Parametrized cobordism categories and the Dwyer-Weiss-Williams index theorem

TL;DR

This paper introduces parametrized cobordism categories as bivariant theories, providing a new proof of the Dwyer-Weiss-Williams index theorem for smooth bundle Euler characteristics.

Contribution

It defines parametrized cobordism categories and characterizes bivariant transformations, leading to a simplified proof of the family index theorem.

Findings

Bivariant transformations are determined by their value at the universal disk bundle.

Provides a short proof of the Dwyer-Weiss-Williams index theorem.

Establishes properties of parametrized cobordism categories as bivariant theories.

Abstract

We define parametrized cobordism categories and study their formal properties as bivariant theories. Bivariant transformations to a strongly excisive bivariant theory give rise to characteristic classes of smooth bundles with strong additivity properties. In the case of cobordisms between manifolds with boundary, we prove that such a bivariant transformation is uniquely determined by its value at the universal disk bundle. This description of bivariant transformations yields a short proof of the Dwyer-Weiss-Williams family index theorem for the parametrized A-theory Euler characteristic of a smooth bundle.