# Project B6: Equivariant homotopy and homology

## Principal investigator

## Participating scientists

- Dr. Noe Barcenas Torres
- apl. Prof. Dr. Michael Joachim
- PhD Arjun Malhotra
- Dr. Gereon Quick
- PD Dr. Jörg Schürmann

## Summary

In this project equivariant homotopy and cohomology is studied for infinite groups. We want to extend the Segal Conjecture for finite groups to infinite groups, provide computational tools for K- and L-groups of group rings or group C*-algebras based on the Farrell-Jones Conjecture and the Baum-Connes Conjecture, and investigate various invariants of equivariant spaces.Investigations of groups and their actions on spaces are often carried out on the homotopy theoretic level. Equivariant homotopy and equivariant (co-)homology are used to attach to groups interesting theories such as K-theory and L-theory via group rings, (equivariant) derived categories, group C*-algebras, group von Neumann algebras, and to investigate groups by applying algebraic topology to all kinds of classifying spaces associated to them. These theories carry many interesting invariants of equivariant spaces such as finiteness obstructions, torsion invariants, surgery obstructions, invariants of Grothendieck groups, indices of operators and characteristic classes. In this project computational and structural aspects and the classification and invariants of equivariant spaces are studied.

One goal is to develop tools for and to carry out explicit calculations of the theories and invariants mentioned above, in particular also for infinite groups. They are very valuable since they give direct applications to manifold theory, singular spaces, group actions, group homology, classification of C*-algebras, representation theory, index theory and many other problems. The Farrell-Jones Conjecture and the Baum-Connes Conjecture identify K- and L-groups of group rings or group C*-algebras with equivariant homology groups of classifying spaces for the family of finite or virtually cyclic subgroups. This motivates why one wants to study such classifying spaces of families of subgroups and their equivariant homotopy and homology theory. Computations of the connective ko-theory of classifying spaces have direct consequences for the Gromov-Lawson-Rosenberg Conjecture.

One structural aspect of this project is to extend notions from equivariant homotopy and homology theories for finite groups to infinite groups. Another goal is to carry out the constructions of natural transformations and the computations of K- and L-groups, which have been previously done for finite groups, also for infinite groups. One needs to investigate the basic framework for equivariant homotopy theory, the stable homotopy category, and develop the theory of equivariant spectra in the setting of infinite groups or parametrized versions. A good test for all these constructions is whether one can extend the Segal Conjecture for finite groups to infinite groups. Another problem yielding a better structural understanding is to describe Lubin-Tate spectra as continuous profinite spectra.

Basic equivariant spaces, besides homogeneous spaces and spaces with finitely many orbits, are classifying spaces for families. Their combinatorics and geometry reflects or reveals properties of groups. They occur in the Baum-Connes and Farrell-Jones Conjecture. The computations mentioned above play a key role in classification problems since relevant obstructions and invariants of equivariant spaces take values in the K-groups, L-groups, and homotopy groups mentioned above. Often characteristic classes or numbers and their equivariant or stratified versions play a key role in the investigation of equivariant or singular spaces. Such a theory of equivariant classes for singular spaces only partially exists and shall be further developed here.