COLLAPSING MANIFOLDS OBTAINED BY KUMMER-TYPE CONSTRUCTIONS

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 ring can be represented by manifolds All with Vol(K) (M) = 0.