Counterexamples to isosystolic inequalities

We explore M. Gromov's counterexamples to systolic inequalities. Does the manifold S² ×S² 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 Sⁿ ×Sⁿ, n ≥ 3. Our point of departure is M. Gromov's metric on S¹ ×S³, and more general examples, due to C. Pittet, of metrics on S¹ ×Sⁿ with 'voluminous' homology. We take the metric product of these metrics with a sphere S^n-1 of a suitable volume, and perform surgery to obtain the desired metrics on Sⁿ ×Sⁿ.