Assembly Maps

TL;DR

This paper explores the concept of assembly maps from a homotopy theoretic perspective, providing interpretations in surgery theory, controlled topology, and index theory, with implications for major conjectures in K- and L-theory.

Contribution

It offers a new homotopy theoretic framework for understanding assembly maps and connects them to key conjectures in topology and operator algebras.

Findings

Assembly maps are analyzed from a homotopy theoretic perspective.

Interpretations in surgery theory, controlled topology, and index theory are provided.

Implications for Farrell-Jones and Baum-Connes conjectures are discussed.

Abstract

We introduce and analyze the concept of an assembly map from the original homotopy theoretic point of view. We give also interpretations in terms of surgery theory, controlled topology and index theory. The motivation is that prominent conjectures of Farrell-Jones and Baum-Connes about K- and L-theory of group rings and group C^*-algebras predict that certain assembly maps are weak homotopy equivalences.