# Starting Position of Research Area F*

## Topological invariants of groups.

Lück, together with Bartels, proved in a sequence of articles (see [F*BL], [F*BLR08a], [F*:BLR08b]) and preprints, partially with additional coauthors, that the class of groups for which the Farrell-Jones conjecture and the Borel conjecture holds, contains hyperbolic groups, CAT(0)-groups, cocompact lattices in almost connected Lie groups, and 3-manifold groups, and is closed under taking subgroups and directed colimits. Bartels-Lück-Weinberger proved that a torsionfree hyperbolic group occurs as fundamental group of a closed aspherical topological manifold, unique up to homeomorphism, if the boundary is a sphere of dimension $\ge 5$ (see [F*:BLW10]).

Ursula Hamenstädt showed the non-triviality of second continuous bounded cohomology with coefficients in the regular representation for the following groups: Closed groups of isometries of a CAT(0)-space containing a rank one element [F*:Ham] and subgroups of ${\rm Out}(F_n)$ which contain a fully irreducible element. She also proved the Novikov conjecture for the mapping class groups [F*:Ham09a].

## Global equivariant homotopy theory.

Schwede contributed to various foundational questions of stable homotopy theory (see [F*:Sch07], [F*:Sch08]), like the comparison of various model categories of spectra. He constructed a $C_2$-equivariant ring spectrum model for the complex cobordism spectrum $MU\mathbb R$. Lück has developed tools in equivariant homotopy and homology theory for infinite groups to calculate the algebraic $K$- and $L$-theory of group rings and the topological $K$-theory of group $C^*$-algebras. He applied these to specific examples motivated by problems from non-commutative geometry and from group actions. Teichner, together with Stolz, developed a precise notion of super symmetric Euclidean field theories, and they showed, partially with Kreck and Hohnhold, that for small space-time dimension these give cocycle theories for de Rham cohomology respectively $K$-theory.

## Categorification, quantum invariants, and low-dimensional topology.

Catharina Stroppel constructed refined knot invariants using the method of categorification (see [F*:FKS06]), a concept which lifts a classical polynomial invariant to a graded Euler characteristic. She constructed a categorification of the Jones polynomial and other quantum group invariants using 2-categories of representations of Lie algebras; these provide first examples of 2-Kac-Moody representations. Teichner, together with Conant and Schneiderman, showed how to measure the failure of the Whitney move in dimension 4 by constructing higher-order intersection invariants of Whitney towers (see [F*:CST11]). They identify some of these new invariants with previously known link invariants like Milnor, Sato-Levine and Arf invariants and define higher-order versions of the latter two types of invariants. Bödigheimer, together with Abhau and Ehrenfried (see [F*:ABE08]), computed the integral homology of the moduli space $\mathfrak{M}_{2,1}$ of surfaces of genus 2 with one boundary curve.