Bonn THDM 2013: Program
Contents |
1 Introduction
1.1 K0 and Wall's finiteness obstruction
Speaker: WL
Abstract: 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.
Exercises: K-group, zeroth (Ex), Finitely dominated CW complexes (Ex).
References: [Lück1987], [Lück1989], [Mislin1995], [Ranicki1985], [Rosenberg1994], [Wall1965a], [Wall1966b].
1.2 K1 and Whitehead torsion
Speaker: WL
Abstract: 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.
Exercises: K-group, first (Ex), Whitehead torsion (Ex), Reidemeister torsion (Ex), K-group, all (Ex).
References: [Cohen1973], [Lück1989], [Lück2002], [Milnor1966], [Ranicki1985a], [Ranicki1987], [Rosenberg1994].
1.3 Normal maps and surgery below the middle dimension
Speaker: SA-F
Abstract:
Exercises:
References:
1.4 L-groups
Speaker: SA-F
Abstract:
Exercises:
References:
1.5 Surgery in the middle dimension
Speaker: SA-F
Abstract:
Exercises:
References:
1.6 The geometric surgery exact sequence
Speaker: SA-F
Abstract:
Exercises:
References:
2 Surgery on smooth manifolds
2.1 Homotopy spheres and other examples
Speaker: DC
Abstract:
Exercises:
References:
2.2 Smoothing and surgery
Speaker: DC
Abstract:
Exercises:
References:
2.3 Classifying spaces for surgery
Speaker: DC
Abstract:
Exercises:
References:
2.4 The Kervaire invariant in surgery
Speaker: DC
Abstract:
Exercises:
References:
3 Algebraic L-theory
Speaker: TM
Abstract:
Exercises:
References:
3.1 L-groups via chain complexes
Speaker: TM
Abstract:
Exercises:
References:
3.2 Signatures
Speaker: TM
Abstract:
Exercises:
References:
3.3 L-groups of categories and assembly maps
Speaker: TM
Abstract:
Exercises:
References:
3.4 Surgery obstructions and assembly maps
Speaker: TM
Abstract:
Exercises:
References:
4 The isomorphism conjectures
4.1 The Isomorphism Conjectures in the torsion-free case
Speaker :WL
Abstract: 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.
Exercises: Smash product (Ex), K-groups of oriented surface groups (Ex), Idempotents in group rings (Ex), Borel conjecture in dimensions 1 and 2 (Ex).
References: [Bartels&Lück&Reich2008], [Baum&Connes&Higson1994], [Farrell&Jones1993a], [Kreck&Lück2005], [Lück&Reich2006].
4.2 The Isomorphism Conjectures in general
Speaker: WL
Abstract: 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
Abstract: 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.
5 References
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- [Baum&Connes&Higson1994] P. Baum, A. Connes and N. Higson, Classifying space for proper actions and -theory of group -algebras, In -algebras: 1943–1993 (San Antonio, TX 1993), (1994), 240–291. MR1292018 (96c:46070) Zbl 0830.46061
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