Bonn THDM 2013: Program
Contents |
1 Introduction
1.1 K0 and Wall's finiteness obstruction
Speaker: WL
We introduce the projective class group . We explain computations for special rings, e.g., fields, complex group rings for finite groups. We state Swan's Theorem which relate the projective class group of the ring of continuous -valued functions to the Grothendieck group of vector bundles over , if is a finite -complex. We discuss Wall's finiteness obstruction that decides whether a finitely dominated -complex is homotopy equivalent to a finite -complex and takes values in the projective class group of the integral group ring of the fundamental group.
References: citeDW. Lück, The geometric finiteness obstruction, Proc. London Math. Soc. (3) 54 (1987), no.2, 367–384. MR872812 (88i:57007) Zbl 0626.57011, [Lück1989], [Mislin1995], [Ranicki1985], [Rosenberg1994], [Wall1965a], [Wall1966].
1.2 K1 and Whitehead torsion
Speaker: WL
We introduce and the Whitehead group . We define the Whitehead torsion of a homotopy equivalence of finite connected -complexes. We discuss the algebraic and topological significance of these notions, in particular the -cobordism theorem. We briefly introduce the surgery program. Finally we introduce negative -theory and the Bass-Heller-Swan decomposition.
1.3 Normal maps and surgery below the middle dimension
1.4 L-groups
1.5 Surgery in the middle dimension
1.6 The geometric surgery exact sequence
2 Surgery on smooth manifolds
2.1 Homotopy spheres and other examples
2.2 Smoothing and surgery
2.3 Classifying spaces for surgery
2.4 The Kervaire invariant in surgery
3 Algebraic L-theory
3.1 L-groups via chain complexes
3.2 Signatures
3.3 L-groups of categories and assembly maps
3.4 Surgery obstructions and assembly maps
4 The isomorphism conjectures
4.1 The Isomorphism Conjectures in the torsion-free case
Speaker :WL
We introduce spectra and how they yield homology theories. We state the Farrell-Jones Conjecture and the Baum-Connes Conjecture for torsion free groups and discuss applications of these conjectures, such as the Kaplansky Conjecture and the Borel Conjecture. We explain that the formulations for torsion free groups cannot extend to arbitrary groups.
4.2 The Isomorphism Conjectures in general
Speaker: WL
We introduce classifying spaces for families. We define equivariant homology theories and explain how they can be construced by spectra over groupoids. Then we state the Farrell-Jones Conjecture and the Baum-Connes Conjecture in general. We discuss further applications, such as the Novikov Conjecture.
4.3 Status and methods of proof;
Speaker: WL
We give a status report of the Farrell-Jones Conjecture, discuss open cases and the search for potential counterexamples, and briefly survey the methods of proof.