# Project B5: Rigidity and K-theory

## Principal investigator

## Participating scientists

- Paul Bubenzer
- apl. Prof. Dr. Michael Joachim
- Daniel Kasprowski
- Svenja Knopf
- Adam Mole
- Dr. Christoph Winges

## Summary

Rigidity phenomena play an important role in mathematics. The famous (and now proven) Poincaré Conjecture is the following rigidity statement: any closed manifold that is homotopy equivalent to a sphere is in fact homeomorphic to a sphere. A central goal of this project is to make progress on the Borel Conjecture:"Any closed manifold that is homotopy equivalent to a closed aspherical manifold M is in fact homeomorphic to M."

This conjecture implies in particular that closed aspherical manifolds are determined up to homeomorphism by their fundamental groups. Aspherical manifolds play an important role in geometry and topology. Examples of aspherical manifolds are manifolds of non-positive curvature. All torsion-free lattices in simply-connected Lie groups appear as fundamental groups of aspherical manifolds. Hyperbolization techniques pioneered by Gromov allow the construction of an abundance of closed aspherical manifolds. For instance, any closed manifold is cobordant to a closed aspherical manifold.

Via surgery theory the Borel Conjecture is related to questions about the algebraic K- and L-theory of group rings. Very important in this context is the Farrell-Jones Conjecture in algebraic K- and L-theory. This conjecture asserts that the algebraic K- and L-theory of a group ring R[G] can each be identified with an equivariant homology group of the universal space with virtually cyclic isotropy. The Borel Conjecture holds for a closed aspherical manifold of dimension ≥ 5 if its fundamental group satisfies the Farrell-Jones Conjecture. The Farrell- Jones Conjecture has further important applications in geometric topology and algebra. For instance, it has applications to the Novikov Conjecture on the homotopy invariance of higher signatures, the Bass Conjecture in algebraic K-theory, computations of Whitehead groups and projective class groups, classification of h-cobordisms, the finiteness obstruction of Wall for finitely dominated spaces, Kaplansky's idempotent conjecture and the conjecture about the homotopy invariance of L

^{2}-torsion.

Presently, no counter-example to any of these conjectures is known. A motivating long-term goal for this project is the verification of these conjectures for all groups G that admit a finite CW-complex as its classifying space. We point out that a special member of this family of conjectures, namely the K-theoretic Novikov Conjecture, is true in this generality. In order to advance toward this long-term goal we will have to start with smaller classes of groups. We will also study further applications of these conjectures and the interplay between them.

Important relatives of the conjectures above are the Baum-Connes Conjecture and the Bost Conjecture, which deal with the topological K-theory of reduced group C*-algebras and group Banach algebras. The Baum-Connes Conjecture and the Bost Conjecture are studied in C3.

Related to the Borel Conjecture is the following question that will also be investigated in this project: which Poincaré duality groups are fundamental groups of aspherical manifolds? This question should be considered as the existence statement corresponding to the uniqueness statement in the Borel Conjecture.

A different rigidity question that will be studied in this project arises in geometric group theory: are two quasi-isometric simply-connected nilpotent Lie groups necessarily isomorphic?

Progress on all these questions will depend on the combination of a large number of methods and concepts from geometric group theory (e.g., non-positively curved groups, boundaries of groups, flow spaces), geometric topology (e.g., controlled topology and algebra, surgery theory, Waldhausen's K-theory of spaces, ANR-homology manifolds), homotopy theory (e.g., topological cyclic homology, the cyclotomic trace, real homotopy theory, ring spectra), and algebra (e.g., algebraic K-theory, L-theory).