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Hyperbolic manifolds with negatively curved exotic triangulations in dimensions greater than five. (English) Zbl 0909.53026

The first two authors proved that if a closed manifold \(M\) of dimension \(\geq 5\) is homotopy equivalent to a closed non-positively curved manifold \(N\), then \(M\) and \(N\) are homeomorphic [F. T. Farrell and L. E. Jones, Proc. Symp. Pure Math. 54, Part 3, 229-274 (1993; Zbl 0796.53043)]. In a related work, they showed that \(M\) and \(N\) are not necessarily diffeomorphic [J. Am. Math. Soc. 2, 899-908 (1989; Zbl 0698.53027)]. Namely, for any \(\varepsilon>0\), they constructed a closed Riemannian manifold of sectional curvatures within \([-1-\varepsilon,-1]\) that is homeomorphic but not diffeomorphic to a hyperbolic manifold. Such examples were constructed in many (but not all) dimensions \(\geq 7\).
More recently P. Ontaneda produced similar examples in dimension six [J. Differ. Geom. 40, 7-22 (1994; Zbl 0817.53026)]. In fact, these \(6\)-manifolds are not even PL homeomorphic to hyperbolic manifolds (where the PL structure on a hyperbolic manifold is induced by its differentiable structure). The paper under review generalizes Ontaneda’s result to all dimensions \(\geq 6\); namely, the following theorem is proved.
Theorem. For each \(n\geq 6\) these exists a closed hyperbolic \(n\)-manifold \(M\) such that, for any \(\varepsilon>0\), \(M\) has a finite cover \(\widetilde M_\varepsilon\) that is homeomorphic but not PL-homeomorphic to a Riemannian manifold of sectional curvatures within \([-1-\varepsilon,-1]\).

MSC:

53C20 Global Riemannian geometry, including pinching
57Q25 Comparison of PL-structures: classification, Hauptvermutung
57Q15 Triangulating manifolds
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