Systolic freedom of orientable manifolds

Abstract

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.