Introduction to systoles

Appearing below is a non-technical introduction to systolic geometry and topology, in ten paragraphs. Here "non-technical" means "containing no formulas"☺ To see some formulas, view a 60-second introduction equipped with formulas (pdf), (html).

1. Abstract. We study the systolic geometry of manifolds and polyhedra, as initially conceived by Charles Loewner, and developed by Mikhael Gromov and others, in its arithmetic, ergodic, and topological manifestations.

2. The systole of a compact metric space X is a metric invariant of X, defined to be the least length of a noncontractible loop in X. In other words, we minimize length over all loops that cannot be contracted to a point in X, i.e. free loops representing nontrivial conjugacy classes in the fundamental group of X. When X is a graph, the invariant is usually referred to as the girth, ever since the '47 article by W. Tutte. Possibly inspired by Tutte's article, Loewner started thinking about systolic questions on surfaces in the late '40's, resulting in a '50 thesis by his student P.M. Pu. The actual term "systole" itself was not coined until a quarter century later, by Marcel Berger. This line of research was, apparently, given further impetus by a remark of the venerable René Thom, in a conversation with Berger in the library of Strasbourg University during the '61-62 academic year, shortly after the publication of the papers of R. Accola and C. Blatter. Referring to these systolic inequalities, Thom reportedly exclaimed: "Mais c'est fondamental!" [These results are of fundamental importance!] Subsequently, Berger popularized the subject in a series of articles and books. A Systolic bibliography currently contains 160 articles. Systolic geometry is a rapidly developing field, featuring a number of recent publications in leading journals. Recently, an intriguing link (msn) has emerged with the Lusternik-Schnirelmann category. The existence of such a link can be thought of as a theorem in "systolic topology".

3. Flavor. To give a preliminary idea of the flavor of the field, one could make the following observations. The main thrust of Thom's remark to Berger quoted above, appears to be the following. Whenever one encounters an inequality relating geometric invariants, such a phenomenon in itself is interesting; all the more so when the inequality is sharp (i.e. optimal). The elegance of the inequalities of Loewner, Pu, and Gromov is indisputable. In systolic questions about surfaces, integral-geometric identities play a particularly important role. Roughly speaking, there is an integral identity relating area on the one hand, and an average of energies of a suitable family of loops, on the other. By the Cauchy-Schwarz inequality, energy is an upper bound for length squared, hence one obtains an inequality between area and the square of the systole. Such an approach works both for Loewner's torus inequality, and for Pu's inequality for the real projective plane. A number of new inequalities of this type have recently been discovered, including universal volume lower bounds.

4. The deepest result in the field is Gromov's inequality for the homotopy 1-systole of an essential n-manifold. A summary of a proof, based on recent results in geometric measure theory by S. Wenger, building upon earlier work by L. Ambrosio and B. Kirchheim, appears in Section 12.2 of the book "Systolic geometry and topology". A completely different approach to the proof of Gromov's inequality was recently proposed by L. Guth.

5. Gromov's stable systolic inequality. A significant difference between 1-systolic invariants (defined in terms of lengths of loops) and the higher, k-systolic invariants (defined in terms of areas of cycles, etc.) should be kept in mind (the definitions of the higher invariants may be found in section 4 of the introduction). While a number of optimal systolic inequalities, involving the 1-systoles, have by now been obtained, just about the only optimal inequality involving purely the higher k-systoles is M. Gromov's optimal stable 2-systolic inequality for complex projective space, where the optimal bound is attained by the symmetric Fubini-Study metric. Just how exceptional Gromov's stable inequality is, only became clear recently. Namely, we discovered (arXiv) that, contrary to expectation, the symmetric metric on the quaternionic projective plane (see Berger prize) is not its systolically optimal metric, in contrast with the complex case.

6. Similarly, just about the only nontrivial lower bound for a k-systole with k=2, results from recent work in gauge theory and J-holomorphic curves. The study of lower bounds for the conformal 2-systole of 4-manifolds has led to a simplified proof of the density of the image of the period map, by Jake Solomon.

7. Schottky problem. Perhaps one of the most striking applications of systoles is in the context of the Schottky problem, by P. Buser and P. Sarnak, who distinguished (msn) the Jacobians of Riemann surfaces among principally polarized abelian varieties, laying the foundation for systolic arithmetic.

8. Asking systolic questions often stimulates questions in related fields. Thus, a numerical invariant, called "systolic category", of a manifold has been defined and investigated (msn), exhibiting a connection to the Lusternik-Schnirelmann (LS) category. The two categories have been shown to coincide for both surfaces and 3-manifolds. Moreover, for orientable 4-manifolds, the systolic category is a lower bound for the LS category. Once the connection is established, the influence is mutual: known results about LS category stimulate systolic questions, and vice versa.

9. Asymptotic phenomena for the systole of surfaces of large genus have been shown to be related to interesting ergodic phenomena, and to properties of congruence subgroups of arithmetic groups. A bibliography for systoles in hyperbolic geometry may be viewed here. Gromov's filling area conjecture has been proved in a hyperelliptic setting. Other systolic ramifications of hyperellipticity have been identified in genus 2.

10. To illustrate the variety of the tools involved, we mention a few connections to other areas: (a) Banaszczyk's functional analytic results on the successive minima of a pair of dual lattices; (b) dynamical systems notion of volume entropy and Katok's inequality; (c) the topological techniques of Lusternik-Schnirelmann category; (d) quaternion algebras and congruence subgroups of arithmetic Fuchsian and Kleinian groups; (e) J-holomorphic curves and Seiberg-Witten invariants.

11. The surveys in the field include M. Berger's '93 survey (msn), as well as M. Gromov's '96 survey (msn), as well as Gromov's '99 book (msn), as well as Berger's '03 panoramic book (msn), as well as the '07 book by Katz (AMS). These references may help a beginner enter the field. They also contain open problems to work on.