Bonn THDM 2013: Program

Revision as of 14:42, 19 July 2013

• Background
• Schedule

1 Introduction

1.1 K₀ and Wall's finiteness obstruction

Speaker: WL

Abstract: We introduce the projective class group $* [[Bonn THDM 2013: Background|Background]] <!--* [[Bonn THDM 2013: Exercises|Exercises]] * [[Bonn THDM 2013: Glossary|Glossary]]--> * [[Bonn THDM 2013: Schedule|Schedule]] ==Introduction== ===K₀ and Wall's finiteness obstruction === <wikitex>; Speaker: [[User:Lueck|WL]] We introduce the ''projective class group $K_0(R)$''. 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 $C(X)$ of continuous $\Rr$-valued functions to the Grothendieck group of vector bundles over $X$, if $X$ is a finite $CW$-complex. We discuss Wall's finiteness obstruction that decides whether a finitely dominated $CW$-complex is homotopy equivalent to a finite $CW$-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: {{cite|Lück1987}}, {{cite|Lück1989}}, {{cite|Mislin1995}}, {{cite|Ranicki1985}}, {{cite|Rosenberg1994}}, {{cite|Wall1965a}}, {{cite|Wall1966b}}. </wikitex> ===K₁ and Whitehead torsion === <wikitex>; Speaker: [[User:Lueck|WL]] We introduce $K_1(R)$ and the Whitehead group $\textup{Wh}(G)$. We define the Whitehead torsion of a homotopy equivalence of finite connected $CW$-complexes. We discuss the algebraic and topological significance of these notions, in particular the $s$-cobordism theorem. We briefly introduce the surgery program. Finally we introduce negative $K$-theory and the Bass-Heller-Swan decomposition. </wikitex> ===Normal maps and surgery below the middle dimension === <wikitex>; [[User:Spiros Adams-Florou|SA-F]] </wikitex> ===L-groups === <wikitex>; [[User:Spiros Adams-Florou|SA-F]] </wikitex> ===Surgery in the middle dimension === <wikitex>; [[User:Spiros Adams-Florou|SA-F]] </wikitex> ===The geometric surgery exact sequence === <wikitex>; [[User:Spiros Adams-Florou|SA-F]] </wikitex> == Surgery on smooth manifolds == <wikitex>; [[User:Diarmuid Crowley|DC]] </wikitex> ===Homotopy spheres and other examples === <wikitex>; [[User:Diarmuid Crowley|DC]] </wikitex> ===Smoothing and surgery === <wikitex>; [[User:Diarmuid Crowley|DC]] </wikitex> ===Classifying spaces for surgery === <wikitex>; [[User:Diarmuid Crowley|DC]] </wikitex> ===The Kervaire invariant in surgery === <wikitex>; [[User:Diarmuid Crowley|DC]] </wikitex> == Algebraic L-theory == <wikitex>; [[User:Tibor Macko|TM]] </wikitex> ===L-groups via chain complexes === <wikitex>; [[User:Tibor Macko|TM]] </wikitex> ===Signatures=== <wikitex>; [[User:Tibor Macko|TM]] </wikitex> ===L-groups of categories and assembly maps === <wikitex>; [[User:Tibor Macko|TM]] </wikitex> ===Surgery obstructions and assembly maps === <wikitex>; [[User:Tibor Macko|TM]] </wikitex> == The isomorphism conjectures == <wikitex>; </wikitex> ===The Isomorphism Conjectures in the torsion-free case === <wikitex>; Speaker :[[User:Lueck|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. </wikitex> ===The Isomorphism Conjectures in general === <wikitex>; Speaker: [[User:Lueck|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. </wikitex> ===Status and methods of proof; === <wikitex>; Speaker: [[User:Lueck|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. </wikitex> == References== {{#RefList:}} [[Category:Bonn THDM 2013]]K_0(R)$. 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 $C(X)$ of continuous $\Rr$-valued functions to the Grothendieck group of vector bundles over $X$, if $X$ is a finite $CW$-complex. We discuss Wall's finiteness obstruction that decides whether a finitely dominated $CW$-complex is homotopy equivalent to a finite $CW$-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 $K_1(R)$ and the Whitehead group $\textup{Wh}(G)$. We define the Whitehead torsion of a homotopy equivalence of finite connected $CW$-complexes. We discuss the algebraic and topological significance of these notions, in particular the $s$-cobordism theorem. We briefly introduce the surgery program. Finally we introduce negative $K$-theory and the Bass-Heller-Swan decomposition.

1.3 Normal maps and surgery below the middle dimension

SA-F

1.4 L-groups

SA-F

1.5 Surgery in the middle dimension

SA-F

1.6 The geometric surgery exact sequence

SA-F

2 Surgery on smooth manifolds

DC

2.1 Homotopy spheres and other examples

DC

2.2 Smoothing and surgery

DC

2.3 Classifying spaces for surgery

DC

2.4 The Kervaire invariant in surgery

DC

3 Algebraic L-theory

TM

3.1 L-groups via chain complexes

TM

3.2 Signatures

TM

3.3 L-groups of categories and assembly maps

TM

3.4 Surgery obstructions and assembly maps

TM

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.

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

• [Lück1987] W. Lück, The geometric finiteness obstruction, Proc. London Math. Soc. (3) 54 (1987), no.2, 367–384. MR872812 (88i:57007) Zbl 0626.57011
• [Lück1989] W. Lück, Transformation groups and algebraic $K$-theory, LNM, 1408, Springer-Verlag, 1989. MR1027600 (91g:57036) Zbl 0709.57024
• [Mislin1995] G. Mislin, Wall's finiteness obstruction, Handbook of algebraic topology, North-Holland, Amsterdam, (1995), 1259–1291. MR1361911 (97g:57029) Zbl 0870.57030
• [Ranicki1985] A. Ranicki, The algebraic theory of finiteness obstruction, Math. Scand. 57 (1985), no.1, 105–126. MR815431 (87d:18014) Zbl 0589.57018
• [Rosenberg1994] J. Rosenberg, Algebraic $K$-theory and its applications, Graduate Texts in Mathematics, 147, Springer-Verlag, 1994. MR1282290 (95e:19001) Zbl 0801.19001
• [Wall1965a] C. T. C. Wall, Finiteness conditions for $CW$-complexes, Ann. of Math. (2) 81 (1965), 56–69. MR0171284 (30 #1515) Zbl 0152.21902
• [Wall1966b] C. T. C. Wall, Finiteness conditions for $CW$ complexes. II, Proc. Roy. Soc. Ser. A 295 (1966), 129–139. MR0211402 (35 #2283)Zbl 0152.21902