George Raptis (Regensburg): Parametrized cobordism categories and bivariant A-theory

Waldhausen's algebraic K-theory of spaces is an extension of algebraic K-theory from rings to spaces (or ring spectra) which also encodes important geometric information about manifolds. Bivariant A-theory is a bivariant extension of algebraic K-theory from spaces to fibrations of spaces. In this talk, I will first recall the definition and basic properties of bivariant A-theory and the A-theory Euler characteristic of Dwyer-Weiss-Williams. I will then introduce a bivariant version of the cobordism category and explain how this may be regarded as a universal space for the definition of additive characteristic classes of smooth bundles. Lastly, I will introduce a bivariant extension of the Dwyer-Weiss-Williams characteristic and discuss the Dwyer-Weiss-Williams smooth index theorem in this context. Time permitting, I will also discuss some ongoing related work on the cobordism category of h-cobordisms. This is joint work with W. Steimle.