By Mikhail G. Katz

The systole of a compact metric house $X$ is a metric invariant of $X$, outlined because the least size of a noncontractible loop in $X$. while $X$ is a graph, the invariant is mostly known as the girth, ever because the 1947 article through W. Tutte. the 1st nontrivial effects for systoles of surfaces are the 2 classical inequalities of C. Loewner and P. Pu, counting on integral-geometric identities, with regards to the two-dimensional torus and actual projective airplane, respectively. at the moment, systolic geometry is a quickly constructing box, which reviews systolic invariants of their relation to different geometric invariants of a manifold. This e-book provides the systolic geometry of manifolds and polyhedra, beginning with the 2 classical inequalities, after which continuing to contemporary effects, together with an evidence of M. Gromov's filling quarter conjecture in a hyperelliptic environment. It then provides Gromov's inequalities and their generalisations, in addition to asymptotic phenomena for systoles of surfaces of enormous genus, revealing a hyperlink either to ergodic conception and to homes of congruence subgroups of mathematics teams. the writer contains effects at the systolic manifestations of Massey items, in addition to of the classical Lusternik-Schnirelmann classification

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