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Homological Methods in Coarse Geometry A The Pennsylvania State University The Graduate School The Eberly College of Science HOMOLOGICAL METHODS IN COARSE GEOMETRY A Dissertation in Mathematics by Steven Hair c 2010 Steven Hair Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2010 The dissertation of Steven Hair was reviewed and approved* by the following: John Roe Professor of Mathematics Dissertation Advisor Chair of Committee Head of the Department of Mathematics Nathanial Brown Associate Professor of Mathematics Nigel Higson Evan Pugh Professor of Mathematics Darrell Velegol Professor of Chemical Engineering Chun Liu Professor of Mathematics Director of Graduate Studies in the Department of Mathematics *Signatures are on file in the Graduate School. ii Abstract The field of coarse algebraic topology involves the application of principles of algebraic topology to coarse spaces, particularly the calculation of the so-called \coarse homology" and \coarse cohomology" of a space. In this dissertation, we present new techniques in coarse algebraic topology, and give applications to studying coarse spaces. We first develop an analogue of the Mayer-Vietoris theorem for coarse homology, discuss a coarse excision principle, and give a \coarse mapping cone" construction for relative coarse cohomology. Next, the notion of homological degree for a coarse map is introduced, and is used to prove a coarse surjectivity theorem as well as a coarse fixed point theorem. Then, we develop an algebraic formalization of underlying concepts in coarse algebraic topology in the form of \geometric modules". These modules are sub- sequently utilized to formulate a coarse analogue of the K¨unneththeorem and Universal Coefficient theorem; these theorems are then applied to compute the coarse homology and cohomology of specific spaces. Finally, we discuss coarse versions of topological CW-complexes and fibrations, use this discussion to introduce a coarse analogue of the Serre spectral sequence, and apply spectral sequence calculations to compute the coarse cohomology of certain nilpotent groups. iii Contents List of Figures vii Acknowledgements viii 1 Introduction 1 1.1 Coarse Spaces . 2 1.2 Examples . 3 1.3 Controlled and Bounded Sets . 4 1.4 Coarse Maps, Equivalence, Homotopy . 5 1.4.1 Gluing of Coarse Spaces . 8 1.5 Coarse Algebraic Topology . 10 1.5.1 Coarse Cohomology . 10 1.5.2 Examples . 11 1.6 Covers and Nerves . 15 1.6.1 Coarse Homology . 18 1.7 Organization . 21 2 Coarse Excision, Relative Cohomology 22 2.1 The Mayer-Vietoris Sequence . 22 2.2 Definition of Relative Coarse Cohomology . 27 2.3 Coarse Excision . 29 iv 2.4 The Mapping Cone Construction . 34 3 Coarse Degree Theory 37 3.1 Degree and Essential Surjectivity . 38 3.2 A Combinatorial Coarse Degree Formula . 42 3.3 Examples . 44 3.4 The Coarse Brouwer Fixed Point Theorem . 46 3.4.1 The Weak Fixed Point Property . 51 4 Geometric Modules 53 4.1 The Controlled and Bounded Coarse Categories . 53 4.2 Geometric Modules . 54 4.2.1 Morphisms of Geometric Modules . 57 4.3 Controlled and Cocontrolled Modules . 59 4.4 Generalized Morphisms . 60 4.4.1 Composition of Morphisms . 67 4.5 Partial Modules and Closures . 70 4.6 Total Modules . 73 4.7 Cocontrolled and Controlled Tensor Products . 76 4.8 Tensor Products . 77 4.8.1 The Cross Product Map . 83 4.8.2 Decomposing elements of CXn(X × Y ) . 87 4.8.3 The Shuffle Map . 89 4.8.4 Construction of Homotopy Equivalences . 91 5 Coarse Universal Coefficient and K¨unnethTheorems 101 5.1 A Coarse Universal Coefficient Theorem . 101 5.2 A Coarse K¨unnethTheorem . 105 5.3 Examples . 117 v 6 Coarse CW-Complexes and Fibrations 119 6.1 Coarse CW-Complexes . 119 6.2 Coarse Cellular Cochain Complexes . 125 6.3 Coarse Fibrations . 128 6.4 The Main Theorem . 137 6.4.1 Construction of the Isomorphism . 138 6.4.2 Construction of the Spectral Sequence . 141 6.4.3 Examples . 144 Bibliography 145 vi List of Figures 3.1 Construction of similar triangles. 50 4.1 Path corresponding to the shuffle µ = f1; 4; 5; 6g, ν = f2; 3; 7g. 92 vii Acknowledgements This dissertation would have never seen the light of day without the contributions of many important people. First, I would like to thank my advisor, John Roe, for his patience and motivation over the years. His influence helped to shape me from an unfocused graduate student to someone who can hopefully contribute positively to the world of mathematics as an educator and researcher. Thanks also to my committee members, Nate Brown, Nigel Higson, and Darrell Velegol, for donating their time. I would also like to thank Becky Halpenny and the rest of the office staff, without whom I would have been lost over the years. Additionally, appreciation goes out to my Master's thesis advisor at Virginia Tech, Peter Linnell, for inspiring me to follow this path in the first place, and my fellow graduate students, whose mathematical conversations proved invaluable over the years. Finally, special thanks go out to Mom and Dad, Jolene and Scott, Kerry, and the rest of the family, friends, and dogs in my life, for their unfailing support and love. viii Chapter 1 Introduction There exists in the current literature a number of algebraic topological methods for analyzing properties of coarse spaces. These methods include calculations of the coarse homology and cohomology groups of these spaces, with associated notions of excision, relative (co)homology, and Mayer-Vietoris sequences. In this dissertation, we introduce additional methods in coarse algebraic topology in order to perform these calculations for a greater number of coarse spaces. These methods will mirror some of the more well-known techniques in classical algebraic topology. In particular, we shall give a coarse mapping cone construction of the coarse coho- mology of a space relative to a subspace, and verify, via coarse excision, that the relative coarse cohomology groups are isomorphic to the absolute coarse cohomology groups of the mapping cone . We also introduce the notion of degree for coarse maps, prove a geometric result for maps of nonzero degree, and provide a combinatorial technique by which degree can be computed, with computations for specific examples. Additionally, by applying results on direct and inverse limits of modules in homolog- ical algebra, we shall develop coarse notions of the Universal Coefficient Theorem and K¨unnethFormula. Then we shall discuss the concept of coarse fibrations: maps between coarse spaces with associated Serre-like spectral sequences in coarse cohomology. This 1 discussion is aided by the presentation of coarse CW-complexes: coarse spaces with a structure analogous to CW-complexes in classical topology. The spectral sequence results will then be applied to compute the coarse cohomology of specific spaces, most notably semidirect products of groups. 1.1 Coarse Spaces Coarse geometry is often thought of as the study of the \large-scale" geometry or the \geometry at infinity" of a space, i.e. the study of what structure survives as one zooms infinitely far away from the space. More precisely, coarse geometry is the study of coarse spaces. Such spaces are the fundamental objects in coarse geometry, analogous to topological spaces in classical topology. A coarse space is a set equipped with a so-called coarse structure, as defined below, which is analogous to the topology on a topological space. Definition 1.1.1. A coarse space (X; E) is a set X accompanied by a collection E of subsets of X × X (the so-called controlled subsets or entourages of X × X) with the following properties: • the diagonal ∆X×X is a member of E; • if E is a member of E, then so is every subset of E; • if E; F 2 E, then so is E [ F ; • if E is a member of E, then so is the inverse E−1 := f(y; x)j(x; y) 2 Eg; • if E and F are members of E, then so is the product E ◦ F := f(x; y)j there exists z 2 X such that (x; z) 2 E; (z; y) 2 F g: 2 The collection E is called the coarse structure of X. Generally, we will use X to refer to the coarse space (X; E). A collection S of subsets of X × X containing ∆X×X is said to generate a coarse structure E if E is the smallest coarse structure containing S. 1.2 Examples 1. If X is a metric space, the metric coarse structure is the structure generated by the set of all metric neighborhoods of the diagonal, i.e. the sets of pairs (x; y) in X × X which are distance at most R apart, for all R ≥ 0. It is well-known (cf. [18], Thm. 2.55) that a coarse structure E on X is countably generated if and only if it is metrizable, i.e. there exists a metric on X for which E is the metric coarse structure. 2. The trivial or indiscrete coarse structure on a set X is the structure consisting of all subsets of X × X. 3. The discrete coarse structure on X is generated by all sets of the form ∆X×X [ F for finite F ⊂ X × X. 4. If A is a subset of a coarse space X, we may equip A with the subspace coarse structure fE \ (A × A) j E ⊂ X × X controlledg. (In the future, when we refer to such a subset A as being a subspace of X, we mean that A is equipped with this particular coarse structure.) 5. If X and Y are coarse spaces, we may equip the set X ×Y with the product coarse structure, consisting of those subsets of (X × Y ) × (X × Y ) whose projections to X × X and Y × Y are controlled under the respective coarse structures.
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