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Systolic geometry and topology. With an appendix by Jake P. Solomon. (English) Zbl 1149.53003

Mathematical Surveys and Monographs 137. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4177-8/hbk). xiv, 222 p. (2007).
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 manifolds.
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 inequality 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 manifestation of Massey products, as well as of the classical Lusternik-Schnirelmann category.
Let me formulate a new theorem from the geometry “in large”, which is closely related to the systolic geometry. Let \(M\) be a centrally-symmetric convex hypersurface in \(E^n\). Given two points \(P\) and \(Q\) in \(M\), which are not symmetric, the inner distance between \(P\) and \(Q\) is less than the half-sum of the inner distances between \(P\), \(Q\) and their antipodal (centrally-symmetric) points \(P^*\), \(Q^*\in M\), i.e., \(d(P,Q)<\frac{1}{2}( d(P,P^*)+d(Q,Q^*))\). Besides the most intrinsically distant points in \(M\) are centrally-symmetric to each other. This theorem appeared because of some intrinsically-geometric generalisations of the well-known Borsuk problem about partitions of convex bodies. It may be extended to Alexandrov metric spaces of bounded curvature. The proof of the cited theorem is realized for the case of \(M^2\in E^3\) only, see [Russ. Math. 36, No. 5, 52–57 (1992); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1992, No. 5(360), 58–63 (1992; Zbl 0853.52015)].

MSC:

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C22 Geodesics in global differential geometry
11R52 Quaternion and other division algebras: arithmetic, zeta functions
28D20 Entropy and other invariants
37C35 Orbit growth in dynamical systems
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
51M25 Length, area and volume in real or complex geometry
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Citations:

Zbl 0853.52015
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