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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 = f(Ai;Bi): i 2 Γg of X is said to be essential if for every T f 2 g 6 ; family Li : i Γ where Li is a separator between Ai and Bi, we have i2Γ 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 Z1-set, which is analogous to Z1-sets of a space. The authors proved that either closed, countable dimensional subspaces or trt-dimensional subspaces in the Hilbert cube are homological Z1 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 2 H (X; Ai), ki > 0, i = 1; 2;::: , Ai ⊂Cl X such that the cup-product k α1 [ αn [···[ αn =6 0 for all n. We call a relative cohomology class α 2 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 2 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.
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