Goals of Research Area F*

Topological invariants of groups.

The Farrell-Jones conjecture predicts the algebraic $K$- and $L$-theory of group rings in terms of equivariant homology theories applied to certain classifying spaces. It implies other prominent conjectures due to Borel (topological rigidity of aspherical manifolds), Kaplansky (idempotents in group rings) and Novikov (homotopy invariance of higher signatures).

One goal in Research Area F* is to extend the class of groups for which the Farrell-Jones conjecture holds. Prominent still open examples are $SL_n(\mathbb{Z})$, the mapping class group and the outer automorphism group ${\rm Out}(F_n)$ of the free group. Here the known techniques do not apply since the familiar contractible spaces on which these groups act and which admit nice invariant compactifications lack certain properties. Bartels-Lück-Reich-Rüping have a promising program for $SL_n(\mathbb{Z})$. Ursula Hamenstädt and Lück will work on the other groups. A better understanding of their geometry and of the associated controlled topology is necessary and will be of independent interest. Particularly challenging is the case of ${\rm Out}(F_n)$ for which even the Novikov conjecture is not known.

The analogue of the Farrell-Jones conjecture for Waldhausen's $A$-theory and for topological Hochschild homology and cyclic homology will be investigated. This has strong implications for the space of automorphisms of closed aspherical manifolds. There is a connection to Research Area A via the Atiyah conjecture about the integrality of $L^2$-Betti numbers which is a spectral invariant of the universal covering of a closed Riemannian manifold: a program for its proof involves the Farrell-Jones conjecture for the projective class group.

The Farrell-Jones conjecture will be used for systematical computations of algebraic $K$- and $L$-groups using the description in terms of equivariant homology. For instance, the answer is not known for semi-direct products $\mathbb{Z}^n \rtimes \mathbb{Z}/m$. This has direct applications to classification problems of manifolds and yields a link to Research Area C, where the Baum-Connes conjecture and its implications are investigated.

Global equivariant homotopy theory.

Equivariant stable homotopy theory is a basic tool that enters at several places in this research area. There is a well-established theory based on actions of one particular finite group or compact Lie group, and where stabilization is performed by suspension with the spheres of orthogonal representations. We need a global version and extensions of the classical theory to more general classes of groups. Here "global" refers to a structure that encodes one equivariant homotopy type for every group.

These simultaneous group actions have to be functorial and compatible with transfer from subgroups. In this area two very different approaches are studied.

The first project, by Schwede, is a systematic study of global equivariant stable homotopy theory where each individual $G$-equivariant homotopy type represents an $RO(G)$-graded equivariant homology theory. The proposed framework is the category of orthogonal spectra, endowed with a new "global" model structure that takes the Mackey functor valued homotopy groups into account. One aim is to lift the global model structure to commutative orthogonal ring spectra; the resulting objects represent highly structured, global multiplicative homology theories equipped with additive transfers and multiplicative norm maps.

The second project, by Teichner, together with Kreck, involves their concept of cocycle theories. These associate to the site of smooth manifolds a family of stacks $A^k$; under specific conditions the concordance classes of stacks give a cohomology theory $hA^k(X)$. All cohomology theories arise this way, but not uniquely: a choice of a cocycle model leads to groups $hA^k(S)$ for any smooth stack $S$ via morphisms from $S$ to $A^k$. In particular, the quotient stacks $X/G$ lead to equivariant cohomology groups simultaneously for all compact Lie groups $G$. One of the future goals is to provide a general frame for geometric cohomology theories, in particular those coming from geometric field theories as studied in Research Area C.

Natural questions to be investigated are how the two approaches are related, how to represent cocycle theories by spectra in the global model structure and how to extend the global theory formalisms to non-compact Lie groups, in particular countable discrete groups. Equivariant homotopy theory for infinite discrete groups or non-compact topological groups is used in connection with the Farrell-Jones conjecture or the Baum-Connes conjecture; in a joint project, Lück and Schwede propose an equivariant stable theory by working over an equivariant base space and suspending fiberwise with respect to sphere bundles of equivariant vector bundles.

Categorification, quantum invariants, and low-dimensional topology.

Khovanov homology is the first successful categorification of knot invariants. Motivated by the problem about the possible values of the higher order Arf invariants constructed in [F*:CST11], Catharina Stroppel and Teichner study certain new quantum invariants arising from Khovanov homology. A key question is how the quantum invariants can be turned into local topological field theories. The best mathematical concept that expresses this locality involves formulating field theories as 3-functors between certain 3-categories. Higher categories are best understood via homotopy theory, leading to interactions with the project "Global equivariant homotopy theory" above and Research Area C.

Khovanov homology exists now in many disguises, their precise connections are still unclear. Using 2- and 3-categories it will be possible to rigidify the constructions and establish concrete equivalences and functors between them. The algebras of 2-morphisms appearing in this context are interesting on their own in representation theory, since they are versions of Hecke algebras, hence deformations and degenerations of the group algebras of (affine) Coxeter groups, symmetric groups and their wreath products with finite groups. The topological origin of the 2-and 3-morphisms can be used to define $\mathbb{Z}$-gradings on these Hecke algebras. These surprising gradings on categories are a very central aspect of modern representation theory, refining decomposition numbers and Jordan-Hölder multiplicities in a crucial way. Catharina Stroppel will investigate their structures and implications.

Catharina Stroppel has a program to extend Khovanov homology to categorified 3-manifold invariants, based on the fact that she was able to lift the complete Reshetikhin-Turaev tangle invariant to a categorical level and to extend this to other quantum invariants. This is expected to provide examples of extended topological field theories, see also Research Area C.

Coxeter groups and wreath products of a fixed group with symmetric groups also arise in the construction of moduli spaces of bundles over surfaces which will be investigated by Bödigheimer. Given a simplicial set $G_{\bullet}$ of normed groups, one glues the strata of fixed norm of all classifying spaces $BG_p$ and obtains spaces with interesting components. For instance, the family of symmetric groups yields the moduli spaces $\mathfrak{M}_{g,1}^m$ of surfaces of genus $g$ with one boundary curve and $m$ permutable punctures.

Another low-dimensional project is due to Ursula Hamenstädt. She aims at a geometric construction of three-manifold invariants using Heegard decompositions and a slice construction for the handlebody group, viewed as a subgroup of the mapping class group of the boundary surface.