Resumen
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 M with Vol K (M)=0.