Collapsing manifolds obtained by Kummer-type constructions. (English) Zbl 1171.53031
Summary: We construct \( \mathcal{F}\)-structures on a Bott manifold and on some other manifolds obtained by Kummer-type constructions. We also prove that if \( M=E\# X\), where \( E\) is a fiber bundle with structure group \( G\) and a fiber admitting a \( G\)-invariant metric of non-negative sectional curvature and \( X\) admits an \( \mathcal{F}\)-structure with one trivial covering, then one can construct on \( M\) a sequence of metrics with sectional curvature uniformly bounded from below and volume tending to zero (i.e. \( \text{Vol}_K(M)=0)\). As a corollary we prove that all the elements in the Spin cobordism ring can be represented by manifolds \( M\) with \( \text{Vol}_K (M)=0\).
MSC:
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 53C20 | Global Riemannian geometry, including pinching |
| 57R17 | Symplectic and contact topology in high or arbitrary dimension |
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