Systolic Geometry and Topology

The systole of a compact metric space $X$ is a metric invariant of $X$, defined as the least length of a noncontractible loop in $X$. When $X$ is a graph, the invariant is usually referred to as the girth, ever since the 1947 article by W. Tutte. The first nontrivial results for systoles of surfaces are the two classical inequalities of C. Loewner and P. Pu, relying on integral-geometric identities, in the case of the two-dimensional torus and real projective plane, respectively. Currently, systolic geometry is a rapidly developing field, which studies systolic invariants in their relation to other geometric invariants of a manifold. This book presents the systolic geometry of manifolds and polyhedra, starting with the two classical inequalities, and then proceeding to recent results, including a proof of M. Gromov's filling area conjecture in a hyperelliptic setting. It then presents Gromov's inequalities and their generalisations, as well as asymptotic phenomena for systoles of surfaces of large genus, revealing a link both to ergodic theory and to properties of congruence subgroups of arithmetic groups. The author includes results on the systolic manifestations of Massey products, as well as of the classical Lusternik-Schnirelmann category.

Table of Contents:
Systolic geometry in dimension 2: Geometry and topology of systoles Historical remarks The theorema egregium of Gauss Global geometry of surfaces Inequalities of Loewner and Pu Systolic applications of integral geometry A primer on surfaces Filling area theorem for hyperelliptic surfaces Hyperelliptic surfaces are Loewner An optimal inequality for CAT(0) metrics Volume entropy and asymptotic upper bounds Systolic geometry and topology in $n$ dimensions: Systoles and their category Gromov's optimal stable systolic inequality for $\mathbb{CP}^n$ Systolic inequalities dependent on Massey products Cup products and stable systoles Dual-critical lattices and systoles Generalized degree and Loewner-type inequalities Higher inequalities of Loewner-Gromov type Systolic inequalities for $L^p$ norms Four-manifold systole asymptotics Period map image density (by Jake Solomon) Open problems Bibliography Index.