Negatively curved manifolds with exotic smooth structures. (English) Zbl 0698.53027
The main result of this paper is the following Theorem: Let M denote a compact Riemannian manifold of dimension \(m\geq 5\) and with sectional curvature \(K\equiv -1\). Let \(\Sigma_ 1,...,\Sigma_ k\) be a complete list of inequivalent exotic spheres of dimension m. Then given any real number \(\delta >0\), there is a finite sheeted covering \(M^*\) of M that satisfies the following properties: (a) No two of the manifolds \(M^*,M^*\#\Sigma_ 1,...,M^*\#\Sigma_ k\) are diffeomorphic, but they all are homeomorphic to one another, where # denotes connected sum. (b) Each of the manifolds \(M^*\#\Sigma_ 1,...,M^*\#\Sigma_ k\) admits a Riemannian metric with sectional curvatures lying in the interval \([-1-\delta,-1+\delta].\)
Remarks: 1) None of the manifolds \(M^*\#\Sigma_ 1,...,M^*\#\Sigma_ k\) admits a metric with constant negative sectional curvature; if one did, it would be diffeomorphic (in fact isometric) to \(M^*\) by the Mostow Rigidity Theorem. 2) In another work [J. Am. Math. Soc. 2, No.2, 257-369 (1989)] the authors have shown that if two manifolds of strictly negative sectional curvature have the same homotopy type, then they are homeomorphic, thereby solving an old problem. 3) As \(\delta\) \(\to 0\) the multiplicity of the covering \(M^*\to \infty\) by the compactness theorem of Cheeger and Gromov. 4) Earlier M. Gromov and W. Thurston [Invent. Math. 89, 1-12 (1987; Zbl 0646.53037)] by another method had found examples of compact manifolds with sectional curvatures arbitrarily close to -1 that admit no metrics of constant negative sectional curvature.
To prove a) one begins with a result of Sullivan, which shows that some finite covering space \(M^*\) of M is stably parallelizable. A further topological argument completes the proof of a). To prove b) one uses residual finiteness of \(\pi_ 1M\) to construct a finite covering \(M^*\) of large multiplicity with the property that every point x of M lies in an imbedded metric ball of prescribed radius \(3\alpha\). Consider such a ball \(B_{3\alpha}=S^{m-1}\times [0,3\alpha].\) Given an exotic m- sphere \(\Sigma_ k\) one constructs \(M^*\#\Sigma_ k\) by glueing \(S^{m-1}\times [\alpha,2\alpha]\) into \(M^*_{\alpha}=M^*-S^{m- 1}\times (\alpha,2\alpha)\) with appropriate identifications along \(\partial M^*_{\alpha}\). On \(S^{m-1}\times (0,2\alpha)\) and \(S^{m-1}\times (2\alpha,3\alpha)\) one introduces appropriate metrics with \(K\equiv -1\) and interpolates between them on [\(\alpha\),2\(\alpha\)]. If \(\alpha\) is sufficiently large, then the interpolation can take place sufficiently slowly to produce sectional curvatures in \([-1-\delta,- 1+\delta]\).
Remarks: 1) None of the manifolds \(M^*\#\Sigma_ 1,...,M^*\#\Sigma_ k\) admits a metric with constant negative sectional curvature; if one did, it would be diffeomorphic (in fact isometric) to \(M^*\) by the Mostow Rigidity Theorem. 2) In another work [J. Am. Math. Soc. 2, No.2, 257-369 (1989)] the authors have shown that if two manifolds of strictly negative sectional curvature have the same homotopy type, then they are homeomorphic, thereby solving an old problem. 3) As \(\delta\) \(\to 0\) the multiplicity of the covering \(M^*\to \infty\) by the compactness theorem of Cheeger and Gromov. 4) Earlier M. Gromov and W. Thurston [Invent. Math. 89, 1-12 (1987; Zbl 0646.53037)] by another method had found examples of compact manifolds with sectional curvatures arbitrarily close to -1 that admit no metrics of constant negative sectional curvature.
To prove a) one begins with a result of Sullivan, which shows that some finite covering space \(M^*\) of M is stably parallelizable. A further topological argument completes the proof of a). To prove b) one uses residual finiteness of \(\pi_ 1M\) to construct a finite covering \(M^*\) of large multiplicity with the property that every point x of M lies in an imbedded metric ball of prescribed radius \(3\alpha\). Consider such a ball \(B_{3\alpha}=S^{m-1}\times [0,3\alpha].\) Given an exotic m- sphere \(\Sigma_ k\) one constructs \(M^*\#\Sigma_ k\) by glueing \(S^{m-1}\times [\alpha,2\alpha]\) into \(M^*_{\alpha}=M^*-S^{m- 1}\times (\alpha,2\alpha)\) with appropriate identifications along \(\partial M^*_{\alpha}\). On \(S^{m-1}\times (0,2\alpha)\) and \(S^{m-1}\times (2\alpha,3\alpha)\) one introduces appropriate metrics with \(K\equiv -1\) and interpolates between them on [\(\alpha\),2\(\alpha\)]. If \(\alpha\) is sufficiently large, then the interpolation can take place sufficiently slowly to produce sectional curvatures in \([-1-\delta,- 1+\delta]\).
Reviewer: P.Eberlein
MSC:
| 53C20 | Global Riemannian geometry, including pinching |
| 57R55 | Differentiable structures in differential topology |
Citations:
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