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

 
 
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