Collapsing manifolds obtained by Kummer-type constructions

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 $\operatorname{Vol}_K (M)=0$.