Teichmüller Spaces and Negatively Curved Fiber Bundles

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 $${\mathbb{S}^k}$$ , the forgetful map $${F_{\mathbb{S}^k}}$$ is not one-to-one. This result follows from Theorem A, which proves that the quotient map $${\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)}$$ is not trivial at some homotopy levels, provided the hyperbolic manifold M satisfies certain conditions. Here $${\mathcal{MET}^{\,\,sec <0 }(M)}$$ is the space of negatively curved metrics on M and $${\mathcal{T}^{\,\,sec <0 }(M) = \mathcal{MET}^{\,\,sec <0 }(M)/ {\rm DIFF}_0(M)}$$ is, as defined in [FO2], the Teichmuller space of negatively curved metrics on M. In particular we conclude that $${\mathcal{T}^{\,\,sec <0 }(M)}$$ is, in general, not connected. Two remarks: (1) the nontrivial elements in $${\pi_{k}\mathcal{MET}^{\,\,sec <0 }(M)}$$ constructed in [FO3] have trivial image by the map induced by $${\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)}$$ ; (2) the nonzero classes in $${\pi_{k}\mathcal{T}^{\,\,sec <0 }(M)}$$ constructed in [FO2] are not in the image of the map induced by $${\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)}$$ ; the nontrivial classes in $${\pi_{k}\mathcal{T}^{\,\,sec <0 }(M)}$$ given here, besides coming from $${\mathcal{MET}^{\,\,sec <0 }(M)}$$ 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 $${\mathbb{S}^1}$$ . 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.