No abstract

In the first part of the paper we introduce the theory of bundles with negatively curved fibers. For a space X there is a forgetful map F X between bundle theories over X, which assigns to a bundle with negatively curved fibers over X its subjacent smooth bundle. Our main result states that, for certain k-spheres S k , the forgetful map F S k is not one-to-one. This result follows from Theorem A, which proves that the quotient map MET sec < 0 (M ) → T sec < 0 (M ) is not trivial at some homotopy levels, provided the hyperbolic manifold M satisfies certain conditions. Here MET sec < 0 (M ) is the space of negatively curved metrics on M and T sec < 0 (M ) = MET sec < 0 (M )/ DIFF 0 (M ) is, as defined in [FO2], the Teichmüller space of negatively curved metrics on M . In particular we conclude that T sec < 0 (M ) is, in general, not connected. Two remarks: (1) the nontrivial elements in π k MET sec < 0 (M ) constructed in [FO3] have trivial image by the map induced by MET sec < 0 (M ) → T sec < 0 (M ); (2) the nonzero classes in π k T sec < 0 (M ) constructed in [FO2] are not in the image of the map induced by MET sec < 0 (M ) → T sec < 0 (M ); the nontrivial classes in π k T sec < 0 (M ) given here, besides coming from MET sec < 0 (M ) and being harder to construct, have a different nature and genesis: the former classes -given in [FO2] -come from the existence of exotic spheres, while the latter classes -given here -arise from the non-triviality and structure of certain homotopy groups of the space of pseudo-isotopies of the circle S 1 . The strength of the new techniques used here allowed us to prove also a homology version of Theorem A, which is given in Theorem B. IntroductionLet M be a closed smooth manifold. We will denote the group of all self-diffeomorphisms of M , with the smooth topology, by DIFF(M ). By a smooth bundle over X, with fiber M , we mean a locally trivial bundle for which the change of coordinates between two local sections over, say, U α , U β ⊂ X is given by a continuous map U α ∩ U β → DIFF(M ). A smooth bundle map between two such bundles over X is bundle map such that, when expressed in a local chart as U × M → U × M , the induced map U → DIFF(M ) is continuous. In this case we say that the bundles GAFA Geometric And Functional Analysis 1398 F.T. FARRELL AND P. ONTANEDA GAFA are smoothly equivalent. Smooth bundles over a space X, with fiber M , modulo smooth equivalence, are classified by [X, B(DIFF(M ))], the set of homotopy classes of (continuous) maps from X to the classifying space B(DIFF(M )).[In what follows we will be considering everything pointed : X comes with a base point x 0 , the bundles come with smooth identifications between the fibers over x 0 and M , and the bundle maps preserve these identifications. Also, classifying maps are base point preserving maps.]If we assume that X is simply connected, then we obtain a reduction in the structural group of these bundles: smooth bundles over a simply connected space X, with fiber M , modulo smooth equivalence, are classified by...

In this paper, we prove the Farrell-Jones Conjecture for the solvable Baumslag-Solitar groups with coefficients in an additive category. We also extend our results to groups of the form, Z[1/p] semidirect product with any virtually cyclic group, where p is a prime number. Date

The Isomorphism Conjecture of Farrell and Jones for L-theory [5] has only been formulated for L −∞ and in this formulation been proved for a large class of groups, for instance for discrete cocompact subgroups of a virtually connected Lie group. The question arises whether the corresponding conjecture is true for L for the decorations 1

Abstract. The paper is devoted to the problem when a map from some closed connected manifold to an aspherical closed manifold approximately fibers, i.e., is homotopic to Manifold Approximate Fibration. We define obstructions in algebraic K-theory. Their vanishing is necessary and under certain conditions sufficient. Basic ingredients are Quinn's Thin h-Cobordism Theorem and End Theorem, and knowledge about the Farrell-Jones Conjectures in algebraic Kand L-theory and the MAF-Rigidity Conjecture by Hughes-Taylor-Williams.