Bonn THDM 2013: Program

• Background
• Schedule

1 Introduction

1.1 K₀ and Wall's finiteness obstruction

Speaker: WL

Abstract: We introduce the projective class group $K_0(R)$$* [[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]] Abstract: 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]] 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. Exercises: [[K-group, first (Ex)]], [[Whitehead torsion (Ex)]], [[Reidemeister torsion (Ex)]], [[K-group, all (Ex)]]. References: \cite{Cohen1973}, \cite{Lück1989}, \cite{Lück2002}, \cite{Milnor1966}, \cite{Ranicki1985a}, \cite{Ranicki1987}, \cite{Rosenberg1994}. </wikitex> ===Normal maps and surgery below the middle dimension === <wikitex>; Speaker: [[User:Spiros Adams-Florou|SA-F]] Abstract: Exercises: References: </wikitex> ===L-groups === <wikitex>; Speaker: [[User:Spiros Adams-Florou|SA-F]] Abstract: Exercises: References: </wikitex> ===Surgery in the middle dimension === <wikitex>; Speaker: [[User:Spiros Adams-Florou|SA-F]] Abstract: Exercises: References: </wikitex> ===The geometric surgery exact sequence === <wikitex>; Speaker: [[User:Spiros Adams-Florou|SA-F]] Abstract: Exercises: References: </wikitex> == Surgery on smooth manifolds == ===Homotopy spheres and other examples === <wikitex>; Speaker: [[User:Diarmuid Crowley|DC]] Abstract: Exercises: References: </wikitex> ===Smoothing and surgery === <wikitex>; Speaker: [[User:Diarmuid Crowley|DC]] Abstract: Exercises: References: </wikitex> ===Classifying spaces for surgery === <wikitex>; Speaker: [[User:Diarmuid Crowley|DC]] Abstract: Exercises: References: </wikitex> ===The Kervaire invariant in surgery === <wikitex>; Speaker: [[User:Diarmuid Crowley|DC]] Abstract: Exercises: References: </wikitex> == Algebraic L-theory == <wikitex>; Speaker: [[User:Tibor Macko|TM]] Abstract: Exercises: References: </wikitex> ===L-groups via chain complexes === <wikitex>; Speaker: [[User:Tibor Macko|TM]] Abstract: Exercises: References: </wikitex> ===Signatures=== <wikitex>; Speaker: [[User:Tibor Macko|TM]] Abstract: Exercises: References: </wikitex> ===L-groups of categories and assembly maps === <wikitex>; Speaker: [[User:Tibor Macko|TM]] Abstract: Exercises: References: </wikitex> ===Surgery obstructions and assembly maps === <wikitex>; Speaker: [[User:Tibor Macko|TM]] Abstract: Exercises: References: </wikitex> == The isomorphism conjectures == ===The Isomorphism Conjectures in the torsion-free case === <wikitex>; Speaker :[[User:Lueck|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: \cite{Bartels&Lück&Reich2008}, \cite{Baum&Connes&Higson1994}, \cite{Farrell&Jones1993a}, \cite{Kreck&Lück2005}, \cite{Lück&Reich2006}. </wikitex> ===The Isomorphism Conjectures in general === <wikitex>; Speaker: [[User:Lueck|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. Exercises: [[Classifying spaces for proper group actions (Ex)]], [[Equivariant homology (Ex)]], [[K-group, zeroth II (Ex)]]. References: </wikitex> ===Status and methods of proof; === <wikitex>; Speaker: [[User:Lueck|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. </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)$$C(X)$ of continuous $\Rr$$\Rr$-valued functions to the Grothendieck group of vector bundles over $X$$X$, if $X$$X$ is a finite $CW$$CW$-complex. We discuss Wall's finiteness obstruction that decides whether a finitely dominated $CW$$CW$-complex is homotopy equivalent to a finite $CW$$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)$$K_1(R)$ and the Whitehead group $\textup{Wh}(G)$$\textup{Wh}(G)$. We define the Whitehead torsion of a homotopy equivalence of finite connected $CW$$CW$-complexes. We discuss the algebraic and topological significance of these notions, in particular the $s$$s$-cobordism theorem. We briefly introduce the surgery program. Finally we introduce negative $K$$K$-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.

Exercises: Classifying spaces for proper group actions (Ex), Equivariant homology (Ex), K-group, zeroth II (Ex).

References:

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

• [Bartels&Lück&Reich2008] A. Bartels, W. Lück and H. Reich, On the Farrell-Jones Conjecture and its applications, Journal of Topology 1 (2008), 57–86. MR2365652 (2008m:19001) Zbl 1141.19002
• [Baum&Connes&Higson1994] P. Baum, A. Connes and N. Higson, Classifying space for proper actions and $K$$K$-theory of group $C^\ast$$C^\ast$-algebras, In $C^\ast$$C^\ast$-algebras: 1943–1993 (San Antonio, TX 1993), (1994), 240–291. MR1292018 (96c:46070) Zbl 0830.46061
• [Cohen1973] M. M. Cohen, A course in simple-homotopy theory, Springer-Verlag, New York, 1973. MR0362320 (50 #14762) Zbl 0261.57009
• [Farrell&Jones1993a] F. T. Farrell and L. E. Jones, Isomorphism conjectures in algebraic $K$$K$-theory, J. Amer. Math. Soc. 6 (1993), no.2, 249–297. MR1179537 (93h:57032) Zbl 0798.57018
• [Kreck&Lück2005] M. Kreck and W. Lück, The Novikov conjecture, Birkhäuser Verlag, Basel, 2005. MR2117411 (2005i:19003) Zbl 1058.19001
• [Lück&Reich2006] W. Lück and H. Reich, Detecting $K$$K$-theory by cyclic homology, Proc. London Math. Soc. (3) 93 (2006), no.3, 593–634. MR2266961 (2007h:19008) Zbl 1116.19002
• [Lück1987] W. Lück, The geometric finiteness obstruction, Proc. London Math. Soc. (3) 54 (1987), no.2, 367–384. MR872812 (88i:57007) Zbl 0626.57011
• [Lück1989] W. Lück, Transformation groups and algebraic $K$$K$-theory, LNM, 1408, Springer-Verlag, 1989. MR1027600 (91g:57036) Zbl 0709.57024
• [Lück2002] W. Lück, $L^2$$L^2$-invariants: theory and applications to geometry and $K$$K$-theory, Springer-Verlag, Berlin, 2002. MR1926649 (2003m:58033) Zbl 1009.55001
• [Milnor1966] J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR0196736 (33 #4922) Zbl 0147.23104
• [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
• [Ranicki1985a] A. Ranicki, The algebraic theory of torsion. I. Foundations, Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math., 1126, Springer (1985), 199–237. MR802792 (87a:18017) Zbl 0591.18007
• [Ranicki1987] A. Ranicki, The algebraic theory of torsion. II. Products, $K$$K$-Theory 1 (1987), no.2, 115–170. MR899919 (89f:18008) Zbl 0591.18007
• [Rosenberg1994] J. Rosenberg, Algebraic $K$$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$$CW$-complexes, Ann. of Math. (2) 81 (1965), 56–69. MR0171284 (30 #1515) Zbl 0152.21902
• [Wall1966b] C. T. C. Wall, Finiteness conditions for $CW$$CW$ complexes. II, Proc. Roy. Soc. Ser. A 295 (1966), 129–139. MR0211402 (35 #2283)Zbl 0152.21902