We show that the space of negatively curved metrics of a closed negatively curved Riemannian n-manifold, n ≥ 10, is highly non-connected. Section 0. Introduction. Let M be a closed smooth manifold. We denote by MET (M ) the space of all smooth Riemannian metrics on M and we consider MET (M ) with the smooth topology. Note that the space MET (M ) is contractible. A subspace of metrics whose sectional curvatures lie in some interval (closed, open, semi-open) will be denoted by placing a superscript on MET (M ). For example, MET sec<ǫ (M ) denotes the subspace of MET (M ) of all Riemannian metrics on M that have all sectional curvatures less that ǫ. Thus saying that all sectional curvatures of a Riemannian metric g lie in the interval [a, b] is equivalent to saying that g ∈ MET a ≤ sec ≤ b (M ). Note that if I ⊂ J then MET sec∈I (M ) ⊂ MET sec∈J (M ). Note also that MET sec = −1 (M ) is the space of hyperbolic metrics Hyp (M ) on M .A natural question about a closed negatively curved manifold M is the following: is the space MET sec<0 (M ) of negatively curved metrics on M path connected? This problem has been around for some time and has been posed several times in the literature. see for instance K. Burns and A. Katok ([2], Question 7.1). In dimension two, Hamilton's Ricci flow [8] shows that Hyp (M 2 ) is a deformation retract of MET sec<0 (M 2 ). But Hyp (M 2 ) fibers over the Teichmüller space T (M 2 ) ∼ = R 6µ−6 (µ is the genus of M 2 ), with contractible fiber D = R + × DIF F (M 2 ) [5]. Therefore Hyp (M 2 ) and MET sec<0 (M 2 ) are contractible.

Let M be a Hadamard manifold, that is, a complete simply connected riemannian manifold with non-positive sectional curvatures. Then every geodesic segment α : [0, a] → M from α(0) to α(a) can be extended to a geodesic ray α : [0, ∞) → M. We say then that the Hadamard manifold M is geodesically complete. Note that, in this case, all geodesic rays are proper maps.CAT(0) spaces are generalizations of Hadamard manifolds. For a CAT(0) space X, all geodesic rays α : [0, ∞) → X are proper maps but, in general, X is not geodesically complete.The following definition of almost geodesic completeness was suggested by M. Mihalik:A geodesic space X, with metric d, is almost geodesically complete if there is a constant C such that for every p, q ∈ X there is a geodesic ray α : [0, ∞) → X, α(0) = p, and d(q, α) ≤ C. by isometries. If H i c (X) = 0, for some i, then X is almost geodesically complete.Theorem B. Let X be a noncompact proper CAT(0) space on which Γ acts cocompactly by isometries with discrete orbits. Then X is almost geodesically complete.Theorem B follows from theorem A and the following two propositions.Proposition A. Let X be a proper CAT(0) space on which Γ acts cocompactly by isometries with discrete orbits. Then X is properly Γ-homotopy equivalent to a Γ-finite Γ-simplicial complex K.Proposition B. Let K be a locally finite contractible simplicial complex which admits a cocompact simplicial action. Then H i c (K) = 0, for some i.Proof of Theorem B from Theorem A and Propositions A and B. Let X be a noncompact proper CAT(0) space on which Γ acts cocompactly, by isometries with discrete orbits. By Proposition A, X is properly Γ-homotopy equivalent to a Γ-finite Γ-simplicial complex K. Since every CAT(0) space is contractible, we have that K is also contractible. Hence, by Proposition A, H i c (X) = H i c (K) = 0. We can now apply Theorem A and conclude that X is almost geodesically complete.It was suggested by R. Geoghegan that proposition A above could be used to prove that that the boundary ∂X of a Γ-cocompact CAT(0) space X is a shape invariant of the Γ action. The next theorem shows that in fact this is true.Theorem C. Let X and Y be proper CAT(0) spaces on which Γ acts cocompactly by isometries with discrete orbits. If X and Y are Γ-homotopy equivalent then ∂X and ∂Y are shape equivalent.Corollary A. Let X and Y be proper CAT(0) spaces on which Γ acts cocompactly by isometries with discrete orbits. If the actions have the same isotropy, i.e. if { G < Γ : X G = ∅ } = { G < Γ : Y G = ∅ } then ∂X and ∂Y are shape equivalent.We say that a group acts on a space with finite isotropy if all isotropy groups are finite.Corollary B. Let X and Y be proper CAT(0) spaces on which Γ acts cocompactly by isometries, with discrete orbits and finite isotropy. Then ∂X and ∂Y are shape equivalent.It is known that if we assume Γ in theorem C to be hyperbolic, then in fact ∂X and ∂Y are homeomorphic (see [12]), but in general ∂X and ∂Y do not have to be homeomorphic (see [8]).Here is a short outline of the paper. In section 1 w...

For a smooth manifold M we define the Teichmüller space -.M / of all Riemannian metrics on M and the Teichmüller space -.M / of -pinched negatively curved metrics on M , where 0 Ä Ä 1. We prove that if M is hyperbolic, the natural inclusion -.M / ,! -.M / is, in general, not homotopically trivial. In particular, -.M / is, in general, not contractible.

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...

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