The double of a hyperbolic manifold and non-positively curved exotic PL structures. (English) Zbl 1043.53026
Given a manifold \(M\), it is a fundamental question how many different differentiable structures or PL structures on \(M\) in geometry or topology exist. There are examples of \(n\)-manifolds, \(n > 4\), with many differentiable structures (some of them inducing different PL structures) such that only one differentiable structure admits a non-positively curved Riemannian metric. For instance the \(n\)-torus \(T^n\), has such properties. Generally, if \(M\) is one of such examples, then \(M \times T^m\) has many different PL structures, some non-positively curved and some other not: one can change the PL structure on \(M\) so that one obtains different non-positively curved PL structures on \(M \times T^m\). If one changes the PL structure on \(T^m\), one can get PL structures that do not admit non-positively curved metrics.
In this paper, the author gives other examples which are better and natural in some sense compared with the examples above. Let \(M\) be a non-compact real hyperbolic \(n\)-manifold of finite volume. Then \(M\) is homeomorphic to the interior of a compact manifold with boundary such that each boundary component is a flat manifold of dimension \(n-1\). The compact manifold with boundary can be doubled along its boundary to form \(\mathcal D(M)\) the double of \(M\). The author proves that there are non-compact finite volume real hyperbolic \(n\)-manifolds \(M\), \(n >5\), such that the double of \(M{\mathcal D}(M)\) has at least three non-equivalent smoothable PL structures. Two of them admit Riemannian metrics with non-positive curvature and negative curvature outside a hypersurface, and the other does not admit a Riemannian metric with non-positive curvature.
To prove this result, the author shows that the doubles of non-compact finite volume real hyperbolic manifolds are differentiable rigid. In other words, if \(f:\mathcal D(M) \to \mathcal D(M)\) is a homeomorphism of the double of a non-compact finite volume real hyperbolic \(n\)-manifold \(M\), \(n\geq 3\), then \(f\) is homotopic to a diffeomorphism \(g : \mathcal D(M) \to \mathcal D(M)\). The author first shows the existence of a non-compact finite volume real hyperbolic manifold to have a cusp diffeomorphic to \(T^{n-1}\times (0, \infty)\) and he obtains the main results by reglue the cusp using an exotic diffeomorphism of \(T^{n-1}\).
In this paper, the author gives other examples which are better and natural in some sense compared with the examples above. Let \(M\) be a non-compact real hyperbolic \(n\)-manifold of finite volume. Then \(M\) is homeomorphic to the interior of a compact manifold with boundary such that each boundary component is a flat manifold of dimension \(n-1\). The compact manifold with boundary can be doubled along its boundary to form \(\mathcal D(M)\) the double of \(M\). The author proves that there are non-compact finite volume real hyperbolic \(n\)-manifolds \(M\), \(n >5\), such that the double of \(M{\mathcal D}(M)\) has at least three non-equivalent smoothable PL structures. Two of them admit Riemannian metrics with non-positive curvature and negative curvature outside a hypersurface, and the other does not admit a Riemannian metric with non-positive curvature.
To prove this result, the author shows that the doubles of non-compact finite volume real hyperbolic manifolds are differentiable rigid. In other words, if \(f:\mathcal D(M) \to \mathcal D(M)\) is a homeomorphism of the double of a non-compact finite volume real hyperbolic \(n\)-manifold \(M\), \(n\geq 3\), then \(f\) is homotopic to a diffeomorphism \(g : \mathcal D(M) \to \mathcal D(M)\). The author first shows the existence of a non-compact finite volume real hyperbolic manifold to have a cusp diffeomorphic to \(T^{n-1}\times (0, \infty)\) and he obtains the main results by reglue the cusp using an exotic diffeomorphism of \(T^{n-1}\).
Reviewer: Gabjin Yun (Yangin)
MSC:
| 53C20 | Global Riemannian geometry, including pinching |
| 57Q25 | Comparison of PL-structures: classification, Hauptvermutung |
| 57R55 | Differentiable structures in differential topology |
Keywords:
differentiable structure; double of hyperbolic manifold; non-positive curvature; PL structure; piecewise linearReferences:
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