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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Collapsing manifolds obtained by Kummer-type constructions


Authors: Gabriel P. Paternain and Jimmy Petean
Journal: Trans. Amer. Math. Soc. 361 (2009), 4077-4090
MSC (2000): Primary 53C23, 53C20
Published electronically: April 1, 2009
MathSciNet review: 2500879
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Abstract: 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\char93 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. $ \operatorname{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 $ \operatorname{Vol}_K (M)=0$.


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Additional Information

Gabriel P. Paternain
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WB, England
Email: g.p.paternain@dpmms.cam.ac.uk

Jimmy Petean
Affiliation: Centro de Investigacón en Matemáticas, A.P. 402, 36000, Guanajuato. Gto., México
Email: jimmy@cimat.mx

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04704-7
PII: S 0002-9947(09)04704-7
Received by editor(s): May 18, 2007
Published electronically: April 1, 2009
Additional Notes: The second author was supported by grant 46274-E of CONACYT
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.