any

ring R with unity. A space which is not cohomologically strongly infinite-dimensional

is called a cohomologically weakly infinite-dimensional space. For cohomologically

weakly infinite-dimensional subsets in the Hilbert cube, we prove that the complement

is acyclic with respect to any ring R with unity. We note that every cohomologically

strongly infinite-dimensional space is strongly infinite-dimensional. Dranishnikov’s [Dra90]

example of a strongly infinite dimensional compactum with finite cohomological dimension

establishes that the class of cohomologically strongly infinite-dimensional compacta

does not coincide with the class of strongly infinite dimensional compacta. Thus making

this new class of spaces having the "strongest" infinite dimension. The fourth section of

Chapter 5 investigates hereditary properties of cohomologically weakly infinite-dimensional

compact, and also contains a discussion on application to the spaces Rk≥0(R2) and

Mk≥0(R2). The results obtained in the above sections was published in [AD17].

In the section 5 of Chapter 4, we obtain a generalization of acyclicity theorem in the

setting of compact Hilbert cube manifolds. Hilbert cube manifolds are separable metric

manifolds that are modeled on the Hilbert cube. We shall show that removing a weakly

infinite-dimensional subspace from a compact Hilbert cube manifold does not change its

homology type.

10

In the final section of Chapter 4, we start with basic definitions of Steenrod homology.

Steenrod homology was first introduced by Steenrod in [Ste40] via regular cycles. A

regular cycle in a compact metric space is a single infinite cycle with the property that

successive approximations tend to zero. For an inverse sequence of spaces Xi with limit

lim←−(Xi), there exists an exact sequence [Fer95]

0→ lim←−1Hst

k+1(Zi)→ Hstk (Z)→ lim←−H

stk (Zi)→ 0

Using this, we prove that the complement of any σ-compact weakly infinite-dimensional

subspace of Q is Steenrod acyclic in dimension zero.

11

CHAPTER 2BACKGROUND

In this chapter, we list some basic results we would recall in the later chapters of this

dissertation.

2.1 Dimension Theory

We will define the topological dimension and characterize it in different settings. The

different classes of infinite dimensional spaces will be introduced, and the known inclusion

relations will be listers. This section will also include some basic theorems in dimension

theory. Every space under consideration is a separable metrizable space.

Definition 2.1.1. A separator between two disjoint closed subsets A and B of a space

X is a closed subset L of X such that X \ L has two components U and V with A ⊂ U

and B ⊂ V . Let X be a space and Γ be an index set. A family of pairs of disjoint, closed

subsets of X, τ = {(Ai, Bi|i ∈ Γ)} is called essential if for every family {Li|i ∈ Γ} where

Li is an arbitrary separator between Ai and Bi for every i, we have∩i∈Γ

Li = ∅. If τ is not

essential, it is called inessential.

It is known that for a compact space X, the set Γ is at most countable [vM02].

Definition 2.1.2. For a space X, define the dimension dimX ∈ {−1, 0, 1, . . . } ∪ {∞} by

dimX = −1 if and only if X = ∅

dimX ≤ n if and only if every family of n+ 1 pairs of disjoint closed subsets of X isinessential.

dimX = n if and only if dimX ≤ n and dimX ≰ n− 1

dimX =∞ if and only if dimX = n for every n ≥ −1

A space X with dimX < ∞ is called finite dimensional. If X is not finite dimensional, it

is called infinite dimensional.

The class of infinitely dimensional spaces have several distinct classes. Some of which

are defined below.

12

Definition 2.1.3. If a space X has an infinite family of pairs of disjoint closed subsets,

we call X strongly infinite-dimensional. If X is not strongly infinite dimensional, it is

called weakly infinite-dimensional. If X can be represented as a countable union of zero

dimensional subsets, X is said to be countable dimensional.

The Hilbert cube, denoted by Q = I∞ is an example of a strongly infinite-dimensional

space. The one point compactification of the disjoint union of cubes⨿

n In is an infinite

dimensional compacta which is an example of a weakly infinite-dimensional space. It

should also be noted that in our convention, every finite dimensional space is a weakly

infinite dimensional space.

Of particular interest is spaces having Haver’s property-C. In the class of compact

metric spaces, this is a class that strictly lies between weakly infinite-dimensional spaces

and strongly infinite-dimensional spaces.

Definition 2.1.4. A topological space X has property C (is a C-space) if for every

sequence G1,G2, . . . of open covers of X, there exists a sequence H1,H2, . . . of families

of pairwise disjoint open subsets of X such that for i = 1, 2, . . . each member of Hi is

contained in a member of Gi and the union∪∞

i=1Hi is a cover of X.

The following Theorem lists all the inclusions between the above classes of infinite-

dimensional compact metric spaces.

Theorem 2.1.1. Every countable dimensional space is a C-space. Every C-space is weakly

infinite dimensional.

Both inverse relations are false in the setting of compacta. [vM02]

There are various different definitions of dimension by Brouwer, Lebesgue, Menger

and other mathematicians. We do not wish to state them all here, but will note that in

the setting of separable metrizable spaces, the Lebesgue covering dimension, small and

large inductive dimension and Brouwer’s dimensiongrad all agree. An extended discussion

about various definitions can be found in [vM02] and [Eng95].

13

However, there is a characterization of dimension, which is due to P. S. Alexandrov

that we will use later in the dissertation. First, one would need to define the concept of

essential maps on to the n-cube.

Definition 2.1.5. A map f : X → In to the n-cube is called essential if it cannot be

deformed to ∂In through the maps of pairs (In, f−1(∂In)) → (In, ∂In). A map f : X → Q

to the Hilbert cube is called essential if the composition pn ◦ f : X → In is essential for

every projection pn : Q→ In onto the factor.

The connection between essential maps and dimension is noted in the theorem below.

Theorem 2.1.2. A compact space X has an essential family of pairs of n disjoint closed

subsets if and only if X admits an essential map onto In. A compact space X is strongly

infinite-dimensional if and only if it admits an essential map on to the Hilbert cube Q.

2.2 Space of Complete Non-negatively Curved Metrics on the Plane.

The spaces of Riemannian metrics with non-negative scalar curvature are subjects of

intensive study. The connectedness properties of such spaces on R2 were studied recently

by Belegradek and Hu in [BH16], which we will use heavily as the base of chapter 3

in this dissertation. The topology under consideration for all the typse of spaces under

consideration will be called the Ck topology. It is the topology of Ck uniform convergence

on compact sets, where 0 ≤ k ≤ ∞, which we will formally define as;

Definition 2.2.1. If M and N are Cγ-manifolds, Cγ(M,N) denotes the sets of Cγ maps

from M to N . The compact-open Cγ topology on Cγ(M,N) is generated by sets defined

as follows. Let f ∈ Cγ(M,N) and let (ϕU), (ψV ) be charts on M,N ; let K ⊂ U be a

compact set such that f(K) ⊂ V ; let 0 < ε ≤ ∞. Define a weak sub-basic neighborhood

N γ(f ; (ϕU), (ψV ), K, ε) to be the set of Cγ maps g : M → N such that g(K) ⊂ V and

∥Dk(ψfϕ−1)(x)−Dk(ψgϕ−1)(x)∥ < ϵ for all x ∈ ϕ(K), k = 0, . . . , r. The compact-open Cγ

topology on Cγ(M,N) is generated by these sets.

The condition on the distance function Dk means that the local representations of

f and g, together with their first k derivatives, are within ε at each point of K. The

14

weak topology on C∞(M,N) is the union of the topologies induced by the inclusion maps

C∞(M,N)→ CγW (M,N) for γ finite.

The Hölder space of order k, α is the space of function from the space of k-differentiable

continuous functions that are “almost” k + 1 differentiable and continuous. With the word

almost, we mean that every (k+1)-differentiable continuous function belongs to the Hölder

space, but not all k-differentiable continuous functions do.

Definition 2.2.2. Let U ⊂ Rn) be a region and let f ∈ Ck(U) for an integer k. We say

that f ∈ Ck,α(U) if and only if

maxn≤k

supx,y∈U ,x =y

|Dnf(y)−Dnf(x)||y − x|α

= [f ]Ck,α(U) <∞

Therefore, the Hölder space Ck,α is a subset of Ck(U), sucht that the functions are Hölder

continuous.

Definition 2.2.3. Rk≥0(R2) is the space of complete non-negatively curved metrics on

the plane. The topology is the Ck compact-open topology when k is an integer or ∞. For

non-integer k, we use Hölder topology defined above. the quotient space of Rk≥0(R2) by the

Diff(R2)-action via pullback is called the moduli space of complete non-negatively curved

metrics and is denoted by Mk≥0(R2).

In [BB17], Belagradek and Banakh proved that if γ is a finite, positive, non-integer,

then Rγ≥0(R2) is homeomorphic to Σω, where Σω is the linear span of Hilbert cube in the

Hilbert space ℓ2. If γ = ∞, Rγ≥0(R2) is homeomorphic to ℓ2. The moduli space Mk

≥0(R2)

is rather pathological. For instance, it is not a Hausdorff space. In [BH16], Belagradek and

Hu exhibit a non-flat metric g ∈ Rk≥0(R2) whose isometry class lies in every neighborhood

of the isometry class of g0, the Euclidean metric.

It should be noted that that any complete non-negatively curved metric on R2 is

conformally equivalent to the standard Euclidean metric g0 isometric to e2ug0 for some

smooth function u, this is a result by Blank-Fiala [BF41]. In [BH16], it was proved that

15

the metric e−2ug0 is complete if and only if α(u) ≤ 1, where α(u) = limr→∞

M(r, u)

log(r)with

M(r, u) = sup{u(z) : |z| = r}.

2.3 Cohomological Dimension of Compact Metric Spaces.

In this section, we will define the cohomological dimension for a compact metric

space, and list some results we will use in Chapter 4 of this dissertation.

Definition 2.3.1. Cohomological dimension of a space X with respect to an abelian

group G is the the largest number n such that there exists a closed subset A ⊂ X with

Hn(X,A;G) = 0. This will be denoted by dimG(X).

Here H∗ denotes Cech cohomology. This definition is good for any space, we

will restrict ourselves when X is a compactum. In the following Theorem H∗c denotes

cohomology with compact support.

Theorem 2.3.1. For any compactum X and abelian G, the following are equivalent.

(1) dimG(X) ≤ n.

(2) Hn+1(X,A;G) = 0 for all closed A ⊂ X.

(3) Hn+1c (U ;G) = 0 for all open U ⊂ X.

(4) For every closed subset A ⊂ X the inclusion homomorphism Hn(X;G) →Hn(A;G) is an epimorphism.

(5) K(G,n) is an absolute extensor for X.

The Alexandroff Theorem gives a relation between cohomological and topological

dimension.

Theorem 2.3.2. For finite dimensional compacta, there is the inequality dimZX = dimX.

When dimX is infinite, the following theorem by Dranishnikov [Dra90] provides a

counterexample.

Theorem 2.3.3. There is a strongly infinite dimensional compactum X having dimZX ≤

3.

16

This example will be exploited later in Chapter 4 where we establish the class

of strongly infinite-dimensional spaces containing the class of cohomologically weakly

infinite-dimensional spaces.

2.4 Z sets, Zn sets and homological Zn sets in Q

The concept of a Z-set was first defined by Anderson in []. Intuitively, A Z-set in a

space X is “small” in the sense of homotopy. It will be used in obtaining a homeomorphism

extension theorem in Q, which we will use later in Chapter 3. Let X and Y be spaces, for

two functions f and g in C(X,Y ), we say f and g are U -close if

Definition 2.4.1. Let X be a space. A closed subset A ⊂ X is called a Z-set in X

provided that for every open cover U of X and every function f ∈ C(Q,X) there is a

function g ∈ C(Q,X) such that f and g are U close and g[Q] ∩ A = ∅.

The collection of Z sets in a space will be denoted by Z(X).

We define the set of bounded functions Crho(X,Y ) = {f ∈ C(X,Y ) : diamρ(f [X]) <

∞} where ρ is an admissible metric on Y . Let X and (Y, ρ) be spaces. For all f, g ∈

C(X,Y ), define ρ(f, g) = sup{ρ(f(x), g(x)) : x ∈ X}. Observe that ρ(f, g) ∈ [0,∞] thus it

need not be a metric on C(X,Y ), but it is a metric on Cρ(X,Y ) defined above. The space

Cρ(X,Y ) will be endowed with the topology induced by ρ.

The main result in Z-sets we will use is the homeomorphism extension Theorem, of

which the proof can be found in [vM02].

Theorem 2.4.1. Let E,F ∈ Z(Q) and let f be a homeomorphism from E onto F such

that ρ(f, 1E) < ε. Then f can be extended to a homeomorphism f : Q → Q such that

ρ(f , 1Q) < ε.

Zn-sets were defined by H. Torunczyk in [Tor78] in LCn-spaces, in which they

coincide with homotopical Zn sets. By definition, a closed subspace A of a space X is

a homotopical Zn-set if for every open cover U of X every map f : In → X can be

approximated by a map f : In → X \ A which is U -homotopic to f .

17

In their paper [BCK11], Banakh, Cauty and Karassev introduced the concept of

homological Zn-set analogous to the characterization of homotopical Zn-sets as locally

n-negligible sets in [Tor78]

Definition 2.4.2. A closed subset A ⊂ X is defined to be a G-homological Zn set, where

G is a coefficient group, if for any open set U ⊂ X and any k < n+1 the relative homology

group Hk(U,U \ A;G) = 0. A homological Zn-set is a Z homological Zn-set in X.

Here n can be any natural number or ∞. We will use the theory of homological

Zn-sets in Chapter 5, and state the Theorems we would use when appropriate. Here, we

would just state the question in section 13 in [BCK11], which we would answer in the

affirmative in Chapter 5 of this dissertation.

Question 2.4.1. Is a closed subset A ⊂ Q a homological Z∞-set in Q if A is a weakly

infinite-dimensional or a C-space?

2.5 Steenrod Homology, Inverse Limits and the First Derived Functor of anInverse Sequence.

Steenrod homology is the homology theory based on regular cycles, introduced by

N. Steenrod in the 1940s. For compact metric spaces, it is the unique homology theory

that satisfies all the seven Eilenberg-Steenrod axioms together with the cluster axiom

and invariance under relative homeomorphism, which we will refer to as nine axioms.

Steenrod homology theory is used in the setting of compact metric spaces having bad local

properties.

A locally finite q-chain of a simplicial complex K is a function defined over the

q-simplices of K, with values in a coefficient group G. Here, the simplicial complex K is a

union of countably many simplices.

Definition 2.5.1. A regular map of a complex K in X is a function f defined over the

vertices of K with values in X such that for any ε > 0, all but a finite number of simplices

have their vertices imaging onto sets of diameter < ε. A regular q-chain of X is a set of

18

three objects: a complex A, a regular map f of A in X, and a locally finite q-chain Cq of

A. If Cq is a q-cycle, (A, f, Cq) is called a regular q-cycle.

Now we define what it means to be homologous in the case of two regular q-chains.

Definition 2.5.2. Two regular q-cycles (A1, f1, C1q ) and (A2, f2, C

2q ) of X are homologous

if there exists a (q + 1)-chain (A, f, Cq+1) such that A1 and A2 are closed (not necessarily

disjoint) subcomplexes of A, f agrees with f1 on A1 and f2 on A2 and ∂Cq+1 = C1q − C2

q .

In modern terminology, the reduced Steenrod homology group Hstq (X) is the abelian

group of homology classes of regular (q + 1)-cycles in X. Observe the shift in dimension.

The use of Steenrod homology will be apparent in Chapter 5, where we use a

continuity property of Steenrod homology to show that the complement of a σ-compact

weakly infinite-dimensional subspace of Q is acyclic. At the heart of the proof we have

inverse sequences of spaces and their limits, which are defined below.

Definition 2.5.3. An inverse sequence of spaces is a collection of spaces Xi called

coordinate spaces together with maps fi : Xi+1 → Xi called bonding maps. The inverse

limit of the inverse sequence (Xi, fi) denoted by lim←−(Xi, fi)i is defined to be the following

subspace of the product of the coordinate spaces.

lim←−(Xi, fi) := {x ∈∞∏i=1

Xi : (∀i ∈ N)(fi(xi+1) = xi)}

Inverse sequences can be defined in any category, and in particular, the category of

topological spaces and the category of abelian groups have inverse limits defined, and the

inverse limit is unique upto homeomorphism and isomorphism, respectively.

For an inverse system of abelian groups (Ai, pi) where pi : Ai+1 → Ai are bonding

homomorphisms, the inverse limit can be defined as the kernel of d, where d :∏

iAi →∏iAi is defined by d(a1, a2, a3, . . . ) = (a1 − p1a2, a2 − p2a3, a3 − p3a4, . . . ). This clearly

coincides with the usual definition. We can now define the first derived functor of the

inverse sequence.

19

Definition 2.5.4. Let (Ai, pi) be an inverse sequence of abelian groups and d as above.

The first derived functor of this sequence, denoted by lim←−1Ai is the cokernel

∏iAi/d

∏iAi

of d.

For an inverse sequence of compact metric spaces, we have the following short exact

sequence, due to Milnor [Mil93].

Theorem 2.5.1. Let H be a homology theory satisfying the nine axioms, and (Xi) be

an inverse system of compact metric spaces with inverse limit X. Then there is an exact

sequence

0→ lim←−1{Hq+1(Xi)} → Hq(X)→ lim←−{Hq(Xi)}to0

for each integer q. A corresponding assertion holds if each space is replaced by a pair.

We end this section by one more proposition we will need throughout the dissertation.

Proposition 2.5.1. If (Ai, pi) is an inverse sequence of abelian groups such that the

bonding homomorphisms pi is surjective for all i, then lim←−1Ai = 0.

20

CHAPTER 3CONNECTEDNESS PROPERTIES OF THE COMPLETE NON-NEGATIVELY

CURVED METRICS ON THE PLANE

The spaces of Riemannian metrics with positive scalar curvature are subjects of

intensive study. The connectedness properties of such spaces on R2 were studied recently

by Belegradek and Hu in [BH16]. They proved that in the space Rk≥0(R2) of complete

Riemannian metrics of non negative curvature on the plane equipped with the topology of

Ck uniform convergence on compact sets, the complement Rk≥0(R2) \ X is connected for

every finite dimensional X. We note that the space Rk≥0(R2) is separable metric [BH16].

In this note we extend Belegradek-Hu’s result to the case of infinite dimensional spaces

X. We recall that infinite dimensional spaces split in two disjoint classes: strongly infinite

dimensional (like the Hilbert cube) and weakly infinite dimensional (like the union of

∪nIn). We prove Belegradek-Hu’s theorem for weakly infinite dimensional X. This

extension is final since strongly infinite dimensional spaces can separate the Hilbert cube.

We note that in [BH16] there is a similar connectedness result with finite dimensional

X for the moduli spaces Mk,c≥0(R2), i.e. the quotient space of Rk

≥0(R2) by the Diff(R2)-action

via pullback. In the case of moduli spaces we manage to extend their connectedness result

to the subsets X ⊂ Mk≥0(R2) with Haver’s property C (called C-spaces in [Eng95]). It is

known that the property C implies the weak infinite dimensionality [Eng95]. There is an

old open problem whether every weakly infinite dimensional compact metric space has

property C. For general spaces these two classes are different [AG78].

3.1 Infinite Dimensional Spaces

We denote the Hilbert cube by Q = [−1, 1]∞ = Π∞n=1In. The pseudo interior of Q is

the set s = (−1, 1)∞ and the pseudo boundary of Q is the set B(Q) = Q \ s. The faces of

Q are the sets W−i = {x ∈ Q|xi = −1} and W+

i = {x ∈ Q|xi = 1}. Every space under

consideration is a separable metric space.

We recall that a space X is continuum connected if every two points x, y ∈ X are

contained in a connected compact subset.

21

The following fact is well known. A proof can be found in [vM02], Corollary 3.7.5.

Lemma 3.1.1. Let X be a compact space, let {Ai, Bi : i ∈ Γ} be an essential family of

pairs of disjoint closed subsets of X and let n ∈ Γ. Suppose that S ⊆ X is such that S

meets every continuum from An to Bn. For each i ∈ Γ \ {n} let Ui and Vi be disjoint closed

neighborhoods of Ai and Bi respectively. Then {(Ui ∩ S, Vi ∩ S)} is essential in S.

Corollary 3.1.1. Let S ⊆ Q be such that S meets every continuum from W+1 to W−

1 .

Then S is strongly infinite dimensional.

A set A ⊆ Q is a Z-set in Q if for every open cover U of X there exists a map of Q

into Q \ A which is U -close to the identity. It should be noted that any face of Q and any

point of Q are Z-sets. The following Theorem is from [Cha76], Theorem 25.2.

Theorem 3.1.1. Let A,B ⊆ Q be Z-sets such that Sh(A) = Sh(B). Then Q \ A is

homeomorphic to Q \B.

We do not use the notion of shape in full generality. We just recall that for homotopy

equivalent spaces A and B we have Sh(A) = Sh(B).

Theorem 3.1.2. Let x, y ∈ Q \ S where S ⊆ Q be such that it intersects every continuum

from x to y. Then S is strongly infinite dimensional.

Proof. Let x, y ∈ Q \ S. Applying the Theorem 3.1.1 to W−1 ∪W+

1 and {x, y} we obtain

a homeomorphism f : Q \ (W−1 ∪W+

1 ) → Q \ {x, y}. In view of the minimality of the

one point compactification, this homeomorphism can be extended to a continuous map of

compactifications f : Q → Q with f(W−1 ) = x, f(W+

1 ) = y and f |Q\(W−1 ∪W

+1 ) = f . Note

that f−1(S) ⊂ Q \ (W−1 ∪W+

1 ) and f defines a homeomorphism between f−1(S) and S.

Since the image f(C) of a continuum C from W+1 to W−

1 is a continuum from x to y and

S ∩ f(C) = ∅, the intersection

f−1(S) ∩ C = f−1(S) ∩ C = f−1(S ∩ f(C)) = f−1(S ∩ f(C))

is not empty. Hence by Corollary 3.1.1, f−1(S) is strongly infinite dimensional, and so is

S.

22

Clearly, some strongly infinite dimensional compacta can separate the Hilbert cube.

Thus, Q×{0} separate the Hilbert cube Q×[−1, 1]. It could be that every strongly infinite

dimensional compactum has this property at least locally. In other words, it is unclear if

the converse to Theorem 3.1.2 holds true:

Question 3.1.1. Does every strongly infinite dimensional compact metric space admit an

embedding into the Hilbert cube Q that separates Q at least locally?

This question will be answered in the negative in Chapter 5.

3.1.0.1 Remark. The preceding theorem in the case when S is compact states precisely

that if S is a weakly infinite dimensional subspace of Q, then Q \ S is path connected.

Definition 3.1.1. A topological space X has property C (is a C-space) if for every

sequence G1,G2, . . . of open covers of X, there exists a sequence H1,H2, . . . of families

of pairwise disjoint open subsets of X such that for i = 1, 2, . . . each member of Hi is

contained in a member of Gi and the union∪∞

i=1Hi is a cover of X.

The following is a theorem on dimension lowering mappings, the proof can be found

in [Eng95] (Chapter 6.3, Theorem 9).

Theorem 3.1.3. If f : X → Y is a closed mapping of space X to C space Y such that

for every y ∈ Y the fibre f−1(y) is weakly infinite dimensional, then X is weakly infinite

dimensional.

If one uses weakly infinite dimensional spaces instead of C spaces the situation is less

clear even in the case of compact Y .

Question 3.1.2. Suppose that a Lie group G admits a free action by isometries on a

metric space X with compact metric weakly infinite dimensional orbit space X/G. Does it

follow that X is weakly infinite dimensional?

This is true for compact Lie groups in view of the slice theorem [Bre72]. It also true

for countable discrete groups [Pol83].

23

3.2 Applications

Now we proceed to generalize two theorems by Belegradek and Hu. We use the

following result proven in [BH16], Theorem 1.3.

Theorem 3.2.1. If K is a countable subset of Rk≥0(R2) and X is a separable metric space,

then for any distinct points x1, x2 ∈ X and any distinct metrics g1, g2 ∈ Rk≥0(R2) \K there

is an embedding of X into Rk≥0(R2) \K that takes x1, x2 to g1, g2 respectively.

Here is our extension of the first Belegradek-Hu theorem.

Theorem 3.2.2. The complement of every weakly infinite dimensional subspace S of

Rk≥0(R2) is continuum connected. If S is closed, Rk

≥0(R2) \ S is path connected.

Proof. Let S be a weakly infinite dimensional subspace of Rk≥0(R2). Fix two metrics

g1, g2 ∈ Rk≥0(R2). Theorem 3.2.1 implies that g1, g2 lies in a subspace Q of Rk

≥0(R2) that is

homeomorphic to Q. Since S∩Q is at most weakly infinite dimensional, Q\S is continuum

connected by Theorem 3.1.2. Then g1, g2 lie in a continuum in Q that is disjoint from S.

Hence Rk≥0(R2) \ S is continuum connected. If S is closed, from Remark 3.1.0.1, we can

conclude that Rk≥0(R2) \ S is path connected.

In view of Theorem 3.1.2 we can state that if a subset S ⊆ ℓ2, the separable Hilbert

space, then S is strongly infinite dimensional. From this fact we derive the following

Theorem 3.2.3. The complement of every weakly infinite dimensional subspace S of

R∞≥0(R2) is locally connected. If S is closed, R∞≥0(R2) \ S is locally path connected.

Proof. Given C∞ topology, the space R∞≥0(R2) is homeomorphic to ℓ2, the separable

Hilbert space [BH16]. Let x ∈ ℓ2, then there is a neighborhood U of x homeomorphic to

ℓ2, and the set U \ S is connected, and path connected if S is closed.

We do not know if the space Rk,c≥0(R2) locally path connected for k <∞.

We prove a similar to Theorem 3.2.2 result for the associated moduli space Mk,c≥0(R2),

when the subspace removed is a Hausdorff space having the property C. This is a

generalization of another Belegradek-Hu theorem.

24

Theorem 3.2.4. If S ⊂ Mk,c≥0(R2) is a closed Hausdorff space with property C then

Mk,c≥0(R2) \ S is path connected.

Proof. Denote by S1 the set of smooth subharmonic functions with α(u) ≤ 1 where

α(u) = limr→∞

sup{u(z) : |z| = r}log r

.

Note that S1 is closed in the Frechét space C∞(R2), it is not locally compact, and is equal

to the set of smooth subharmonic functions u such that the metric e−2ug0 is complete,

where g0 is the standard Euclidean metric [BH16]. Let q : S1 → Mk,c≥0(R2) denote the

continuous surjection sending u to the isometry class of e−2ug0. Let S = q−1(S). Fix two

points g1, g2 ∈ Mk,c≥0(R2) \ S. which are q images of u1, u2 in S, respectively. By Theorem

3.2.1 we may assume that u1, u2 lie in an embedded copy Q of Hilbert cube. It suffices to

show that Q \ S is path connected.

The set Q ∩ S is compact, hence q, the restriction of q to Q ∩ S is a continuous

surjection. The map q : Q ∩ S → q(Q) ∩ S is a map between compact spaces, and

in particular, it is a closed map. The set q(Q ∩ S) has property C. We have each fiber

q−1(y) to be finite dimensional [BH16], and hence Q ∩ S is weakly infinite dimensional

by Theorem 3.1.3. Therefore, Q \ S is path connected by the discussion following

Theorem 3.1.2, and so is Mk≥0(R2) \ S .

It should be noted that the Hausdorff condition is essential in Theorem 3.2.4. If S

is not Hausdorff, the map q above ceases to be a map between compact metric spaces.

In the original paper [BH16] the authors did not require the Hausdorff condition in the

formulation of their Theorem 1.6 when S is finite dimensional.

Proposition 3.2.1. Suppose that Problem 3.1.2 has an affirmative answer for the Lie

group G = conf(R2). Then in Theorem 3.2.4 one can replace the property C condition to

the weak infinite dimensionality of S.

25

Proof. We use the same setting as in the proof of theorem 3.2.4. As stated in the proof of

theorem 1.6 in [BH16], two functions u and v of S1 lie in the same isometry class if and

only if v = u ◦ ψ − log |a| for some ψ ∈ conf(R2). i.e, they lie in the same orbit under the

action of conf(R2) on the space C∞(R2) given by (u, ψ) 7→ u ◦ ψ − log |a|. The subspace

S1 of C∞(R2) is invariant under this action. Let π : S1 → S1/conf(R2) be the projection

onto the orbit space of this action. Also we note that the action of conf(R2) on S1 is a free

action by isometries.

Let S be a closed, weakly infinite dimensional Hausdorff subset of Mk≥0(R2). Let f

and g be two elements in the complement of S. Then there are functions u and v mapping

to f and g respectively by q. As noted above, u and v lend in the same class if and only if

v = u ◦ ψ − log |a| for some ψ ∈ conf(R2). Theorem 1.4 of [BH16] shows that u and v lie in

an embedded copy Q of Hilbert cube. Denote q−1(S) = S. It suffices to prove that Q ∩ S

is weakly infinite dimensional, so we would have a path joining u and v in Q \ S, which

transforms to a path joining g and f in Mk≥0(R2) \ S.

The set S is closed hence Q ∩ S is compact. So the restriction of q to Q ∩ S is a

continuous surjection q : Q ∩ S → q(Q) ∩ S of compact separable metric spaces. Define the

map η : S1/conf(R2) by uG 7→ u∗, the isometry class of e−2ug0. This map is injective by

definition and the diagram

S1π //

q !!CCC

CCCC

C S1/conf(R2)

ηxxrrr

rrrrrr

r

Mk≥0

commutes. Let Y be the η preimage of q(Q) ∩ S in S1/conf(R2). The action restricted

to the preimage π−1(Y ) of S1 is an action of conf(R2) on π−1(Y ) with orbit space Y ,

and Q ∩ S ⊆ π−1(Y ), and the set Y is weakly infinite dimensional. Assuming that the

Problem 3.1.2 has an affirmative answer for the Lie group G = C∗ ⋊ C, we can say that

Problem 3.1.2 has an affirmative answer for the Lie group conf(R2). By this, we can

26

conclude that π−1(Y ) is weakly infinite dimensional. Hence q(Q) ∩ S is weakly infinite

dimensional, therefore Mk≥0(R2) \ S is path connected.

27

CHAPTER 4ACYCLICITY OF COMPLEMENTS OF WEAKLY INFINITE-DIMENSIONAL SPACES

IN HILBERT CUBE

4.1 Preliminaries

The content in this chapter was motivated in part by a paper by Belegradek and Hu

[BH16] about connectedness properties of the space Rk≥0(R2) of non-negative curvature

metrics on R2 where k stands for Ck topology. In particular, they proved that the

complement Rk≥0(R2) \ X is continuum connected for every finite dimensional X. In

[Ama17] this result was extended to weakly infinite dimensional spaces X. The main

topological idea of these results is that a weakly infinite dimensional compact set cannot

separate the Hilbert cube.

Banakh and Zarichnyi posted in [Pea07] a problem (Problem Q1053) whether the

complement Q \ X to a weakly infinite dimensional compactum X in the Hilbert cube

Q is acyclic. They were motivated by the facts that Q \ X is acyclic in the case when

X is finite dimensional [Kro74] and when it is countably dimensional (see [BCK11]). It

turns out that the problem was already answered affirmatively by Garity and Wright in

the 80s [GW87]. It follows from Theorem 4.5 [GW87] which states that finite codimension

closed subsets of the Hilbert cube are strongly infinite dimensional. In this chapter

we will find some sufficient cohomological conditions on compact metric spaces for the

acyclicity of the complement in the Hilbert cube which give an alternative solution for the

Banakh-Zarichnyi problem.

We introduce a cohomological version of the concept of strongly infinite dimensional

spaces. We call a spaces which is not cohomologically strongly infinite dimensional as

cohomologically weakly infinite-dimensional spaces. The main result of this chapter is an

acyclicity statement for the complement of compacta which are cohomologically weakly

infinite-dimensional subsets in the Hilbert cube.

28

4.2 Main Theorem

4.2.1 Alexander Duality

We recall that for a compact set X ⊂ Sn there is a natural isomorphism ADn :

Hn−k−1(X) → Hk(Sn \ X) called the Alexander duality. The naturality means that for a

closed subset Y ⊂ X there is a commutative diagram (1):

Hn−k−1(X)ADn−−−→ Hk(S

n \X)y yHn−k−1(Y )

ADn−−−→ Hk(Sn \ Y ).

Here we use the singular homology groups and the Čech cohomology groups. We note the

Alexander Duality commutes with the suspension isomorphism s in the diagram (2):

Hn−k−1(X)ADn−−−→ Hk(S

n \X)

s

y ∼=y

Hn−k(ΣX)ADn+1−−−−→ Hk(S

n+1 \ ΣX).

This the diagram can be viewed as a special case of the Spanier-Whitehead duality [Whi70].

Since Bn/∂Bn = Sn, the Alexander Duality in the n-sphere Sn and its property can

be restated verbatim for the n-ball Bn for subspaces X ⊂ Bn with X ∩ ∂Bn = ∅ in terms

of relative cohomology groups

Hn−k−1(X,X ∩ ∂Bn)AD−→ Hk(IntB

n \X).

Note that for a pointed space X the reduced suspension ΣX is the quotient space

(X × I)/(X × ∂I ∪ x0 × I) and the suspension isomorphism s : H i(X, x0) → H i+1(ΣX) =

H i+1(X × I,X × ∂I ∪ x0 × I) is defined by a cross product with the fundamental class

ϕ ∈ H1(I, ∂I), s(α) = α × ϕ. Note that α × ϕ = α∗ ∪ ϕ∗ in H i+1(X × I,X × ∂I ∪ x0 × I)

where α∗ ∈ H i(X × I, x0 × I) is the image of α under the induced homomorphism for the

projection X × I → X and ϕ∗ ∈ H1(X × I,X × ∂I) is the image of ϕ under the induced

homomorphism defined by the projection X × I → I. Thus in view of an isomorphism

H∗(IntBn \X) → H∗(B

n \X), the commutative diagram (2) stated for Bn turns into the

29

following (2′)

Hn−k−1(X, ∂X)ADn−−−→ Hk(B

n \X)

−∪ϕ∗y ∼=

yHn−k(X × I, ∂X × I ∪X × ∂I) ADn+1−−−−→ Hk(B

n+1 \ (X × I))

where ∂X = X ∩ ∂In.

4.2.2 Strongly Infinite Dimensional Spaces

First, we recall some classical definitions which are due to Alexandroff [AP73]. A map

f : X → In to the n-dimensional cube is called essential if it cannot be deformed to ∂In

through the maps of pairs (In, f−1(∂In)) → (In, ∂In). A map f : X → Q =∏∞

i=1 I

to the Hilbert cube is called essential if the composition pn ◦ f : X → In is essential

for every projection pn : Q → In onto the factor. A compact space X is called strongly

infinite dimensional if it admits an essential map onto the Hilbert cube. A space which is

not strongly infinite dimensional is called weakly infinite dimensional. Observe that in this

definition, finite-dimensional spaces are also weakly infinite-dimensional.

We call a relative cohomology class α ∈ Hk(X,A) pure relative if j∗(α) = 0 for the

inclusion of pairs homomorphism j∗ : Hk(X,A) → Hk(X). This means that α = δ(β) for

β ∈ Hk−1(A). In the case when k = 1 the cohomology class β can be represented by a map

f ′ : A→ S0 = ∂I. Let f : X → I be a continuous extension of f ′. Then α = f ∗(ϕ) for the

map f : (X,A)→ (I, ∂I) of pairs where ϕ is the fundamental class.

4.2.2.1 Proposition. Suppose that for a compact metric space X there exists a sequence

of pure relative classes αi ∈ H1(X,Ai), i = 1, 2, . . . , Ai ⊂Cl X such that the cup-product

α1 ∪ αn ∪ · · · ∪ αn = 0 for all n. Then X is strongly infinite dimensional.

Proof. Let fi : (X,Ai) → (I, ∂I) be representing αi maps. Note that the homomorphism

f ∗n : Hn(In, ∂In)→ Hn(X,∪ni=1Ai) induced by the map fn = (f1, f2, . . . , fn) : X → In takes

the product of the fundamental classes ϕ1 ∪ · · · ∪ ϕn to α1 ∪ αn ∪ · · · ∪ αn = 0. Therefore,

30

fn is essential. Thus, the map

f = (f1, f2, . . . ) : X →∞∏i=1

I = Q

is essential.

We call a space X cohomologically strongly infinite dimensional if there is a sequence

of pure relative classes αi ∈ Hki(X,Ai), ki > 0, i = 1, 2, . . . , Ai ⊂Cl X such that the

cup-product α1 ∪ αn ∪ · · · ∪ αn = 0 for all n. Compacta which are not cohomologically

strongly infinite dimensional will be called as cohomologically weakly infinite dimensional

compacta or relative cohomology weak compacta. One can take any coefficient ring R to

define cohomologically weakly infinite-dimensional compacta with respect to R.

Thus, the class of cohomologically weakly infinite-dimensional compacta contains

weakly infinite dimensional compacta (Proposition 4.3.0.2) as well as all compacta with

finite cohomological dimension.

4.2.3 Proof of the Main Theorem

4.2.3.1 Proposition. Let X ⊂ Q be a closed subset and let a ∈ Hi(Q \ X) be a nonzero

element for some i. Then there exists an n ∈ N such that the image an of a under the

inclusion homomorphism

Hi(Q \X)→ Hi(Q \ p−1n pn(X))

is nontrivial.

Proof. This follows from the fact that

Hi(Q \X) = lim→Hi(Q \ p−1n pn(X)).

4.2.3.2 Theorem. Let X ⊂ Q be a cohomologically weakly infinite-dimensional compact

subset with respect to a ring with unit R. Then Hi(Q \X;R) = 0 for all i.

Proof. We assume everywhere below that the coefficients are taken in R.

31

We note that Q \ p−1n pn(X) = (In \ pn(X)) × Qn where Qn is the Hilbert cube

complementary to In in Q = In ×Qn.

Assume the contrary, Hi(Q \ X) = 0. Let a ∈ Hi(Q \ X) be a nonzero element.

We use the notation Xk = pk(X). By Proposition 4.2.3.1 there is n such that ak =

0, ak ∈ Hi((Ik \ Xk) × Qk) for all k ≥ n. Thus, we may assume that ak lives in

Hi((Ik \Xk)× {0}) ∼= Hi((I

k \Xk)×Qk).

Let βk ∈ Hk−i−1(Xk, Xk ∩ ∂Ik) be dual to ak for the Alexander duality applied in

(Ik, ∂Ik). By the suspension isomorphism for the Alexander duality (see diagram (2′)), ak

is dual in Ik+1 to β∗k ∪ ϕ∗ where β∗k is the image of βk under the homomorphism induced by

the projection Xk × I → Xk and ϕ∗ is the image of the fundamental class ϕ ∈ H1(I, ∂I)

under the homomorphism induced by the projection Xk × I → I.

By the naturality of the Alexander duality (we use the Bn-version of the diagram (1))

applied to the inclusion Xk+1 ⊂ Xk × I we obtain that ak+1 is dual to β′k ∪ ϕk+1 where

β′k+1 is the restriction of β∗k to Xk+1 and ϕk+1 is the restriction of ϕ∗ to Xk+1. Note that

β′k+1 = (qk+1k )∗(βk) where qk+1

k : Xk+1 → Xk is the projection pk+1k : Ik+1 → Ik restricted to

Xk+1. Thus, βk+1 = (qk+1k )∗(βk) ∪ ϕk+1.

By induction on m we obtain

(∗) βn+m = (qn+mn )∗(βn) ∪ ψ1 ∪ · · · ∪ ψm

where ψj = (qn+mn+j )

∗(ϕn+j), j = 1, . . . ,m.

We define a sequence αj = (qn+j)∗(ϕn+j) of relative cohomology classes on X where

qn = pn|X : X → Xn. Since X = lim←Xr, for the Čech cohomology we have

Hℓ(X, q−1n+k(A)) = lim→{Hℓ(Xn+m, (q

n+mn+k )

−1(A))}m≥k

for any A ⊂Cl Xn+k. Hence for fixed k the product α1 ∪ · · · ∪ αk is defined by the thread

{(qn+mn+1 )

∗(ϕn+1) ∪ · · · ∪ (qn+mn+k )

∗(ϕn+k)}m≥k

32

in the above direct limit. Since βn+m = 0, in view of (∗) we obtain that

(qn+mn+1 )

∗(ϕn+1) ∪ · · · ∪ (qn+mn+k )

∗(ϕn+k) = ψ1 ∪ · · · ∪ ψk = 0.

Hence, α1 ∪ · · · ∪ αk = 0 for all k. Therefore, X is cohomologically strongly infinite

dimensional. This contradicts to the assumption.

4.2.3.3 Corollary. Let X ⊂ Q be a compactum with finite cohomological dimension. Then

Hi(Q \X) = 0 for all i.

We recall that the cohomological dimension of a space X with integers as the

coefficients is defined as

dimZX = sup{n | ∃ A ⊂Cl X with Hn(X,A) = 0}.

4.2.3.4 Remark. In view of the above corollary, the answer to question 3.1.1 is in the

negative. If X ⊂ Q is a strongly infinite dimensional subset with dimZ < ∞ (such spaces

exist by [Dra90]) then it does not admit an embedding into Q that separates Q, even

locally.

4.2.3.5 Corollary. Let X ⊂ Q be weakly infinite dimensional compactum. Then

Hi(Q \X) = 0 for all i.

Proof. This is rather a corollary of the proof of Theorem 4.2.3.2 where we constructed a

nontrivial infinite product of pure relative 1-dimensional cohomology classes α1 ∪ α2 ∪ . . . .

on X. Then the corollary holds true in view of Proposition 4.2.2.1.

4.3 On Weakly Infinite Dimensional Compacta

Let ϕ ∈ Hm−1(K(G,m − 1);G) be the fundamental class. Let ϕ denote the image

δ(ϕ) ∈ Hm(Cone(K(G, k − 1)), K(G, k − 1);G) of the fundamental class under the

connecting homomorphism in the exact sequence of pair.

33

4.3.0.1 Proposition. For any pure relative cohomology class α ∈ Hm(X,A;G) there is a

map of pairs

f : (X,A)→ (Cone(K(G,m− 1)), K(G,m− 1))

such that α = f ∗(ϕ).

Proof. From the exact sequence of the pair (X,A) it follows that α = δ(β) for some β ∈

Hm−1(A;G). Let g : A→ K(G,m− 1) be a map representing β. Since Cone(K(G,m− 1))

is an AR, there is an extension f : X → Cone(K(G,m − 1)). Then the commutativity

of the diagram formed by f ∗ and the exact sequence of pairs implies δg∗(ϕ) = f ∗δ(ϕ) =

f ∗(ϕ).

Corollary 4.2.3.5 can be derived formally from the following:

4.3.0.2 Theorem. Every weakly infinite dimensional compactum is cohomologically

weakly infinite-dimensional for any coefficient ring R.

Proof. Let αi ∈ Hki(X,Ai;R) be a sequence of pure relative classes with a nontrivial

infinite cup-product. We may assume that ki > 1. We denote Ki = K(R, ki). By

Proposition 4.3.0.1 there are maps fi : (X,Ai)→ (Cone(Ki), Ki) representing αi in a sense

that f ∗i (ϕi) = αi. Let

qi = Cone(c) : Cone(Ki)→ Cone(pt) = [0, 1]

be the cone over the constant map c : Ki → {0}. We show that g = (q1f1, q2f2, . . . ) : X →∏∞i=1 I is essential. Assume that for some n the map

gn = (q1f1, q2f2, . . . , qnfn) : X →n∏

i=1

I = In

is inessential. Then there is a deformation H of gn to a map h : X → Nϵ(∂In) rel

g−1n (Nϵ(∂In)) to the ϵ-neighborhood of the boundary ∂In. Since the map q1 × · · · × qn is a

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fibration over Int In, there is a lift H of H that deforms

fn = (f1, . . . , fn) : X →n∏

i=1

Cone(Ki)

to an ϵ-neighborhood of M = (q1 × · · · × qn)−1(∂In). Since M is an ANR we may assume

that fn can be deformed to M rel f−1n (M). Since α1 ∪ · · · ∪ αn = f ∗n(ϕ1 ∪ · · · ∪ ϕn), to

complete the proof it suffices to show that the restriction of ϕ1 ∪ · · · ∪ ϕn to M is zero.

We denote

D =n∏

i=1

Cone(Ki) \n∏

i=1

Ocone(Ki)

where Ocone(K) stands for the open cone. Since∏Cone(Ki) is contractible, ϕ1∪· · ·∪ϕn =

δω for some ω ∈ Hk1+···+kn−1(D;R).

We show that D is homeomorphic to K1 ∗ · · · ∗Kn. Note that

n∏i=1

Cone(Ki) = {t1x1 + · · ·+ tnxn | xi ∈ Ki, ti ∈ [0, 1]}

andn∏

i=1

Ocone(Ki) = {t1x1 + · · ·+ tnxn | xi ∈ Ki, ti ∈ [0, 1)}

with the convention that 0x = 0x′. Then the set D consists of all t1x1 + · · · + tnxn such

that ti = 1 for some i. We recall that

K1 ∗ · · · ∗Kn = {t1x1 + · · ·+ tnxn | xi ∈ Ki, ti ∈ [0, 1]∑

ti = 1}

with the same convention. Clearly, the projection of of the unit sphere in Rn for the

max{|xi|} norm onto the unit sphere for the |x1| + · · · + |xn| norm is bijective. Thus, the

renormalizing map ρ : D → K1 ∗ · · · ∗Kn,

ρ(t1x1 + · · ·+ tnxn) =t1∑tix1 + · · ·+

tn∑tixn

is bijective and, hence, is a homeomorphism.

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The space M can be defined as follows:

M = {t1x1 + · · ·+ tnxn | xi ∈ Ki, ti ∈ [0, 1] ∃i : ti ∈ 0, 1}.

Therefore, M = D ∪ L where

L = {t1x1 + · · ·+ tnxn | xi ∈ Ki, ti ∈ [0, 1] ∃i : ti = 0}.

We note that L is the cone over

L∗ = {t1x1 + · · ·+ tnxn ∈ K1 ∗ · · · ∗Kn | ∃i : ti = 0}.

Thus, the space M is obtained by attaching to D the cone over L∗.

We show that the inclusion L∗ → D is null-homotopic. Note that

L∗ =n∪

i=1

K1 ∗ · · · ∗ Ki ∗ · · · ∗Kn

where Ki means that the i-factor is missing. Fix points ei ∈ Ki and the straight-line

deformation

hit : K1 ∗ · · · ∗ Ki ∗ · · · ∗Kn → K1 ∗ · · · ∗Kn

to the point ei. We recall that the reduced joint product is defined as follows

K1∗ · · · ∗Kn = {t1x1 + · · ·+ tnxn | xi ∈ Ki, ti ∈ [0, 1],∑

ti = 1}

with convention that 0x = 0x′ and ei = ej. Let q : K1 ∗ · · · ∗ Kn → K1∗ · · · ∗Kn be the

quotient map. Note that q has contractible fibers and hence is a homotopy equivalence.

Then x0 = q(∆n−1) is the base point in the reduced joint product where the simplex ∆n−1

is spanned by e1, . . . , en.

The deformation hit defines a deformation hit of K1∗ · · · ∗Ki∗ · · · ∗Kn to the base

point∑tiei in the reduced joint product K1∗ · · · ∗Kn. We note that for any i < j

the deformations hit and hjt agree on the common part K1∗ · · · ∗Ki∗ · · · ∗Kj ∗ · · · ∗Kn.

Therefore, the union (∪hit) ◦ q|L∗ is a well-defined deformation of L∗ to the base point in

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the reduced joint product where q : K1 ∗ · · · ∗Kn → K1∗ · · · ∗Kn is quotient map. Since q

is a homotopy equivalence, L∗ is null-homotopic in K1 ∗ · · · ∗Kn.

Since there exist strongly infinite dimensional compacta with finite cohomological

dimension (see [Dra90],[DW04]) the converse to Theorem 4.3.0.2 does not hold true.

The following can be easily derived from from the definition of the cup product for

the singular cohomology and the definition of the Čech cohomology.

4.3.0.3 Proposition. Let A,A′ ⊂ Y , Y,B,B′ ⊂ X be closed subsets with B ∩ Y = A and

B′ ∩ Y = A′. Then for any ring R there is a commutative diagram generated by inclusions

and the cup product

Hk(X,B;R)×H l(X,B′;R)∪−−−→ Hk+l(X,B ∪B′;R)y y

Hk(Y,A;R)×H l(Y,A′;R)∪−−−→ Hk+l(Y,A ∪ A′;R).

4.3.0.4 Theorem. Let Y be a closed subset of a cohomologically weakly infinite-

dimensional compactum X over a ring R. Then Y is cohomologically weakly infinite-

dimensional over R.

Proof. Assume that Y is cohomologically strongly infinite dimensional. Let αi ∈

Hki(Y,Ai;R) be pure relative cohomology classes with nonzero product. Let

fi : (Y,Ai)→ (Cone(K(R, ki − 1)), K(R, ki − 1))

be a map from Proposition 4.3.0.1 representing αi. Let

gi : X → Cone(K(R, ki − 1))

be an extension of fi and let Bi = g−1i (K(R, ki−1)). We consider βi = g∗i (ϕi) ∈ Hki(X,Bi).

Note that αi = ξ∗i (βi) where ξi : (Y,Ai) → (X,Bi) is the inclusion. By induction,

Proposition 4.3.0.3, and the fact αi ∪ · · · ∪ αn = 0 we obtain that β1 ∪ · · · ∪ βn = 0 for all

n. Then we can conclude that X is cohomologically strongly infinite dimensional.

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4.4 Applications

Let Σ denote the linear span of the standard Hilbert cube Q in the Hilbert space

ℓ2 and let Σω denote the product of countably many copies of Σ. The following is

well-known:

4.4.0.1 Proposition. Let H be either ℓ2 or Σω. Then for any closed subset A ⊂ Q of the

Hilbert cube any embedding ϕ : A→ H can be extended to an embedding ϕ : Q→ H.

Let Rγ(M) denote the space of all Riemannian metrics on a manifold M with the Cγ

topology. For the definition of such topology for finite γ which are not integers we refer

to [BB17]. The following theorem was proved in [BB17]:

4.4.0.2 Theorem. For a connected manifold M the space Rγ(M) is homeomorphic to Σω

when γ <∞ and R∞(M) is homeomorphic to ℓ2.

The space Rγ≥0(R2) is homeomorphic to Σω when γ <∞ and it is not an integer. The

space R∞≥0(R2) is homeomorphic to ℓ2.

This theorem with our main result implies the following

4.4.0.3 Corollary. Let X ⊂ Rγ(M) be a closed weakly infinite dimensional subset for a

connected manifold M . Then Rγ(M) \X is acyclic.

Let X ⊂ Rγ≥0(R2) be a closed weakly infinite dimensional subset when γ = ∞ or

γ <∞ and it is not an integer. Then the complement Rγ≥0(R2) \X is acyclic.

Proof. In either of the cases we denote the space of metrics by H. Let f : K → H \X be

a singular cycle. By Proposition 4.4.0.1 there is an embedding of the Hilbert cube Q ⊂ H

such that f(K) ⊂ Q. By Theorem 4.2.3.2 f is homologous to zero in Q \X and, hence, in

H \X.

4.5 Weakly Infinite-dimensional Compacta in Compact Q-manifolds.

The purpose of this section is to extend the corollary 4.2.3.5 to a theorem about

Q-manifolds. We shall show that removing a compact, weakly infinite-dimensional subset

will not change the homology type of a compact Q-manifold. Hilbert cube manifolds, or

Q-manifolds, are separable metric manifolds that are modeled on Q. It is just a locally

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compact metric space which is locally homeomorphic to Q × [0, 1). Some examples of Q

manifolds are open subsets of Q and Mn ×Q where Mn is a finite dimensional topological

n-manifold [Cha76]. The following theorem presents a generalization of corollary 4.2.3.5 to

compact Q-manifolds.

Theorem 4.5.1. Let K be a Q-manifold and X ⊂ K be a weakly infinite-dimensional

compactum. Then Hi(K \X) = Hi(K) for all i.

Proof. K is a finite union∪n

i=1Qi of copies of Hilbert cubes {Qi}ni=1, with each non-empty

intersection Qi ∩ Qi is homeomorphic to Q [Cha76]. We prove the statement by induction

on n.

When n = 1, the assertion is true by theorem 4.2.3.4. Assume the assertion is true for

any Q-manifold that can be expressed as a union of n copies of Q. Let L be a Q-manifold

that can be expressed as a union of n + 1 copies of Q, so that L = Q1 ∪Q2 ∪ · · · ∪Qn+1 =

K ∪Q for some Q manifold K and K ∩Qn+1 is homeomorphic to Q.

Let X ⊂ L be a closed, weakly infinite-dimensional subset. Mayer-Vietoris sequence

for L \X and L fits into the following commutative diagram, with vertical homomorphisms

induced by inclusion L \X → L.

Hi((K ∩Qn+1) \X)

��

// Hi(K \X)⊕ Hi(Qn+1 \X)

��

// Hi(L \X)

��

// Hi−1((K ∩Qn+1) \X)

��

// Hi−1(K \X)⊕ Hi(Qn+1 \X)

��

Hi((K ∩Qn+1)) // Hi(K)⊕ Hi(Qn+1) // Hi(L) // Hi−1((K ∩Qn+1)) // Hi−1(K)⊕ Hi−1(Qn+1)

The first and the fourth arrows are isomorphisms by theorem 4.2.3.4, the second and

the fifth arrows are isomorphisms by the induction hypothesis and by Theorem 4.2.3.4.

Hence, by the five lemma, the middle arrow is an isomorphism. So we get Hi(L \ X) =

Hi(L) for all i ≥ 0.

4.6 Zero-dimensional Steenrod Acyclicity of Complement of a σ-compactSubset in the Hilbert Cube

In Chapter 3, we proved that the complement of any weakly infinite dimensional

subset in the Hilbert cube is continuum connected. In this section we will prove that the

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zero-dimensional Steenrod homology of the complement of a σ compact subset in the

Hilbert cube is trivial. This is a stronger result for σ-compact spaces compared to theorem

3.3.3 as there are continua with non-trivial Steenrod homology [Ste40]. For compact

metric spaces, Steenrod homology is the unique ordinary homology theory that satisfies

the seven Eilenberg-Steenrod axioms together with the cluster axiom and invariance

under relative homeomorphism [Mil93]. We will denote Steenrod homology of a compact

metric space X by Hst∗ (X). It should be noted that the arguments in the proof of theorem

4.2.3.4. does not change if we replace singular homology by Steenrod homology, as the

Alexander duality theorem can be restated verbatim for Steenrod homology, and the

rest of the argument follows. For non compact metric spaces, the Steenrod homology is

defined to be the direct limit of Steeenrod homology groups taken over all compact subsets

[Mil93].

Theorem 4.6.1. Let X =∪

iXi be σ-compact weakly infinite-dimensional subset of Q.

Then Hst0 (Q \X) is trivial.

Proof. Let S = {p, q} ⊂ Q \X define a Steenrod 0-cycle in the complement of X. We shall

show that S bounds in a compact subset in Q \X.

Since X1 is a weakly infinite-dimensional compactum, Q \X1 is path connected, hence

there is a path J1 in Q \ X connecting p and q. Since X1 and J1 are both compact, there

exists ε1 > 0 such that a closed ε1 neighborhood of J1, denoted by K1, that does not meet

X1. Observe that K1 is a homeomorphic image of the Hilbert cube.

To continue the construction, we have, by theorem 4.2.3.4 again, that K1 \X2 is path

connected, hence a path J2 in K1 \X2 connects p and q, and there is an ε2 > 0 such that

a closed ε2 neighborhood of J2 in K1 \ X2 does not meet X2. Let this neighborhood be

K2, yet again a homeomorphic image of Q. Continuing this construction, we get a nested

sequence Ki of copies of Q, with lim←−Ki =∩

iKi = K non empty, containing p and q, and

in the complement of X. By the Milnor’s short exact sequence for Steenrod homology, we

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have,

0→ lim←−1(Hst

1 (Ki))→ Hst0 (K)→ lim←−(H

st0 (Ki))→ 0

Where the first and the third terms are zero, hence Hst0 (K) is trivial. Which imply S

bounds in K, hence in Q \X.

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BIOGRAPHICAL SKETCH

Ashwini Amarasinghe grew up in Colombo, Sri Lanka. He reveived a bachelor’s

degree in mathematics and physics from the University of Peradeniya, Sri Lanka in

2011. He attended University of Florida in Gainesville to pursue his doctoral degree in

mathematics.

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