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Homotopy Theory with Bornological Coarse Spaces

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Homotopy theory with bornological coarse spaces Ulrich Bunke∗ Alexander Engely Abstract We propose an axiomatic characterization of coarse homology theories defined on the category of bornological coarse spaces BornCoarse. We construct a stable 1-category SpX of motivic coarse spectra which is the target of the universal coarse homology theory BornCoarse ! SpX . Besides the development of the foundations of the theory our focus is the classifi- cation of coarse homology theories and the construction of examples. As a guiding example we show that if a transformation between coarse homology theories induces an equivalence on all discrete bornological coarse spaces, then it is an equivalence on bornological coarse spaces of finite asymptotic dimension. The main homotopy theoretic method of construction of coarse homology theories is the coarsification of locally finite homology theories. The construction and the classification of locally finite homology theories defined on the category of topological bornological spaces TopBorn will be studied in detail. We will develop two versions of coarse K-homology KX and KXql as examples of coarse homology theories. They will be modeled on corresponding versions of the Roe algebras. One version is defined using finite propagation operators, the other version using quasi-local operators. Applying our comparison theorem we will be able deduce that these two versions agree on all bornological coarse spaces of finite asymptotic dimension. We discuss the construction of the coarse assembly map from the coarsification of analytic locally finite K-homology to coarse K-homology. As an application of our classification of locally finite homology theories we show that on bornological coarse spaces of locally bounded geometry the domain of the coarse assembly map can be identified with the coarsification of the homotopy theorist's version of locally finite K-homology. ∗Fakult¨atf¨urMathematik, Universit¨atRegensburg, 93040 Regensburg, GERMANY [email protected] yFakult¨atf¨urMathematik, Universit¨atRegensburg, 93040 Regensburg, GERMANY [email protected] 1 Contents 1. Introduction 3 1.1. Overview . .3 1.2. Motives of bornological coarse spaces . .6 1.3. Coarse homology theories . .8 1.4. Locally finite homology theories . .9 1.5. Reduction to discrete bornological coarse spaces . 10 1.6. Coarse K-homology . 10 I. Motivic coarse spaces and spectra 12 2. Bornological coarse spaces 12 2.1. Basic definitions . 12 2.2. Examples . 14 2.3. Categorical properties of BornCoarse .................... 16 3. Motivic coarse spaces 18 3.1. Descent . 19 3.2. Coarse equivalences . 22 3.3. Flasque spaces . 24 3.4. u-continuity and motivic coarse spaces . 26 3.5. Coarse excision and further properties . 28 4. Motivic coarse spectra 30 4.1. Stabilization . 30 4.2. Further properties of Yos ............................ 34 4.3. Homotopy invariance . 37 5. Merging coarse and uniform structures 41 5.1. The hybrid structure . 41 5.2. Decomposition theorem . 46 5.3. Homotopy theorem . 52 5.4. Flasque hybrid spaces . 57 5.5. Decomposition of simplicial complexes . 61 5.6. Flasqueness of the coarsening space . 65 II. Coarse homology theories 73 6. Coarse homology theories 73 6.1. Definition and fundamental properties . 74 6.2. Additivity and coproducts . 77 6.3. Coarse ordinary homology . 79 6.4. Coarsification of stable homotopy . 84 2 6.5. Locally finite homology theories . 93 6.6. Coarsification of locally finite theories . 108 6.7. Analytic locally finite K-homology . 111 6.8. Coarsification spaces . 117 6.9. Comparison of coarse homology theories . 122 7. Coarse K-homology 125 7.1. X-controlled Hilbert spaces . 126 7.2. Ample X-controlled Hilbert spaces . 128 7.3. Roe algebras . 133 7.4. K-theory of C∗-algebras . 139 7.5. C∗-categories . 140 7.6. Coarse K-homology . 149 7.7. Comparison with the classical definition . 157 7.8. Additivity and coproducts . 165 7.9. Dirac operators . 169 7.10. Assembly map . 175 References 178 Index 182 1. Introduction 1.1. Overview In coarse geometry we study the large scale structure of spaces. In most cases this structure is provided by a metric which encodes local and global geometric information at the same time. In order to stripe off the local part, we will encode the global structure in the abstract notion of a coarse structure associated to the metric. Another source of coarse structures are control maps as considered in controlled topology. If we start with a metric space, then the associated coarse structure still allows us to reconstruct the bornology of metrically bounded subsets. But in other cases it turns out to be useful to consider more general bornologies as well. In the present paper we introduce a very general category of bornological coarse spaces which subsumes all examples known to the authors. Bornological coarse spaces are studied and distinguished by invariants. Examples of yes/no invariants are locally bounded geometry or embeddability into a Hilbert space. A typical numerical invariant is the asymptotic dimension. The main examples of invariants relevant for the present paper are classical coarse homology theories. There are various examples known which are usually constructed and studied under special assumptions. For example, coarse K-homology is usually defined for proper metric spaces. 3 In this paper we provide a very general axiomatic framework for coarse homology theories. The challenge is to develop examples of coarse homology theories with good properties defined on most general bornological coarse spaces. The goal of this paper is threefold. First we introduce the category BornCoarse of bornological coarse spaces and the stable 1-category SpX of motivic coarse spectra. The Yoneda functor Yos : BornCoarse ! SpX encodes the common properties of classical coarse homology theories. Our construction can be visualized by the following diagram: bornological coarse spaces classical C-valued coarse homology theory e.g. spectra / (1.1) BornCoarse 9 C motive C-valued coarse homology theory ' motivic coarse spectra SpX where C is some cocomplete stable 1-category, e.g., the category of spectra Sp. By definition, a C-valued coarse homology theory is a colimit-preserving functor SpX! C. The category of motivic coarse spectra is constructed such that it has the universal property that every classical C-valued coarse homology theory has an essentially unique factorization over a C-valued coarse homology theory. We will often use the shorter phrase coarse homology theory if the category C is clear from the context. The basic construction of examples of coarse homology theories by homotopy theoretic methods is given by the process of coarsification locally finite homology theories −−−−−−−!coarsification coarse homology theories : This process and its input, locally finite homology theories, will be studied thoroughly in the present paper. Our second theme concerns the comparison of different coarse versions of a homology theory. More precisely, if F is a spectrum we say that a classical coarse homology theory E is a coarse version of F if E(∗) ' F . It frequently appears that different geometric constructions produce coarse versions of the same spectrum. This raises the natural question how different these versions are. To this end we consider the cocomplete full subcategory SpX hAdisci ⊆ SpX generated by the set of motives Adisc of discrete bornological coarse spaces. A discrete bornological coarse space has the simplest possible coarse structure on its underlying set, but its bornology might be interesting. 4 Since coarse homology theories are colimit preserving functors it is clear that a transfor- mation between two classical coarse homology theories which is an equivalence on discrete spaces is clearly an equivalence on all bornological coarse spaces whose motives belong to SpX hAdisci. So it is an interesting question to formulate sufficient geometric conditions on a bornological coarse space which ensure that its motive belongs to SpX hAdisci. Our main result in this direction is Theorem 6.114: Theorem 1.1. If X is a bornological coarse space of finite asymptotic dimension, then s Yo (X) 2 SpX hAdisci. Corollary 1.2. If a transformation between classical coarse homology theories induces an equivalence on all discrete bornological coarse spaces, then it is an equivalence on bornological coarse spaces of finite asymptotic dimension. We refer to the statement of Theorem 6.114 for a precise formulation of the assumptions on the space X in the above theorem. Under more restrictive assumptions we also have the following result which is closer to the classical uniqueness results in ordinary algebraic topology. Corollary 1.3. If a transformation between additive classical coarse homology theories induces an equivalence on the point, then it is an equivalence on path-metric spaces of finite asymptotic dimension. For the proof see Theorem 6.116 and its subsequent paragraph. Our third goal is to develop coarse K-homology as a classical coarse homology theory. The construction of coarse K-homology can be visualized by the following picture bornological coarse space Roe C∗-category universal C∗-algebra C∗-algebra K-theory ∗ f ∗ f ∗ (1.2) X C?(X) A (C?(X)) K(A (C?(X))) ∗ We will define two versions KX and KXql using two versions of the Roe C -category ∗ ∗ ∗ denoted by C and Cql. The Roe C -categories are categories of (locally finite) X-controlled Hilbert spaces, and their morphisms are generated by bounded operators with controlled propagation, resp. quasi-local operators. Our main result is Theorem 7.53: Theorem 1.4. The functors KX ;KXql : BornCoarse ! Sp defined by (1.2) are classical coarse homology theories. The coarse homology theories KX and KXql are both versions of usual complex K- homology. Since the quasi-locality condition is weaker than controlled propagation we ∗ ∗ have an inclusion C ,! Cql and therefore a natural transformation KX! KXql : It is easy to see (Lemma 7.61) that this transformation is an equivalence on discrete bornological coarse spaces.
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