  In 1972, Marcel Berger defined a metric invariant that captures the `size' of
k-dimensional homology of a Riemannian manifold. This invariant came to be
called the k-dimensional SYSTOLE. He asked if the systoles can be constrained
by the volume, in the spirit of the 1949 theorem of C. Loewner. We construct
metrics, inspired by M. Gromov's 1993 example, which give a negative answer for
large classes of manifolds, for the product of systoles in a pair of
complementary dimensions. An obstruction (restriction on k modulo 4) to
constructing further examples by our methods seems to reside in the free part
of real Bott periodicity. The construction takes place in a split neighborhood
of a suitable k-dimensional submanifold whose connected components (rationally)
generate the k-dimensional homology group of the manifold. Bounded geometry
(combined with the coarea inequality) implies a lower bound for the k-systole,
while calibration with support in this neighborhood provides a lower bound for
the systole of the complementary dimension. In dimension 4 everything reduces
to the case of S^2 x S^2.
