?WHAT IS... a Coarse Space? John Roe Imagine a new student of analysis. In Calculus I, she E ◦ F := {(x, z): ∃y, (x, y) ∈ E, (y,z) ∈ F}. hears about limits and continuity, probably at first The other axioms require that the diagonal should in a quite informal way: “the limit is what happens be a controlled set, and that subsets, transposes, on the small scale”. Later, this idea is formalized and (finite) unions of controlled sets should be in terms of the classical -δ definition, and soon it controlled. The appearance of subsets of X × X, becomes apparent that the natural domain of this rather than of X itself, is related to the word “uni- definition is the world of metric spaces. Then, per- formly” in our informal description of coarse equiv- haps in the first graduate course, the student takes alence at the end of the previous paragraph. In the final step in this journey of abstraction: she fact, it is more accurate to say that a coarse struc- learns that what really matters in understanding ture is the large-scale counterpart of a uniformity limits and continuity is not the numerical value of than of a topology. the metric, just the open sets that it defines. This These axioms are modeled on the behavior of realization leads naturally to the abstract notion the fundamental example, the bounded coarse struc- of topological space, but it also enhances under- ture on a metric space, where we say that a set is standing even in the metrizable world—for in- controlled if and only if the distance function stance, there is only one natural topology on a d: X × X → R+ is bounded on it. A coarse space is finite-dimensional (real) vector space, though there a set with a coarse structure, and a coarse map is are many metrics that give rise to it. The notion of coarse space arises through a sim- a proper map that sends controlled sets to con- ilar process of abstraction starting with the infor- trolled sets. Finally, two coarse spaces X and Y are mal idea of studying “what happens on the large coarsely equivalent if there exist coarse maps → → ◦ scale”. To understand this idea, consider the met- f : X Y and g: Y X such that the graphs of f g ◦ × × ric spaces Zn and Rn. Their small-scale structure— and g f are controlled subsets of Y Y and X X their topology—is entirely different, but on the respectively. The reader can easily check that the Zn → Rn large scale they resemble each other closely: any inclusion and the “integer part” function Rn → Zn geometric configuration in Rn can be approximated implement a coarse equivalence between Zn Rn by one in Zn, to within a uniformly bounded error. and . As another exercise, say that a coarse × We think of such spaces as “coarsely equivalent”. space X is bounded if X X is controlled. Verify Formally speaking, a coarse structure on a set that this corresponds to metric boundedness, and X is defined to be a collection of subsets of X × X, that if X is bounded and nonempty, the inclusion called the controlled sets or entourages for the of any point into X is a coarse equivalence. coarse structure, which satisfy some simple ax- Here are some more examples of coarse spaces ioms. The most important of these states that if E underlying classical constructions in algebra, geom- and F are controlled then so is etry, and topology. Let Gbe a locally compact topological group. John Roe is professor of mathematics at Pennsylvania The sets g∈G gK, as K ranges over compact sub- State University. His email address is [email protected]. sets of G × G, generate a canonical translation- 668 NOTICES OF THE AMS VOLUME 53, NUMBER 6 invariant coarse structure on G. When G is discrete an obstruction group depending on the control and finitely generated, this coincides with the space. Continuously controlled coarse spaces X ⊆ Y bounded coarse structure coming from any word- are particularly useful as parameter spaces here, length metric on G. Thanks to the work of Gromov because the relevant obstruction groups can be and others, geometric group theorists know that shown to be generalized homology groups of Y \ X. many interesting properties of infinite discrete Not unrelated to the previous example, coarse groups depend only on the large-scale properties spaces have appeared in the index theory of ellip- of their word-length metrics: that is, on their coarse tic partial differential operators on noncompact structure. For instance, it can be shown that such complete Riemannian manifolds. An elliptic oper- a group is coarsely equivalent to Zn if and only if ator D on a compact manifold has the Fredholm it actually contains Zn as a subgroup of finite index. property: the kernel has finite dimension, the range More generally let G act on a space V, with has finite codimension, and the index compact quotient; for instance, V might be the Index(D) = dim ker D − codim im D universal cover of a compact manifold M, and G the fundamental group of M. The sets g∈G gK, K is a topological invariant of D. On a noncompact compact in V × V, generate a coarse structure on manifold the Fredholm property does not hold in V; in the example of a universal cover, this is the the usual form. Nevertheless, one can define an coarse structure associated to the lift to V of any “index group” (actually the K-theory of a certain C∗- Riemannian metric on M. It is not hard to see that algebra), which only depends on the coarse struc- if the action is proper, then the map g → gx (for ture and which allows the index of D to be well- any fixed x ∈ V) gives a coarse equivalence G → V. defined as an element of this group. (Any compact This is the abstract form of an old result of Milnor manifold is coarsely equivalent to a point, so the and Svarc, which states that the orbit map g → gx, index group for all compact manifolds is the same. from the fundamental group G of a compact man- In fact it is Z, and one recovers the ordinary index.) ifold M to its universal cover V, is a coarse equiv- This construction allows the Atiyah-Singer index alence. Taking M to be a torus, we recover our theorem and its applications to be generalized to Zn → Rn original example of the inclusion . noncompact manifolds. An important task remains, Let X be a dense open subset of a compact however: to compute the index group in particu- metrizable topological space Y. One can define a lar cases. coarse structure on X by declaring that a subset Such computations have applications to differ- ⊆ × E X X is controlled if, whenever (un,vn) is a se- ential topology, in particular to the question of quence in E and one of the sequences un, vn con- which characteristic numbers are invariants of ho- \ verges to a point of Y X, the other sequence con- motopy type (the Novikov conjecture proposes an verges also to the same point. (To see where this answer to this question). A very general theorem Rn curious definition comes from, think of X as and of Yu computes the index group for a manifold that Y as the compactification of X by the “sphere at can be coarsely embedded in a Hilbert space. It fol- infinity”. Then every boundedly controlled set has lows that the Novikov conjecture is true for a com- the property indicated.) It can be shown that this pact manifold whose fundamental group coarsely continuously controlled coarse structure is not (ex- embeds into Hilbert space. This is a very large class cept in trivial cases) the bounded structure asso- of groups, including all hyperbolic groups, all lin- ciated to any metric. ear groups, and all amenable groups. In fact, any We have already mentioned the importance of discrete group that acts amenably on a compact the canonical coarse structure on an infinite dis- space must coarsely embed into Hilbert space. crete group. A different application occurs in con- It is now natural to ask whether every metric trolled topology, a method for addressing homeo- space, or every discrete group, can be coarsely em- morphism questions about manifolds that is rooted bedded into Hilbert space. Unfortunately, the an- in the work of Quinn and others. A typical con- swer is negative: some counterexamples are fur- trolled construction on a manifold will carry out nished by expander graphs. A systematic infinitely many of the basic “moves” of differen- understanding of all possible counterexamples and tial topology (connected sums, surgeries, handle at- their connection with geometry and index theory tachments, and so on). This infinite process must remains elusive. be “controlled” in such a way that the result con- verges in the topological, although perhaps not in Further reading the differentiable category. One way to achieve [1] MARTIN BRIDSON and ANDRÉ HAEFLIGER, Metric Spaces of this is to keep track of the sizes of the moves per- Non-Positive Curvature, Springer, 2000. formed by parameterizing them over a coarse [2] JOHN ROE, Lectures on Coarse Geometry, American space, called the control space. Typically, the con- Mathematical Society, 2003. struction can be carried out provided that some al- [3] SHMUEL WEINBERGER, The Topological Classification of gebraic invariant vanishes: an invariant that lies in Stratified Spaces, University of Chicago Press, 1994. JUNE/JULY 2006 NOTICES OF THE AMS 669.