Coarse Philosophies of Coarse Embeddabilities: Metric Space Conjectures Act Algorithmically On Manifolds – Thought of the Day 145.0

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A coarse structure on a set X is defined to be a collection of subsets of X × X, called the controlled sets or entourages for the coarse structure, which satisfy some simple axioms. The most important of these states that if E and F are controlled then so is

E ◦ F := {(x, z) : ∃y, (x, y) ∈ E, (y, z) ∈ F}

Consider the metric spaces Zn and Rn. Their small-scale structure, their topology is entirely different, but on the large scale they resemble each other closely: any geometric configuration in Rn can be approximated by one in Zn, to within a uniformly bounded error. We think of such spaces as “coarsely equivalent”. The other axioms require that the diagonal should be a controlled set, and that subsets, transposes, and (finite) unions of controlled sets should be controlled. It is accurate to say that a coarse structure is the large-scale counterpart of a uniformity than of a topology.

Coarse structures and coarse spaces enjoy a philosophical advantage over coarse metric spaces, in that, all left invariant bounded geometry metrics on a countable group induce the same metric coarse structure which is therefore transparently uniquely determined by the group. On the other hand, the absence of a natural gauge complicates the notion of a coarse family, while it is natural to speak of sets of uniform size in different metric spaces it is not possible to do so in different coarse spaces without imposing additional structure.

Mikhail Leonidovich Gromov introduced the notion of coarse embedding for metric spaces. Let X and Y be metric spaces.

A map f : X → Y is said to be a coarse embedding if ∃ nondecreasing functions ρ1 and ρ2 from R+ = [0, ∞) to R such that

  • ρ1(d(x,y)) ≤ d(f(x),f(y)) ≤ ρ2(d(x,y)) ∀ x, y ∈ X.
  • limr→∞ ρi(r) = +∞ (i=1, 2).

Intuitively, coarse embeddability of a metric space X into Y means that we can draw a picture of X in Y which reflects the large scale geometry of X. In early 90’s, Gromov suggested that coarse embeddability of a discrete group into Hilbert space or some Banach spaces should be relevant to solving the Novikov conjecture. The connection between large scale geometry and differential topology and differential geometry, such as the Novikov conjecture, is built by index theory. Recall that an elliptic differential operator D on a compact manifold M is Fredholm in the sense that the kernel and cokernel of D are finite dimensional. The Fredholm index of D, which is defined by

index(D) = dim(kerD) − dim(cokerD),

has the following fundamental properties:

(1) it is an obstruction to invertibility of D;

(2) it is invariant under homotopy equivalence.

The celebrated Atiyah-Singer index theorem computes the Fredholm index of elliptic differential operators on compact manifolds and has important applications. However, an elliptic differential operator on a noncompact manifold is in general not Fredholm in the usual sense, but Fredholm in a generalized sense. The generalized Fredholm index for such an operator is called the higher index. In particular, on a general noncompact complete Riemannian manifold M, John Roe (Coarse Cohomology and Index Theory on Complete Riemannian Manifolds) introduced a higher index theory for elliptic differential operators on M.

The coarse Baum-Connes conjecture is an algorithm to compute the higher index of an elliptic differential operator on noncompact complete Riemannian manifolds. By the descent principal, the coarse Baum-Connes conjecture implies the Novikov higher signature conjecture. Guoliang Yu has proved the coarse Baum-Connes conjecture for bounded geometry metric spaces which are coarsely embeddable into Hilbert space. The metric spaces which admit coarse embeddings into Hilbert space are a large class, including e.g. all amenable groups and hyperbolic groups. In general, however, there are counterexamples to the coarse Baum-Connes conjecture. A notorious one is expander graphs. On the other hand, the coarse Novikov conjecture (i.e. the injectivity part of the coarse Baum-Connes conjecture) is an algorithm of determining non-vanishing of the higher index. Kasparov-Yu have proved the coarse Novikov conjecture for spaces which admit coarse embeddings into a uniformly convex Banach space.

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