Isomorphismusvermutungen und 3-Mannigfaltigkeiten

TL;DR

This paper provides an axiomatic framework to determine when the Meta-Isomorphism-Conjecture holds for fundamental groups of 3-manifolds, confirming the Farrell-Jones conjectures for certain classes of these groups.

Contribution

It establishes conditions under which the Meta-Isomorphism-Conjecture applies to 3-manifold groups, extending the validity of Farrell-Jones conjectures to new cases.

Findings

Proves the conjectures for fundamental groups of 3-manifolds under specific conditions.

Shows the conjectures hold if true for semidirect products of Z^2 with Z.

Provides an axiomatic summary based on prior results by S.K. Roushon.

Abstract

Based on results by S.K. Roushon (math.KT/0408243 and math.KT/0405211) this thesis summarizes in an axiomatic way when a Meta-Isomorphism-Conjecture in the sense of Lueck and Reich (math.KT/0402405) is true for fundamental groups of 3-dimensional manifolds. In particular we prove that the fibered Farrell-Jones isomorphism conjectures for L-theory and algebraic K-theory are true for this class of groups if they are true for semidirect products of $\mathbb{Z^2}$ with $\mathbb{Z}$.