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Lecture Course on Coarse Geometry

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Lecture Course on Coarse Geometry Lecture course on coarse geometry Ulrich Bunke∗ February 8, 2021 Contents 1 Bornological spaces2 2 Coarse spaces9 3 Bornological coarse spaces 26 4 Coarse homology theories 34 5 Coarse ordinary homology 46 6 Coarse homotopy 54 7 Continuous controlled cones and homology of compact spaces 57 8 Controlled objects in additive categories 67 9 Coarse algebraic K-homology 75 10 The equivariant case 77 11 Uniform bornological coarse spaces 87 12 Topological K-theory 109 References 123 ∗Fakult¨atf¨urMathematik, Universit¨atRegensburg, 93040 Regensburg, [email protected] regensburg.de 1 1 Bornological spaces In this section we introduce the notion of a bornological space. We show that the category of bornological spaces and proper maps is cocomplete and almost cocomplete. We explain how bornologies can be constructed and how they are applied. Let X be a set. By PX we denote the power set of X. Let B be a subset of PX . Definition 1.1. B is called a bornology if it has the following properties: 1. B is closed under taking subsets. 2. B is closed under forming finite unions. S 3. B2B B = X. The elements of the bornology are called the bounded subsets of X. 2 in Definition 1.1 Definition 1.2. B is a generalized bornology if it satisfies the Conditions 1 and 2 in Definition 1.1. Thus we we get the notion of a generalized bornology by dropping Condition 3 in Definition 1.1. Remark 1.3. Let x be in X. Then the singleton fxg belongs to any bornology. Indeed, by Condition 3 there exists an element B in B such that x 2 B. Then fxg ⊆ B and hence fxg 2 B by Condition 1. If B is a generalized bornology, then a point x in X is called bounded if fxg 2 B. Otherwise it is called unbounded. A generalized bornology is a bornology if and only all points of X are bounded. If X is a generalized bornological space, then we have a disjoint decomposition X = Xb tXu into the subsets of bounded and unbounded points. Then B becomes a bornology on Xb. Remark 1.4. Let A be an abelian group and consider the abelian group AX of functions from X to A. For f in AX we let supp(f) := fx 2 X j f(x) 6= 0g be the support of f. We have supp(f + f 0) ⊆ sup(f) [ supp(f 0). If B is a generalized bornology on X, then we can consider the subset X CB(X; A) := ff 2 A j supp(f) 2 Bg : 2 X 0 We shall see that this is a subgroup of A . Indeed, if f; f are in CB(X; A), then supp(f) and supp(f) belong to B. Since supp(f + f 0) ⊆ supp(f) [ supp(f 0) we see that also 0 0 supp(f + f ) belongs to B so that f + f is in CB(X; A). Similarly, supp(−f) = supp(f) so that with f also −f belongs to CB(X; A). In this way B determines a subgroup of AX of functions with bounded support. X 0 If A is a ring, then A is also a ring and CB(X; A) is a subring since supp(ff ) ⊆ supp(f) \ supp(f 0). We now turn to examples and constructions of bornologies. Let X be a set and (Bi)i2I be a family of (generalized) bornologies on X. T Lemma 1.5. The intersection i2I Bi is a (generalized) bornology. T 0 Proof. Let B := i2I Bi. Consider B in B and let B be a subset of B. Then B 2 Bi for 0 0 0 all i in I and hence B 2 Bi for all i in I. Hence B 2 B . 0 0 0 Assume that B; B belong to B. Then B 2 Bi and B 2 Bi for all i in I. Hence B [B 2 Bi for all i in I and hence B [ B0 2 B. This finishes the case of generalized bornologies. For the case of bornologies we consider x S in X. Then fxg 2 Bi for every i in I and hence fxg 2 B. Hence x 2 B2B B. Let A be a subset of PX . Then there is a smallest bornology containing A given by \ BhAi := B ; B; A⊆B where the intersection runs over all bornologies B on X. Similarly there is a smallest generalized bornology containing A \ BhAi~ = B~ ; B; A⊆B~ where the intersection runs over all generalized bornologies B on X containing A. S We can describe BhAi explicitly. We will assume that A2A A = X. Since singletons belong to every bornology (see Example 1.3) we can enlarge A by singletons without changing BhAi in order to ensure this condition. S Lemma 1.6. Assume that A = X. Then a subset B in PX belongs to BhAi if and A2A S only if of there exists a finite family (Ai)i2I in A such that B ⊆ i2I Ai. 3 Proof. We consider the subset B0 of P of subsets B of X such that there exists a finite S family (Ai)i2I in A with B ⊆ i2I Ai. One checks that B0 is a bornology which contains A. Therefore BhAi ⊆ B0. On the other hand, B0 is contained in any other bornology which contains A. This implies that B0 ⊆ BhAi. ~ Remark 1.7. The generalized bornology BhAi has a similar description. We let Xb := S 0 A2A A. Then A generates a bornology B on Xb whose elements are described as in Lemma 1.6. We have BhAi~ = B0. Example 1.8. If X is a set, then it has the minimal bornology Bmin of finite subsets and the maximal bornology Bmax of all subsets. It has the empty generalized bornology. Example 1.9. Let X be a topological space. The set Bqc of subsets of quasi-compact subsets of X is a bornology. The set Brc of relatively quasi compact subsets (subsets whose closures are quasi compact) is a generalized bornology. It might happen that the closure of a point is not quasi compact. This can happen of X is not Hausdorff. If we replace the condition \quasi compact" by \compact", then in general we do not get a bornology since the union of two compact subsets is not necessarily compact. Again this can happen since compactness includes the condition of being Hausdorff and a union of two Hausdorff subsets need not be Hausdorff. Example 1.10. Let d be a quasi-metric (infinite distances are allowed) on X. Then the metrically bounded subsets generate a bornology Bd. Note that we must say \generate" since the union of two bounded sets might be unbounded. But if d is a metric, then the bounded sets form a bornology since Condition 2 follows from the triangle inequality for the metric. The bornology Bd is generated by the set of balls fB(x; r) j x 2 X r 2 [0; 1)g. Example 1.11. Let Y be a topological space, A be a subset, and X := Y n A. Let B := fZ ⊆ X j Z¯ \ A = ;g. This is a generalized bornology. If Y is Hausdorff then it is a bornology. Example 1.12. Recall that a filter on a set X is a subset F of PX with the following properties: 1. ; 2= F and X 2 F. 2. F is closed under forming finite intersections. 4 3. F is closed under taking supersets. If F is a filter on X, then the set of complements F c := fX n F j F 2 Fg is a generalized bornology on X. Vice versa the complements of a generalized bornology B is a filter if and only if X 62 B, i.e., B is not the maximal bornology. The generalized bornology F c is a bornology if and only if the filter is free, i.e., if and only T if F 2F F = ;. Hence, upon taking complements, non-maximal bornologies correspond to free filters on X. Example 1.13. Let F be a subset of PX . Then F ? := fY ⊆ X j (8L 2 F j jL \ Y j < 1g is a bornology. It is called the dual bornology to F. We have an obvious inclusion F ⊆ (F ?)?. Definition 1.14. A (generalized) bornological space is a pair (X; B) of a set with a (generalized) bornology. Usually we write X for generalized bornological spaces and let BX denote its generalized bornology. Example 1.15. For a set X we write Xmin and Xmax for X equipped with the minimal and maximal bornology. We write X; for the generalized bornological space with the empty bornology. Example 1.16. We consider a bornological space X and a subset L: Definition 1.17. L is called locally finite if jL \ Bj < 1 for all B in BX . We let LF(X) denote the set of locally finite subsets of X. By Example 1.13, the subset LF(X) of PX is a bornology on X. ? ? ? We write (X; LF(X)) =: X . With this notation we have (Xmin) = Xmax and (Xmax) = Xmin. Let f : X ! Y be a map between the underlying sets of generalized bornological spaces. Definition 1.18. f is called: −1 1. proper if f (BY ) ⊆ BX . 2. bornological, if f(BX ) ⊆ BY . 5 Let A be a subset of PY . By the following lemma we can check properness of a map on generators. Lemma 1.19. S 1.
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