SURV/137: Systolic Geometry and Topology 收缩几何学与拓扑学
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作者： Mikhail G. Katz著
出 版 社：
出版时间： 2007-4-1字数：版次： 1页数： 222印刷时间： 2007/04/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780821841778包装： 精装内容简介
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.
目录
Part.1 Systolic geometry in dimension 2
Chapter.1 Geometry and topology of systoles
Chapter.2 Historical remarks
Chapter.3 The theorema egregium of Gauss
Chapter.4 Global geometry of surfaces
Chapter.5 Inequalities of Loewner and Pu
Chapter.6 Systolic applications of integral geometry
Chapter.7 A primer on surfaces
Chapter.8 Filling area theorem for hyperelliptic surfaces
Chapter.9 Hyperelliptic surfaces are Loewner
Chapter.10 An optimal inequality for CAT(0) metrics
Chapter.11 Volume entropy and asymptotic upper bounds
Part.2 Systolic geometry and topology in n dimensions
Chapter.12 Systoles and their category
Chapter.13 Gromov's optimal stable systolic inequality for CP[superscript n]
Chapter.14 Systolic inequalities dependent on Massey products
Chapter.15 Cup products and stable systoles
Chapter.16 Dual-critical lattices and systoles
Chapter.17 Generalized degree and Loewner-type inequalities
Chapter.18 Higher inequalities of Loewner-Gromov type
Chapter.19 Systolic inequalities for L[superscript p] norms
Chapter.20 Four-manifold systole asymptotics
App.A Period map image density by Jake Solomon
App.B Open problems