Counterexamples to Isosystolic Inequalities

We explore M. Gromov's counterexamples to systolic inequalities. Does the manifold S 2 × S 2 admit metrics of arbitrarily small volume such that every noncontractible surface inside it has at least unit area? This question is still open, but the answer is affirmative for its analogue in the case of S '~ × S n, n > 3. Our point of departure is M. Gromov's metric on S 1 x S 3, and more general examples, due to C. Pittet, of metrics on S ~ × S ' ~ with 'voluminous' homology. We take the metric product of these metrics with a sphere S '~-1 of a suitable volume, and perform surgery to obtain the desired metrics on S ' ~ × S '~. In 1949 Loewner proved that for any metric g on the 2-toms, one has area(g) sys (g)-5-' where the 1-systole sysl(g) is defined to be the length of the shortest noncontractible curve in g. Berger ([3]) introduced the notion of the k-systole of an n-dimensional Rie-mannian manifold (V~,g) in 1972. Here sysk(g) is the infimum of volumes of k-dimensional integer cycles representing nonzero homology classes (for k = 1, this gives the same definition as above, in the case of abelian fundamental group). Loewner's inequality leads one to consider an analogous inequality for the 1-systole of an n-dimensional manifold, and an inequality for the middle-dimensional systole: A. vol~(g_______~) >__ positive constant, and sys~(g) B. vol2.(g) sys (g)->_ positive constant. Gromov ([10]) proved inequality A for n-dimensional manifolds V admitting a map to a K(Tr, 1) space such that the induced homomorphism in n-dimensional