Collapsing manifolds obtained by Kummer-type constructions

Details Collapsing manifolds obtained by Kummer-type constructions Collapsing manifolds obtained by Kummer-type constructions

2005-07-05

arxiv.org/abs/math/0507099v1

Mathematics

Differential Geometry

Scientific paper

We construct 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 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. $Vol_K$ (M)=0). As a corollary we prove that all the elements in the Spin cobordism group can be represented by manifolds M with $Vol_K$ (M)==0.

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