Topological rigidity for FJ by the infinite cyclic group

We call a group FJ if it satisfies the [Formula: see text]- and [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text]. We show that if [Formula: see text] is FJ, then the simple Borel conjecture (in dimensions [Formula: see text]) holds for every group of the form [Formula: see text]. If in addition [Formula: see text], which is true for all known torsion-free FJ groups, then the bordism Borel conjecture (in dimensions [Formula: see text]) holds for [Formula: see text]. One of the key ingredients in proving these rigidity results is another main result, which says that if a torsion-free group [Formula: see text] satisfies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text], then any semi-direct product [Formula: see text] also satisfies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text]. Our result is indeed more general and implies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in additive categories is closed under extensions of torsion-free groups. This enables us to extend the class of groups which satisfy the Novikov conjecture.

Topological rigidity problems

We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and applications of the Farrell-Jones Conjecture for algebraic \(K\)-and \(L\)-theory for a group ring $RG$ and coefficients in an additive category. These conjectures imply many other well-known and important conjectures. Examples are the Borel Conjecture about the topological rigidity of closed aspherical manifolds, the Novikov Conjecture about the homotopy invariance of higher signatures and the Conjecture for vanishing of the Whitehead group. We here present the status of the Borel, Novikov and vanishing of the Whitehead group Conjectures.

Borel's conjecture in topological groups

We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal numberκ, let BCκdenote this generalization. Then BCℕ0is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, ¬BCℕ1is equivalent to the existence of a Kurepa tree of height ℕ1. Using the connection of BCκwith a generalization of Kurepa's Hypothesis, we obtain the following consistency results:(1) If it is consistent that there is a 1-inaccessible cardinal then it is consistent that BCℕ1.(2) If it is consistent that BCℕ1, then it is consistent that there is an inaccessible cardinal.(3) If it is consistent that there is a 1-inaccessible cardinal withωinaccessible cardinals above it, then ¬BCℕω+ (∀n<ω)BCℕnis consistent.(4) If it is consistent that there is a 2-huge cardinal, then it is consistent that BCℕω(5) If it is consistent that there is a 3-huge cardinal, then it is consistent that BCκfor a proper class of cardinalsκof countable cofinality.