Enlargeability and index theory

Enlargeability and index theory

Let M be a closed enlargeable spin manifold. We show nontriviality of the universal index obstruction in the K-theory of the maximal C-algebra of the fundamental group of M . Our proof is independent from the injectivity of the Baum-Connes assembly map for π1(M) and relies on the construction of a certain infinite dimensional flat vector bundle out of a sequence of finite dimensional vector bun...

Enlargeability and Index Theory: Infinite Covers

In [5] we showed non-vanishing of the universal index elements in the K-theory of the maximal C∗-algebras of the fundamental groups of enlargeable spin manifolds. The underlying notion of enlargeability was the one from [3], involving contracting maps defined on finite covers of the given manifolds. In the paper at hand, we weaken this assumption to the one in [4] where infinite covers are allo...

Index Theory

Lecture Notes taken by Nilufer Koldan In this lecture I will describe one of the most significant achievements of the second half of the XX century – the Atiyah-Singer index theorem. I will also discuss some more recent developments in the area as well as some open problems. 1. The definition of the index 1.1. The finite-dimensional case. Let A : V 1 → V 2 be a linear map between finite dimensi...

Equivariant KK-theory and noncommutative index theory

Conley Index Theory and Novikov-Morse Theory

We derive general Novikov-Morse type inequalities in a Conley type framework for flows carrying cocycles, therefore generalizing our results in [FJ2] derived for integral cocycle. The condition of carrying a cocycle expresses the nontriviality of integrals of that cocycle on flow lines. Gradient-like flows are distinguished from general flows carrying a cocycle by boundedness conditions on thes...