The Teichmüller space of pinched negatively curved metrics on a hyperbolic manifold is not contractible

Abstract

For a smooth manifold $M$ we define the Teichmüller space $\mathcal{T}(M)$ of all Riemannian metrics on $M$ and the Teichmüller space $\mathcal{T}^\epsilon(M)$ of $\epsilon$-pinched negatively curved metrics on $M$, where $0\leq\epsilon\leq\infty$. We prove that if $M$ is hyperbolic, the natural inclusion $\mathcal{T}^\epsilon(M)\hookrightarrow\mathcal{T}(M)$ is, in general, not homotopically trivial. In particular, $\mathcal{T}^\epsilon(M)$ is, in general, not contractible.

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      author = {Farrell, F. T. and Ontaneda, P.},
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Authors

F. Thomas Farrell

Department of Mathematical Sciences
Binghamton University
Binghamton, NY 13902-6000
United States

Pedro Ontaneda

Department of Mathematical Sciences
Binghamton University
Binghamton, NY 13902-6000
United States