Summer School

1. Definition and basic properties of Outer space and Teichmüller space
2. Lipschitz distance, optimal maps and train tracks
3. Classification of automorphisms of free groups
4. Folding paths and negatively curved features

Notes are available here.

1. Lattices and their homomorphisms to mapping class groups and automorphism groups of free groups
2. The symmetries of Outer Space, Teichmuller space, and related objects
3. Mappings between outer automorphism groups of free groups, and between mapping class groups
4. Actions of mapping class groups and automorphism groups of free groups on non-positively curved and acyclic spaces

1. Topological versus smooth rigidity
2. Best of all possible maps
3. Space of negatively curved metrics
4. Bundles with negatively curved fibers

1. Middle algebraic K-theory
   
   We introduce the projective class group $K_0(R)$ and discuss its algebraic and topological significance e.g., the finiteness obstruction. We define $K_1(R)$ and the Whitehead group $Wh(G)$ and explain their algebraic and topological significance, e.g., s-cobordism theorem. Finally we state the Bass-Heller-Swan decomposition and discuss homological behavior of K-theory of group rings.
   
   
2. The Isomorphism Conjectures in the torsionfree case
   
   We introduce spectra and how they yield homology theories. We state the Farrell-Jones Conjecture and the Baum-Connes Conjecture for torsionfree groups. We discuss applications of these conjectures, such as the Kaplansky Conjecture and the Borel Conjecture. We explain that the formulations for torsionfree groups cannot extend to arbitrary groups.
   
   
3. The Isomorphism Conjectures in general
   
   We introduce classifying spaces for families and equivariant homology theories. We state the Farrell-Jones-Conjecture and the Baum-Connes Conjecture in general. We discuss further applications, such as the Novikov Conjecture.
   
   
4. Status and Methods of Proof
   
   We give a status report of the Full Farrell-Jones Conjecture which is known for instance for hyperbolic groups, CAT(0)-groups, cocompact lattices in almost connected Lie groups, three-manifold groups and arithmetic groups over number fields. We discuss open cases and potential candidates for counterexamples. We explain methods of proof, that are, besides homotopy theory, controlled topology and flow spaces.
   
   Slides available at: http://131.220.77.52/lueck/publications.php#slides

1. Computing group cohomology using the spine of Auter space
2. Lie graph homology and Kontsevich's theorem
3. Morita's cocycles for the group of outer automorphisms of free groups
4. Hairy graph homology

1. Rational Pontryagin classes and smoothing theory
   
   For many years I have been obsessed with a few questions about the rational cohomology of $BTOP(n)$, the classifying space for fiber bundles with $n$-dimensional euclidean space as fiber (no vector bundle structure in the fibers assumed). Thom and Novikov showed that the rational Pontryagin classes for $n$-dimensional vector bundles can also be defined for these more general bundles with $n$-dimensional fibers. Do the generalized rational Pontryagin classes of Thom and Novikov also satisfy certain relations that we know for vector bundles? Do they vanish in cohomology of dimensions $>2n$? For $n$ even, is it true that the Pontryagin class in dimension $2n$ is the square of the twisted Euler class? One way to approach these questions is through smoothing theory, which relates the coset space $TOP(n)/O(n)$ to the space of diffeomorphisms rel boundary from the $n$-disk to itself. This talk would try to give an idea of what the above questions look like after they have gone through smoothing theory.
   
   
2. Characteristic classes for homotopy theorists
   
   This describes a homotopy theoretic approach to characteristic classes based on orthogonal calculus, a variation on Goodwillie's homotopy functor calculus.
   Orthogonal calculus is concerned with (continuous) functors from the category of finite dimensional real vector spaces with inner product to spaces. It is needed to complete the translation of one conjecture (on the Thom-Novikov rational Pontryagin classes) into another conjecture (on spaces of regular smooth maps to the plane).
   
   
3. Singularity theory, multisingularity theory, h-principles and generalizations
   
   It is a tradition coming from concordance theory, as practiced by J Cerf and K Igusa, that one tries to understand spaces of regular (=nonsingular) smooth maps by viewing them as subspaces of spaces of not-quite-regular maps. These not-quite-regular maps are smooth maps having only a few well-understood singularity types. In thinking specifically about smooth maps to the plane, it is helfpul to impose conditions not only on the singularities in the source, but also on the singularities in the target. That is, where a critical value has several critical points mapping to it, we like to impose conditions on the set of those critical points which have no meaning for each critical point on its own. Again following the tradition of concordance theory, we try to avoid all maps whose singularities and multisingularities do not satisfy our conditions. This phobia leads us very naturally to h-principles and generalizations.
   
   
4. Spaces of smooth maps to the plane and applications to Pontryagin classes
   
   This talk should give an idea of how cohomological information about spaces of not-quite-regular smooth maps, coming from h-principles and generalizations, can lead to cohomological information about spaces of smooth regular maps. By talks 1) and 2), enough cohomological information about spaces of smooth regular maps implies answers to some of those questions about rational Pontryagin classes of fiber bundles with $n$-dimensional euclidean space as fiber.
   
   
   Here are the notes for the lecture series.