CONTINUOUS COMBINATORICS OF A LATTICE GRAPH IN THE CANTOR SPACE Edward Krohne Dissertation Prepared for the Degree of DOCTOR OF PHILOSOPHY UNIVERSITY OF NORTH TEXAS May 2016 APPROVED: Su Gao, Major Professor and Chair of the Department of Mathematics Steve Jackson, Major Professor Charles Conley, Committee Member Krohne, Edward. Continuous Combinatorics of a Lattice Graph in the Cantor Space. Doctor of Philosophy (Mathematics), May 2016, 85 pp., 28 figures, references, 4 titles. We present a novel theorem of Borel Combinatorics that sheds light on the types of continuous functions that can be defined on the Cantor space. We specifically consider the part X=F(2 ) from the Cantor space, where the group G is the additive group of integer pairs ². That is, X is the set of aperiodic {0,1} labelings of the two- dimensional infinite lattice graph. We give X the Bernoulli shift action, and this action induces a graph on X in which each connected component is again a two-dimensional lattice graph. It is folklore that no continuous (indeed, Borel) function provides a two- coloring of the graph on X, despite the fact that any finite subgraph of X is bipartite. Our main result offers a much more complete analysis of continuous functions on this space. We construct a countable collection of finite graphs, each consisting of twelve "tiles", such that for any property P (such as "two-coloring") that is locally recognizable in the proper sense, a continuous function with property P exists on X if and only if a function with a corresponding property P' exists on one of the graphs in the collection. We present the theorem, and give several applications. Copyright 2016 by Edward Krohne ii ACKNOWLEDGMENTS I would like to thank Tamara Knox, a dear and old friend who has supported me through both a thesis and a dissertation, and from whom I have learned so much. iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS iii LIST OF FIGURESv CHAPTER 1 INTRODUCTION1 1.1. Basic Definitions and Notations3 1.2. The Borel and Continuous Chromatic Numbers of F (2Z) 14 CHAPTER 2 MAIN RESULT 19 2 2.1. Passing from F (2Z ) to Finite Graphs 22 2 Z 2.2. Passing from Γn;p;q to F (2 ) 30 2.3. Main Result in Summary 43 CHAPTER 3 EXAMPLES AND IMMEDIATE COROLLARIES 44 CHAPTER 4 UNDIRECTED GRAPH HOMOMORPHISMS 54 2 CHAPTER 5 SUBSETS OF F (2Z ) WITH CONTINUOUS GROUP ACTIONS 75 CHAPTER 6 CONCLUSIONS AND FUTURE WORK 83 BIBLIOGRAPHY 85 iv LIST OF FIGURES Page Figure 1.1. An illustration of pullbacks.6 Figure 2.1. A simplified notation for a quotient Z2-graph of a Z2-graph consisting of rectangular grid graphs. 20 Figure 2.2. The first four \tiles" of Γn;p;q. 21 Figure 2.3. The next four \tiles" of Γn;p;q. 22 Figure 2.4. The long horizontal \tiles" of Γn;p;q. 23 Figure 2.5. The long vertical \tiles" of Γn;p;q. 24 Figure 2.6. Constructing the portion of the hyper-aperiodic point that shows how to 0 define ' on Tca=ac. 26 Figure 2.7. Constructing the portion of the hyper-aperiodic point that shows how to color Tdca=acd. 27 Figure 2.8. Edge boundaries for marker regions. 33 Figure 2.9. \Downward" propagation of an instance of Rd using tile Tda=ad through a region mostly tiled with tile Tca=ac. 34 Figure 2.10. \Diagonal" propagation of an instance of Rd using tile Tdca=acd, through a region mostly tiled with Tca=ac 35 Figure 2.11. An illustration of the construction of R 36 Figure 3.1. The four-coloring of Γ1;2;3. 44 Figure 3.2. The five-edge-coloring of Γ1;2;3. 48 Figure 3.3. A function of ':Γ1;5;2 ! 2 inducing a continuous function 2 2 Z 'F (2Z ) : F (2 ) ! 2 which has no infinite monochromatic graph component. 49 Figure 3.4. Toast. 50 Figure 4.1. The complete graph of order 3, K3, with a weighting function. 61 v Figure 4.2. The Petersen graph ΓP . 62 Figure 4.3. Steve Jackson's \Clamshell" graph ΓJ . 63 ∗ Figure 4.4. A graph ΓK such that ΓK is a cell-complex homeomorphic to the Klein bottle. 64 Figure 4.5. An illustration of how to \connect" two copies of γ using a winding path. 68 Figure 4.6. A process for eliminating spurs in a cycle one at a time. 69 Figure 4.7. Construction of a hub. 70 Figure 4.8. The Chv´atalGraph. 71 Figure 4.9. The Gr¨otzsch Graph. 71 Figure 4.10. A rectangular grid graph Tr ⊆ Γ establishing the homotopy group relation ∗ r = 1 in π1(Γ ). 73 2 Figure 5.1. On a Z -pullback 'Z2 of 'Γ, with 'Z2 ◦ σ = 'Γ. 77 Figure 5.2. Almost lined-up marker regions. 81 vi CHAPTER 1 INTRODUCTION We are interested in viewing familiar graph theoretic questions such as chromatic number and edge chromatic number under the lens of definability. For example, it is a simple fact that any finite (indeed, any well-ordered) acyclic graph has chromatic number at most two, i.e., each node in the graph can be assigned a color from a palette of two colors such that no two adjacent nodes have the same color. Of interest in this work is the fact that, if the nodes of the acyclic graph are drawn from a standard Borel space, even in a sensible way, the two-coloring function may not be reasonably definable. The two-coloring may be at least non-Borel, and it may not be clear that a two-coloring exists at all without the well-ordering principle. This research grew out of the study of Borel equivalence relations on standard Borel spaces, e.g., [4]. One powerful result in this area is that of Feldman-Moore [1], which states that any countable Borel equivalence relation (i.e., any equivalence relation in which the equivalences are countable) on a standard Borel space X is the orbit equivalence relation of some Borel action of some countable group on X. That is, broadly speaking, countable Borel equivalence relations can be studied one group at a time. We may begin this program of study with the space f0; 1gZ = 2Z (henceforth we denote k = f0; 1; : : : ; k − 1g), under the product topology. This space is homeomorphic to the Cantor space. Our action will be the action of Z acting on 2Z by the Bernoulli shift or 1 left-shift action \·". That is, for x 2 2Z; g; h 2 Z, (g · x)(h) = x(g−1 · h): Consider the orbits of this action on 2Z, where Z is considered a multiplicative group and generated by z. Each point x of 2Z is a map x: Z ! 2. If x is periodic with period 1The left-shift action is so named because g appears on the left of h, not because it is a left action. Indeed, there is another action, called the right-shift action, which is nonetheless a left action as well; namely (g ·right x)(h) = x(h · g). All actions considered in this work will act on the left. 1 g 2 Z, the orbit [x] of x is finite with cardinality equal to jkj with zk = g. If x is aperiodic, then Z acts freely on [x] and [x] is countably infinite. It is natural to consider the unit-shifts z±1 · x to be \adjacent" to x; this gives rise to ±1 a graph structure (Γ2Z ; ∼) with x ∼ y iff z · x = y, and we may study the properties of this graph. Then each connected component of Γ2Z is simply an orbit in the action; finite orbits are cycles, and infinite orbits are infinite chains. Henceforth, we will abusively abbreviate Z (Γ2Z ; ∼) as 2 when such usage is unambiguous. We are interested specifically in graph-theoretic behavior arising out of the require- ment that all functions and constructions be definable; in our case either Borel or continuous. Therefore, we remove the finite graph components from consideration, and only consider the set F (2Z) of aperiodic points in 2Z. This leaves an acyclic graph, but it is folklore (the proof to be presented shortly) that there exists no Borel two-coloring of F (2Z); this justifies our decision to drop cycles from consideration as the acyclic components are quite interesting on their own. We also present a folklore construction of a three-coloring on F (2Z) which is not only Borel, it is continuous. Therefore both the Borel and continuous chromatic numbers of F (2Z) are three. As the case of F (2Z) is relatively well-understood, we focus our attention in this work 2 on the next most complicated group action, F (2Z ). Indeed, a continuous four-coloring of 2 F (2Z ) has long been known [2, Thm. 4.2], but the question of the existence of a Borel or continuous three-coloring remained open [2,4]; the present work gives a proof of nonexistence 2 of a continuous three-coloring of F (2Z ). In addressing this problem, we needed to consider intricate intermediate structures, called marker structures, which serve as a “scaffolding" for building Borel or continuous functions. The questions of chromatic number, existence of particular flavors of marker structures, and existence of other combinatorial constructions on 2 F (2Z ) turn out to be tightly related, especially in the continuous case. Our main theorem 2 unifies these ideas for the continuous case of F (2Z ). Specifically, given a discrete space D and a property P that can be witnessed locally in the proper sense (e.g., the property of being a k-coloring) our theorem provides a necessary and sufficient condition for the existence of 2 2 a continuous map ': F (2Z ) ! D with property P .