Geometry & Topology RTG

In this series of two lectures, I will introduce and motivate the stable homotopy category, AKA the homotopy category of spectra. I will then give basic constructions and examples. Very little background in homotopy theory will be assumed.

The curvature measures, first introduced by H. Minkowski under the name of quermassintegrals, have turned up in various unexpected places such as Weyl's tube formula and various Crofton-type formulas. The modern definition of these measures, is through the normal cycle, an object Morse theoretic in nature, associated to "reasonably singular" subsets of an Euclidean space and introduced independently and by different means by J. Fu and Kashiwara-Schapira. In these lectures I hope to give you a tour of the ecclectic world of normal cycles, curvature measures and singular Morse theory.

In this first talk on the Index Theorem we define elliptic differential operators and their principal symbols. Then we recall topological K-theory and show that the principal symbol of an elliptic operator D on a manifold X represents an element in the relative K-theory group K(T*X,(T*X)_0) where T*X is the cotangent bundle of X, and (T*X)_0 is the subspace of non-zero cotangent vectors. A key statement is that the index of an elliptic operator D on a compact manifold X depends only on that K-theory element given by its principal symbol.

In this lecture we state the Index Theorem in its K-theory formulation. The key player here is the "topological index homomorphism" from the K-theory group K(T*X,(T*X)_0) to the integers, which is constructed in a purely topological way that has nothing to do with differential operators. According to the Index Theorem the index of a elliptic differential operator D on a compact manifold X is equal to the image of the element in K(T*X,(T*X)_0) represented by the principal symbol of D under the topological index homomorphism.

The principal symbol of an elliptic operator on a manifold X represents an element in the K-theory group K(T*X,(T*X)_0) where T*X is the cotangent bundle of X, and (T*X)_0 is the subspace of non-zero cotangent vectors. A spin structure on X determines a Thom class U in K(T*X,(T*X)_0) and hence every element of this K-theory group is of the form U times the pullback of a vector bundle V over X. In this lecture we describe the elliptic operator whose principal symbol is that K-theory element; this operator is called the Dirac operator on X twisted by V.

The topic of this lecture is the cohomological version of the Index Theorem which describes the index of Dirac operators on spin manifold X in terms of characteristic numbers of X. More precisely, the index is given by evaluating the A-roof genus of the tangent bundle TX (a polynomial in the Chern classes of the complexification of TX) on the fundamental class of X. More generally, the index of the Dirac operator twisted by a vector bundle V is given by the characteristic number obtained by evaluating the product of the A-roof genus of TX and the Chern character of V on the fundamental class of X. No prior knowledge of characteristic classes is needed for this talk.

We present a variety of easy to understand interesting results and prominent conjectures from different areas of mathematics, e.g., group theory, algebra, topology, K-theory, differential geometry and algebraic geometry. These are on the first glance not related to L^2-Betti numbers. We will later see in the other lectures how L^2-methods can be used to prove these results or to confirm the conjectures in rather general cases. The remainder of the first talk consists of an introduction to the L^2-setting from the topological point of view using von Neumann algebras and Hilbert modules and explaining the basic properties of L^2-Betti numbers.

In the second lecture we present more advanced properties of L^2-Betti numbers. This includes their analytic interpretation in terms of heat kernels and the L^2-Hodge-de Rham Theorem. We compute the L^2-Betti numbers for the universal coverings of hyperbolic manifolds and manfolds of dimension less or equal to 3. Finally we deal with the Atiyah-Conjecture which predicts that the L^2-Betti numbers of the universal coverings of closed Riemannian manifolds with torsionfree fundamental groups are always integers. We link it to a problem about projective class groups of groups rings and other rings associated to von Neumann algebras and explain its consequences.

In the third lecture we will explain how one can approximate L^2-Betti number by classical Betti numbers. Then we will introduce the generalized Murray-von Neumann dimension which allows to define L^2-Betti numbers for arbitrary G-spaces and in particular for arbitrary groups. This is used to explain how some of the results stated in the first lecture can be proven using L^2-methods. Finally we give a brief outlook of the overwhelming variety of other L^2-invariants, prominent results and open problems.