# Bonn THDM 2013: Program

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## 1 Introduction

### 1.1 K0 and Wall's finiteness obstruction

Speaker: WL

Slides: Lecture 1

Abstract: We introduce the projective class group $K_0(R)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}K_0(R)$. We explain computations for special rings, e.g., fields, complex group rings for finite groups. We state Swan's Theorem which relates 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 which 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.

References: [Lück1987], [Lück1989], [Mislin1995], [Ranicki1985], [Rosenberg1994], [Wall1965a], [Wall1966b]

### 1.2 K1 and Whitehead torsion

Speaker: WL

Slides: Lecture 2

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.

References: [Cohen1973], [Lück1989], [Lück2002], [Milnor1966], [Ranicki1985a], [Ranicki1987], [Rosenberg1994]

### 1.3 Normal maps and surgery below the middle dimension

Speaker: SA-F

Slides: Lecture 3

Abstract: We begin with a discussion of the surgery programme, motivating the definition of a degree one normal map. The surgery method proceeds via a two stage obstruction theory. We will primarily concentrate on the second $L$$L$-theoretic obstruction which corresponds to the question of when a degree one normal map is normal bordant to a homotopy equivalence. Introducing surgery and the surgery step we explain how surgery can always be performed below the middle dimension. If surgery can also be done in the middle dimension Poincaré duality and Whitehead's theorem give us a homotopy equivalence. As we shall see in the next two lectures this cannot always be done... .

References: [Ranicki2002, Chapters 1 & 10], [Lück2001, Chapter 3], [Wall1970b, Chapter 1], [Kervaire&Milnor1963], [Browder1972]

### 1.4 The even-dimensional surgery obstruction

Speaker: SA-F

Slides: Lecture 4

Abstract: We begin by explaining how we cannot necessarily do surgery on an element $x\in \pi_{k+1}(f)$$x\in \pi_{k+1}(f)$ in the middle dimension if $n=2k$$n=2k$ as immersions may not be regularly homotopic to embeddings. For $k\geqslant 3$$k\geqslant 3$ we can do surgery if the self-intersection number of the immersion is trivial. This motivates us to study intersections and self-intersections which leads to the definition of the $Q$$Q$-groups and Wall's $\mu$$\mu$ form. From the geometry we extract the algebraic obstruction to surgery in the middle dimension which lies in the even dimensional $L$$L$-groups. For $k\geqslant 3$$k\geqslant 3$ the surgery obstruction vanishes if and only if we can do finitely many surgeries on our degree one normal map to obtain a degree one normal map covering a homotopy equivalence.

References: [Ranicki2002, Chapter 11], [Lück2001, Chapter 4, 4.1-4.3], [Wall1970b, Chapters 5 & 8]

### 1.5 The odd-dimensional surgery obstruction

Speaker: SA-F

Slides: Lecture 5

Abstract: We begin by discussing how even though we can always do surgery on an element $x \in \pi_{k+1}(f)$$x \in \pi_{k+1}(f)$ for $n= 2k+1$$n= 2k+1$ the surgery does not necessarily make the surgery kernel smaller as different choices of framing have different results. We discuss Heegaard splittings and from the geometry motivate the definition of quadratic formations. We introduce the kernel formation associated to a Heegaard splitting of a degree one normal map and the notion of a presentation. We characterise algebraically when we can do surgery to kill $\pi_*(f)$$\pi_*(f)$ and this leads to the definition of the odd-dimensional $L$$L$-groups; in dimensions at least five the surgery obstruction vanishes if and only if our degree one normal map is normal bordant to a homotopy equivalence.

References: [Ranicki2002, Chapter 12], [Lück2001, Chapter 4, 4.4-4.6], [Wall1970b, Chapters 6-7], [Wall1963], [Novikov1970], [Ranicki1973]

### 1.6 The geometric surgery exact sequence

Speaker: SA-F

Slides: Lecture 6

Abstract: We begin with a discussion of Wall realisation: if we permit ourselves to consider manifolds with boundary then every surgery obstruction can be realised. We then reach the culmination of the classical surgery programme, namely the surgery exact sequence - the main theoretical tool for the classification of high-dimensional manifolds. We also discuss the simple surgery exact sequence.

References: [Ranicki2002, Chapter 13], [Lück2001, Chapter 5]

## 2 Surgery on smooth manifolds

### 2.1 Homotopy spheres and other examples

Speaker: DC

Slides: Lecture 7

Abstract: We review the Kervaire-Milnor classification of the group $\Theta_n$$\Theta_n$ of homotopy spheres, that is the group of diffeomorphism classes oriented closed smooth manifolds homotopy equivalent to the $n$$n$-sphere. This classification is equivalent to computing the structure set of $S^n$$S^n$. We report on what is known about the group $\Theta_n$$\Theta_n$. We then consider the problem of classifying homotopy equivalent manifolds up to diffeomorphism and show how this is equivalent to computing the action of the group of self-homotopy equivalences on the structure set. Finally, we present the structure sets of products of spheres and certain Stiefel manifolds as illustrative examples.

References: [Kervaire&Milnor1963], [Levine1983], [Brumfiel1968], [Crowley&Lück&Macko2013, Ch.6], [Crowley2010]

### 2.2 Normal invariants in detail

Speaker: DC

Slides: Lecture 8

Abstract: We present the Browder-Sullivan theory of normal invariants of degree one normal maps in detail and in particular show how Spanier-Whitehead duality is used to convert normal bordism classes into maps to certain classifying spaces. We then give several examples including self-homotopy equivalences with non-trivial normal invariant which lead to Poincaré complexes whose Spivak normal fibration does not admit a vector bundle reduction.

References: [Browder1972, II, §4], [Madsen&Taylor&Williams1980], [Crowley&Hambleton2013, §6, §7]

### 2.3 Smoothing and surgery

Speaker: DC

Slides: Lecture 9, Slides: Lecture 10

Abstract: We review the main results of smoothing theory, in particular the isomorphism $\Theta_n \cong \pi_n(PL/O)$$\Theta_n \cong \pi_n(PL/O)$, and then state the relationship of smoothing theory to the surgery exact sequence. We give examples of non-smoothable topological manifolds due to Kervaire and also smooth manifolds with a unique smooth structure due to Kreck. We then define the geometric Kervaire-Milnor braid and the homotopy Kervaire-Milnor braid and sketch the proof that these braids are isomorphic. Finally, we look at the action of the surgery exact sequence of the sphere on the surgery exact sequence of a general manifold and define the inertia group and homotopy inertia group.

References: [Hirsch&Mazur1974], [Kervaire1960a], [Kreck1984], [Crowley&Lück&Macko2013, §6.7], [Schultz1987]

### 2.4 Classifying spaces for surgery

Speaker: DC

Abstract: We review Sullivan's analysis of the homotopy type of $G/PL$$G/PL$ and state its extension to $G/TOP$$G/TOP$ provided by Kirby and Siebenmann. In particular, we review the cohomology classes $\kappa_{4i+2} \in H^{4i+1}(G/PL; \Zz/2)$$\kappa_{4i+2} \in H^{4i+1}(G/PL; \Zz/2)$ and $l_{4i} \in H^{4k}(G/PL; \Zz_{(2)})$$l_{4i} \in H^{4k}(G/PL; \Zz_{(2)})$. We then give the simply connected surgery obstruction formula, and state fundamental results of Brumfiel, Madsen and Milgram concerning the canonical homomorphism $H^*(G/TOP; \Zz/2) \to H^*(G; \Zz/2)$$H^*(G/TOP; \Zz/2) \to H^*(G; \Zz/2)$. Finally, we formulate the role of the classes $\kappa_{4k+2}$$\kappa_{4k+2}$ in the Kervaire invariant problem.

References: [Madsen&Milgram1979, Ch.4], [Kirby&Siebenmann1977, Essay V], [Brumfiel&Madsen&Milgram1973]

## 3 Algebraic L-theory

### 3.1 L-groups via chain complexes

Speaker: TM

Slides: L-Theory I

Abstract: We introduce symmetric and quadratic structures on chain complexes which are analogues of symmetric and quadratic forms on modules. A non-degeneracy condition yields Poincaré complexes. Further we generalize to structures on chain maps obtaining the notion of a symmetric pair and a quadratic pair and a non-degeneracy assumption on these produces the notion of a cobordism of symmetric and quadratic Poincaré complexes. Cobordism classes then define the promised $L$$L$-groups. We introduce the technique of algebraic surgery which is the main tool in showing that the $L$$L$-groups defined via chain complexes agree with the $L$$L$-groups defined via forms and formations.

References: [Ranicki1980], [Ranicki1980a], [Ranicki2002a], [Kühl&Macko&Mole2011], [Crowley&Lück&Macko2013, Chapter 8].

### 3.2 Signatures

Speaker: TM

Slides: L-Theory II

Abstract: We first show that an $n$$n$-dimensional geometric Poincaré complex defines an $n$$n$-dimensional Poincaré symmetric structure on its singular chain complex. Its class in an appropriate $L$$L$-group is called the symmetric signature of that complex. Secondly we show that a degree one normal map between $n$$n$-dimensional geometric Poincaré complexes defines an $n$$n$-dimensional Poincaré quadratic structure on the mapping cone of the Umkehr map, again its class in an appropriate $L$$L$-group is called the quadratic signature of that map. To obtain it, we introduce the suspension of a symmetric complex and of a quadratic complex and using it we state a condition for refining symmetric structures to quadratic structures. We use $S$$S$-duality from stable homotopy theory to see that the condition is fulfilled for such a degree one normal map. In case our Poincaré complexes are manifolds the resulting complex represents the surgery obstruction in the $L$$L$-group.

References: [Ranicki1980], [Ranicki1980a], [Ranicki2002a], [Kühl&Macko&Mole2011], [Crowley&Lück&Macko2013, Chapter 8] and [Browder1972, I §4] for S-duality; see also [Adams1974], [Switzer2002, Ch.14]

### 3.3 L-groups of categories and assembly maps

Speaker: TM

Slides: L-Theory III

Abstract: We generalize the notions from the first two talks to complexes over an arbitrary additive category with chain duality. We define such a chain duality on the category of modules over a simplicial complex. This yields on one hand a construction of the $L$$L$-spectra. On the other hand imposing a certain local Poincaré condition produces homology of a space with coefficients in an $L$$L$-theory spectrum. Relaxing the local Poincaré duality to global Poincaré duality produces the assembly map.

References: [Ranicki1992], [Kühl&Macko&Mole2011].

### 3.4 Surgery obstructions and assembly maps

Speaker: TM

Slides: L-Theory IV

Abstract: We generalize the signatures from the second talk in the case when the Poincaré complexes are manifolds to signatures over the categories from the third talk. This produces a map (which turns out to be an isomorphism) from the geometric surgery exact sequence to the algebraic surgery exact sequence. This map identifies the surgery obstruction map with the assembly map.

References: [Ranicki1992], [Kühl&Macko&Mole2011]

## 4 The isomorphism conjectures

### 4.1 The Isomorphism Conjectures in the torsion-free case

Speaker: WL

Slides: Lecture 3

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.

References: [Bartels&Lück&Reich2008], [Baum&Connes&Higson1994], [Farrell&Jones1993a], [Kreck&Lück2009], [Lück&Reich2006]

### 4.2 The Isomorphism Conjectures in general

Speaker: WL

Slides: Lecture 4

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.

References: [Bartels&Lück&Reich2008], [Bartels&Lück&Weinberger2010], [Baum&Connes&Higson1994], [Kreck&Lück2005], [Lück2002a], [Lück2002b], [Lück2005], [Farrell&Jones1993a], [Lück&Reich2006]

### 4.3 Status and methods of proof

Speaker: WL

Slides: Lecture 5

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.

References: [Bartels2012], [Bartels&Farrell&Lück2011], [Bartels&Lück2012], [Bartels&Lück&Reich2008a], [Bartels&Lück&Reich2008], [Bartels&Lück&Reich&Rueping2012], [Lück2008a]