On Acyclicity Properties of Complements of Subsets in the Hilbert Cube

ON ACYCLICITY PROPERTIES OF COMPLEMENTS OF SUBSETS IN THE HILBERT CUBE

By ASHWINI K. AMARASINGHE

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2017 ⃝c 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 CONTENTS page ACKNOWLEDGMENTS ...... 4 ABSTRACT ...... 6

CHAPTER 1 INTRODUCTION ...... 8 2 BACKGROUND ...... 12 2.1 Dimension Theory ...... 12 2.2 Space of Complete Non-negatively Curved Metrics on the Plane...... 14 2.3 Cohomological Dimension of Compact Metric Spaces...... 16 2.4 Z sets, Zn sets and homological Zn sets in Q ...... 17 2.5 Steenrod Homology, Inverse Limits and the First Derived Functor of an Inverse Sequence...... 18 3 CONNECTEDNESS PROPERTIES OF THE COMPLETE NON-NEGATIVELY CURVED METRICS ON THE PLANE ...... 21 3.1 Infinite Dimensional Spaces ...... 21 3.2 Applications ...... 24 4 ACYCLICITY OF COMPLEMENTS OF WEAKLY INFINITE-DIMENSIONAL SPACES IN HILBERT CUBE ...... 28 4.1 Preliminaries ...... 28 4.2 Main Theorem ...... 29 4.2.1 Alexander Duality ...... 29 4.2.2 Strongly Infinite Dimensional Spaces ...... 30 4.2.3 Proof of the Main Theorem ...... 31 4.3 On Weakly Infinite Dimensional Compacta ...... 33 4.4 Applications ...... 38 4.5 Weakly Infinite-dimensional Compacta in Compact Q-manifolds...... 38 4.6 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 School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ON ACYCLICITY PROPERTIES OF COMPLEMENTS OF SUBSETS IN THE HILBERT CUBE By Ashwini K. Amarasinghe August 2017 Chair: Alexander N. Dranishnikov Major: 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 1 INTRODUCTION 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,

Rk R2 non-negatively curved Riemannian metrics on the plane, ≥0( ) cannot be separated by removing a ﬁnite-dimensional subspace. We shall show, by using a theorem on

Rk R2 continuum-wise separation by Mazurkiewicz [Eng95], that the space ≥0( ) 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 R2 Rk R2 R2 ≥0( ) is defined as the quotient space of ≥0( ) under the action of Diff( ) 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

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

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

k α1 ∪ αn ∪ · · · ∪ αn ≠ 0 for all n. We call a relative cohomology class α ∈ H (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 ∈ H˜ k−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

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

→ 1 ˜ st → ˜ st → st → 0 ←−lim Hk+1(Zi) Hk (Z) ←−lim Hk (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 2 BACKGROUND 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 Li ≠ ∅. If τ is not i∈Γ 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 dim X ∈ {−1, 0, 1,... } ∪ {∞} by

dim X = −1 if and only if X = ∅

dim X ≤ n if and only if every family of n + 1 pairs of disjoint closed subsets of X is inessential.

dim X = n if and only if dim X ≤ n and dim X ≰ n − 1

dim X = ∞ if and only if dim X ≠ n for every n ≥ −1 A space X with dim X < ∞ 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 ⨿ n space. The one point compactification of the disjoint union of cubes n I 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 ∪ G ∞ H contained in a member of i and the union i=1 i 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

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

n every projection pn : Q → I 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 ∞ → γ C (M,N) CW (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

|Dnf(y) − Dnf(x)| max sup = [f] k,α ¯ < ∞ ≤ | − |α C (U) n k x,y∈U,x¯ ≠ y y x

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

Rk R2 Deﬁnition 2.2.3. ≥0( ) 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

Rk R2 non-integer k, we use Hölder topology deﬁned above. the quotient space of ≥0( ) by the Diﬀ(R2)-action via pullback is called the moduli space of complete non-negatively curved

Mk R2 metrics and is denoted by ≥0( ). In [BB17], Belagradek and Banakh proved that if γ is a ﬁnite, positive, non-integer,

Rγ R2 ω ω then ≥0( ) is homeomorphic to Σ , where Σ is the linear span of Hilbert cube in the 2 ∞ Rγ R2 2 Mk R2 Hilbert space ℓ . If γ = , ≥0( ) is homeomorphic to ℓ . The moduli space ≥0( ) is rather pathological. For instance, it is not a Hausdorﬀ space. In [BH16], Belagradek and

∈ Rk R2 Hu exhibit a non-ﬂat metric g ≥0( ) 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

2u conformally equivalent to the standard Euclidean metric g0 isometric to e g0 for some smooth function u, this is a result by Blank-Fiala [BF41]. In [BH16], it was proved that

15 −2u M(r, u) the metric e g0 is complete if and only if α(u) ≤ 1, where α(u) = lim with r→∞ log(r) 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

ˇ n H (X,A; G) ≠ 0. This will be denoted by dimG(X). Here Hˇ ∗ denotes Cech˘ cohomology. This deﬁnition is good for any space, we

∗ will restrict ourselves when X is a compactum. In the following Theorem Hc 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) Hˇ n+1(X,A; G) = 0 for all closed A ⊂ X.

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

(4) For every closed subset A ⊂ X the inclusion homomorphism Hˇ n(X; G) → Hˇ n(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 dimZ X = dim X. When dim X is inﬁnite, the following theorem by Dranishnikov [Dra90] provides a counterexample.

Theorem 2.3.3. There is a strongly inﬁnite dimensional compactum X having dimZ X ≤ 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 ρˆ(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 ρˆ. 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 ρˆ(f, 1E) < ε. Then f can be extended to a homeomorphism f : Q → Q such that ¯ ρˆ(f, 1Q) < ε.

n Zn-sets were deﬁned by H. Torunczyk in [Tor78] in LC -spaces, in which they coincide with homotopical Zn sets. By deﬁnition, a closed subspace A of a space X is

n a homotopical Zn-set if for every open cover U of X every map f : I → 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 an Inverse 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.

1 2 Deﬁnition 2.5.2. Two regular q-cycles (A1, f1,Cq ) and (A2, f2,Cq ) of X are homologous

if there exists a (q + 1)-chain (A, f, Cq+1) such that A1 and A2 are closed (not necessarily 1 − 2 disjoint) subcomplexes of A, f agrees with f1 on A1 and f2 on A2 and ∂Cq+1 = Cq Cq . ˜ st In modern terminology, the reduced Steenrod homology group Hq (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. ∏∞ { ∈ ∀ ∈ N } ←−lim(Xi, fi) := x Xi :( i )(fi(xi+1) = xi) i=1 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 : i Ai ∏ − − − i Ai 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. ∏ ∏ 1 The ﬁrst derived functor of this sequence, denoted by ←−lim Ai is the cokernel i Ai/d i Ai 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 → 1{ } → → { } 0 ←−lim 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

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

20 CHAPTER 3 CONNECTEDNESS 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

Rk R2 by Belegradek and Hu in [BH16]. They proved that in the space ≥0( ) of complete Riemannian metrics of non negative curvature on the plane equipped with the topology of

k Rk R2 \ C uniform convergence on compact sets, the complement ≥0( ) X is connected for Rk R2 every finite dimensional X. We note that the space ≥0( ) 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

n ∪nI ). 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

Mk,c R2 Rk R2 R2 X for the moduli spaces ≥0( ), i.e. the quotient space of ≥0( ) by the Diﬀ( )-action via pullback. In the case of moduli spaces we manage to extend their connectedness result

⊂ Mk R2 C C to the subsets X ≥0( ) with Haver’s property (called -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 − ∞ ∞ I We denote the Hilbert cube by Q = [ 1, 1] = Πn=1 n. 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 Wi = x Q xi = 1 and Wi = 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 W1 to W1 . 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 W1 W1 and x, y we obtain \ − ∪ + → \{ } a homeomorphism f : Q (W1 W1 ) 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 ) = x, f(W ) = y and f| \ −∪ + = f. Note 1 1 Q (W1 W1 ) ¯−1 ⊂ \ − ∪ + ¯−1 that f (S) Q (W1 W1 ) and f defines a homeomorphism between f (S) and S. ¯ + − Since the image f(C) of a continuum C from W1 to W1 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 ∪ G ∞ H contained in a member of i and the union i=1 i 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.

Rk R2 Theorem 3.2.1. If K is a countable subset of ≥0( ) and X is a separable metric space, ∈ ∈ Rk R2 \ then for any distinct points x1, x2 X and any distinct metrics g1, g2 ≥0( ) K there Rk R2 \ is an embedding of X into ≥0( ) 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 R2 Rk R2 \ ≥0( ) is continuum connected. If S is closed, ≥0( ) S is path connected.

Rk R2 Proof. Let S be a weakly inﬁnite dimensional subspace of ≥0( ). Fix two metrics ∈ Rk R2 ˆ Rk R2 g1, g2 ≥0( ). Theorem 3.2.1 implies that g1, g2 lies in a subspace Q of ≥0( ) that is homeomorphic to Q. Since S ∩Qˆ is at most weakly inﬁnite dimensional, Qˆ \S is continuum ˆ connected by Theorem 3.1.2. Then g1, g2 lie in a continuum in Q that is disjoint from S. Rk R2 \ Hence ≥0( ) S is continuum connected. If S is closed, from Remark 3.1.0.1, we can Rk R2 \ conclude that ≥0( ) 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∞ R2 R∞ R2 \ ≥0( ) is locally connected. If S is closed, ≥0( ) S is locally path connected.

∞ R∞ R2 2 Proof. Given C topology, the space ≥0( ) is homeomorphic to ℓ , 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.

Rk,c R2 ∞ We do not know if the space ≥0( ) locally path connected for k < . Mk,c R2 We prove a similar to Theorem 3.2.2 result for the associated moduli space ≥0( ), when the subspace removed is a Hausdorﬀ space having the property C. This is a generalization of another Belegradek-Hu theorem.

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

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

sup{u(z): |z| = r} α(u) = lim . r→∞ log r

∞ 2 Note that S1 is closed in the Frechét space C (R ), it is not locally compact, and is equal

−2u to the set of smooth subharmonic functions u such that the metric e g0 is complete, → Mk,c R2 where g0 is the standard Euclidean metric [BH16]. Let q : S1 ≥0( ) denote the −2u ˆ −1 continuous surjection sending u to the isometry class of e g0. Let S = q (S). Fix two ∈ Mk,c R2 \ points g1, g2 ≥0( ) S. which are q images of u1, u2 in S, respectively. By Theorem ˆ 3.2.1 we may assume that u1, u2 lie in an embedded copy Q of Hilbert cube. It suffices to show that Qˆ \ Sˆ is path connected. The set Qˆ ∩ Sˆ is compact, hence qˆ, the restriction of q to Qˆ ∩ Sˆ is a continuous surjection. The map qˆ : Qˆ ∩ Sˆ → q(Qˆ) ∩ S is a map between compact spaces, and in particular, it is a closed map. The set qˆ(Qˆ ∩ Sˆ) has property C. We have each fiber q−1(y) to be finite dimensional [BH16], and hence Qˆ ∩ Sˆ is weakly infinite dimensional by Theorem 3.1.3. Therefore, Qˆ \ Sˆ is path connected by the discussion following

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

It should be noted that the Hausdorff condition is essential in Theorem 3.2.4. If S is not Hausdorff, the map qˆ 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

∞ 2 2 S1 of C (R ) is invariant under this action. Let π : S1 → S1/conf(R ) be the projection

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

Mk R2 Let S be a closed, weakly infinite dimensional Hausdorff subset of ≥0( ). 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) = Sˆ. It suffices to prove that Q ∩ Sˆ is weakly infinite dimensional, so we would have a path joining u and v in Q \ Sˆ, which

Mk R2 \ transforms to a path joining g and f in ≥0( ) S. The set Sˆ is closed hence Q ∩ Sˆ is compact. So the restriction of q to Q ∩ Sˆ is a continuous surjection qˆ : Q ∩ Sˆ → q(Q) ∩ S of compact separable metric spaces. Deﬁne the

2 ∗ −2u map η : S1/conf(R ) by uG 7→ u , the isometry class of e g0. This map is injective by deﬁnition and the diagram

π / R2 S1 C S1/conf( ) CC rr CC rrr CC rr q C! rx rr η Mk ≥0

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

−1 2 −1 to the preimage π (Y ) of S1 is an action of conf(R ) on π (Y ) with orbit space Y , and Q ∩ Sˆ ⊆ 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) ∩ Sˆ is weakly inﬁnite

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

27 CHAPTER 4 ACYCLICITY 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

Rk R2 [BH16] about connectedness properties of the space ≥0( ) of non-negative curvature metrics on R2 where k stands for Ck topology. In particular, they proved that the

Rk R2 \ complement ≥0( ) 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

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

n−k−1 n H (X) → Hk(S \ 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 H˜ (Sn \ X)  k    y y

n−k−1 ADn ˜ n H (Y ) −−−→ Hk(S \ 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 H˜ (Sn \ X)  k   ∼ sy =y

n−k ADn+1 ˜ n+1 H (ΣX) −−−−→ Hk(S \ 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

n−k−1 n AD n H (X,X ∩ ∂B ) −→ Hk(Int B \ X).

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

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

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

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

∗ i where α ∈ H (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

n n n H∗(Int B \ X) → H∗(B \ X), the commutative diagram (2) stated for B turns into the

29 following (2′)

Hn−k−1(X, ∂X) −−−→ADn H˜ (Bn \ X)  k    ∗ ∼ −∪ϕ y =y

n−k ADn+1 ˜ n+1 H (X × I, ∂X × I ∪ X × ∂I) −−−−→ Hk(B \ (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 ∏ n −1 n → n n → ∞ through the maps of pairs (I , f (∂I )) (I , ∂I ). A map f : X Q = i=1 I

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

n for every projection pn : Q → I 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 ∈ H˜ k−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

1 of pure relative classes αi ∈ H (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 ∗ n n n → n ∪n ¯ → n fn : H (I , ∂I ) H (X, i=1Ai) induced by the map fn = (f1, f2, . . . , fn): X I 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 = Q i=1 is essential.

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

ki of pure relative classes αi ∈ H (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 \ → \ −1 Hi(Q X) Hi(Q pn pn(X)) is nontrivial.

Proof. This follows from the fact that

H (Q \ X) = limH (Q \ p−1p (X)). i → i n n

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

n n complementary to I in Q = I × 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 ≠ ˜ k 0, ak ∈ Hi((I \ Xk) × Qk) for all k ≥ n. Thus, we may assume that ak lives in ˜ k ∼ ˜ k Hi((I \ Xk) × {0}) = Hi((I \ Xk) × Qk).

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

k k ′ (I , ∂I ). By the suspension isomorphism for the Alexander duality (see diagram (2 )), ak k+1 ∗ ∪ ∗ ∗ is dual in I to βk ϕ where βk is the image of βk under the homomorphism induced by

∗ 1 the projection Xk × I → Xk and ϕ is the image of the fundamental class ϕ ∈ H (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 βk ϕk+1 where ′ ∗ ∗ βk+1 is the restriction of βk to Xk+1 and ϕk+1 is the restriction of ϕ to Xk+1. Note that ′ k+1 ∗ k+1 → k+1 k+1 → k βk+1 = (qk ) (βk) where qk : Xk+1 Xk is the projection pk : I I restricted to k+1 ∗ ∪ Xk+1. Thus, βk+1 = (qk ) (βk) ϕk+1. By induction on m we obtain

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

n+m ∗ where ψj = (qn+j ) (ϕn+j), j = 1, . . . , m. ∗ We deﬁne a sequence αj = (qn+j) (ϕn+j) of relative cohomology classes on X where q = p | : X → X . Since X = limX , for the Čech cohomology we have n n X n ← r

ℓ −1 ℓ n+m −1 H (X, q (A)) = lim{H (X , (q ) (A))} ≥ n+k → n+m n+k m k

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

{ n+m ∗ ∪ · · · ∪ n+m ∗ } (qn+1 ) (ϕn+1) (qn+k ) (ϕn+k) m≥k

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

n+m ∗ ∪ · · · ∪ n+m ∗ ∪ · · · ∪ ̸ (qn+1 ) (ϕn+1) (qn+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

n dimZ X = sup{n | ∃ A ⊂Cl X with H (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.

ki Proof. Let αi ∈ H (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 fi (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 ∏n n gn = (q1f1, q2f2, . . . , qnfn): X → I = I i=1

n is inessential. Then there is a deformation H of gn to a map h : X → Nϵ(∂I ) rel −1 n n × · · · × gn (Nϵ(∂I )) to the ϵ-neighborhood of the boundary ∂I . Since the map q1 qn is a

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

∏n ¯ fn = (f1, . . . , fn): X → Cone(Ki) i=1

−1 n to an ϵ-neighborhood of M = (q1 × · · · × qn) (∂I ). Since M is an ANR we may assume ¯ ¯−1 ∪ · · · ∪ ¯∗ ¯ ∪ · · · ∪ ¯ that fn can be deformed to M rel fn (M). Since α1 αn = fn(ϕ1 ϕn), to ¯ ¯ complete the proof it suﬃces to show that the restriction of ϕ1 ∪ · · · ∪ ϕn to M is zero. We denote ∏n ∏n D = Cone(Ki) \ Ocone(Ki) i=1 i=1∏ 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 Cone(Ki) = {t1x1 + ··· + tnxn | xi ∈ Ki, ti ∈ [0, 1]} i=1 and ∏n Ocone(Ki) = {t1x1 + ··· + tnxn | xi ∈ Ki, ti ∈ [0, 1)} i=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,

∑t1 ∑tn ρ(t1x1 + ··· + tnxn) = x1 + ··· + xn ti ti is bijective and, hence, is a homeomorphism.

35 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

∪n ∗ ˆ L = K1 ∗ · · · ∗ Ki ∗ · · · ∗ Kn i=1 ˆ where Ki means that the i-factor is missing. Fix points ei ∈ Ki and the straight-line deformation i ∗ · · · ∗ ˆ ∗ · · · ∗ → ∗ · · · ∗ ht : 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.

n−1 n−1 Then x˜0 = q(∆ ) is the base point in the reduced joint product where the simplex ∆

is spanned by e1, . . . , en. i ˜i ∗ · · · ∗ ˆ ∗ · · · ∗ The deformation ht deﬁnes a deformation ht of K1˜ ˜Ki˜ ˜Kn to the base ∑ point tiei in the reduced joint product K1∗˜ · · · ∗˜Kn. We note that for any i < j ˜i ˜j ∗ · · · ∗ ˆ ∗ · · · ∗ ˆ ∗ · · · ∗ the deformations ht and ht agree on the common part K1˜ ˜Ki˜ ˜Kj˜ ˜Kn. ∗ ∪˜i ◦ | ∗ Therefore, the union ( ht) q L is a well-deﬁned deformation of L to the base point in

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

∪ Hk(Y,A; R) × Hl(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 ∈

ki H (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))

− ∗ 1 − ¯ ∈ ki be an extension of fi and let Bi = gi (K(R, ki 1)). We consider βi = gi (ϕi) H (X,Bi). ∗ → Note that αi = ξi (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.

37 4.4 Applications

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

ω ℓ2 and let Σ denote the product of countably many copies of Σ. The following is well-known:

ω 4.4.0.1 Proposition. Let H be either ℓ2 or Σ . 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. Rγ R2 ω ∞ The space ≥0( ) is homeomorphic to Σ when γ < and it is not an integer. The R∞ R2 space ≥0( ) 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.

⊂ Rγ R2 ∞ Let X ≥0( ) be a closed weakly inﬁnite dimensional subset when γ = or ∞ Rγ R2 \ γ < and it is not an integer. Then the complement ≥0( ) 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

38 compact metric space which is locally homeomorphic to Q × [0, 1). Some examples of Q manifolds are open subsets of Q and M n × Q where M n 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. ∪ n { }n Proof. K is a finite union i=1 Qi of copies of Hilbert cubes Qi i=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-compact Subset 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

39 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

st metric space X by H∗ (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 = i Xi be σ-compact weakly inﬁnite-dimensional subset of Q. st \ Then H0 (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 = i Ki = K non empty, containing p and q, and in the complement of X. By the Milnor’s short exact sequence for Steenrod homology, we

40 have, → 1 st → st → st → 0 ←−lim (H1 (Ki)) H0 (K) ←−lim(H0 (Ki)) 0

˜ st Where the ﬁrst and the third terms are zero, hence H0 (K) is trivial. Which imply S bounds in K, hence in Q \ X.

41 REFERENCES [Ama17] A. Amarasinghe, On separators of the space of complete non-negatively curved metrics on the plane, Topology Proc. 50, 2017 Pp 13-19 [AD17] A. Amarasinghe, A. N. Dranishnikov On homology of complements of compact sets in Hilbert Cube, Topology and its applications, Volume 221, Pp:262-269 [AP73] P.S. Aleksandrov, B.A. Pasynkov, Introduction to dimension theory (Russian), Nauka, Moscow, 1973. [AG78] D. F. Addis and J. H. Gresham, A class of inﬁnite-dimensional spaces, Part I: dimension theory and Alexandroﬀ’s problem, Fund. Math. 101 (1978), 195–205. [Bro11] von BROUWER, L. E. J, Beweis der Invarianz der Dimensionenzahl Math. Ann. 70 (1911) [BB17] T. Banakh, I. Belegradek, Spaces of nonpositively curved spaces, preprint arXiv: 1510.07269v1. [BF41] C. Blanc, F. Fiala Le type dune surface et sa courbure totale, Comment. Math. Helv. 14 (1941), 230233. [Bre72] G. E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, (1972).

[BCK11] T. Banakh, R. Cauty, A. Karassev. On homotopical and homological Zn-sets, Topology Proc. 38 (2011), 29-82. [BH16] I. Belegradek, J. Hu, Connectedness properties of the space of complete non negatively curved planes, Math. Ann., 364 (2016), no. 1-2, 711-712. [Cha76] T. A. Chapman, Lectures on Hilbert cube manifolds, C. B. M. S. Regional Conference Series in Mathematics; No 28. (1976) [Dra90] A. Dranishnikov, Generalized cohomological dimension of compact metric spaces, Tsukuba J. Math. 14 (1990) no. 2, 247-262. [DW04] A. Dranishnikov, J. West, Compact group actions that raise dimension to infinity, Topology Appl. 80 (1997), no. 1-2, 101-114 (Correction in Topology Appl. 135 (2004), no. 1-3, 249-252). [Eng95] R. Engelking, Theory of dimension: finite and infinite, Heldermann Verlag, 1995. [Fer95] Ferry, S. Remarks On the Steenrod homology theory Novikov conjectures, index theorems, and rigidity. Vol.2 (CUP, 1995) [GW87] D. Garity, D. Wright, Vertical order in the Hilbert cube, Illinois Journ. of Math. 31 No 3 (1987), 446-452.

42 [Hat02] A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. [HW41] Hurewicz, W., Wallman, H Dimension Theory Princeton University Press, 1941 [HY89] Y. Hattori and K. Yamada, Closed preimages of C-spaces, Math. Japonica 34 (1989), 555–561. [Kro74] N. Kroonenberg, Characterization of finite-dimensional Z-sets. Proc. Amer. Math. Soc. 43 (1974), 421–427. [Kel31] O. H. Keller, Die Homoiomorphie der kompakten konvexen Mengen in Hilbertschen Raum, Math. Ann. 105 (1) (1931), 748-758 [Mil93] Milnor, J. On the Steenrod homology Theory. Novikov conjectures, index theorems and rigidity. Vol 1 (1993) [Pea07] E. Pearl, Open Problems in Topology II, Elsevier, 2007. [Pol83] L. Polkowsky, Some theorems on invariance of infinite dimension under open and closed mappings, Fnd. Math. 119.1 (1983), 11–34. [Ros07] J. Rosenberg, Manifolds of positive scalar curvature: a progress report, Surv. Differ. Geom. 11, (2007), 259–294. [Rud87] Y.B. Rudyak, "Realization of homology classes of PL-manifolds with singularities" Math. Notes , 41 : 5 (1987) pp. 417421 Mat. Zametki , 41 : 5 (1987) pp. 741749 [Ste40] N. E. Steenrod, Regular cycles of compact metric spaces. Annals of Mathematics Second Series, Vol. 41, No. 4 (Oct., 1940), pp. 833-851 [Tho54] R. Thom, "Quelques propriétés globales des variétés differentiable". Comm. Math. Helv. (1954). 28: 1786. [Tor78] H.Torunczyk, Concerning locally homotopy negligible sets and characterization of l2-manifolds, Fund. Math. 101:2 (1978), 93110 [vM02] J. van Mill, The infinite dimensional topology of function spaces,North-Holland Mathematical Library (2002) [Whi70] G. W. Whitehead. Recent Advances in homotopy theory. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 5. American Mathematical Society, Providence, R.I., 1970.

43 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.