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SURV/137: Systolic Geometry and Topology 收缩几何学与拓扑学

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SURV/137: Systolic Geometry and Topology 收缩几何学与拓扑学

最 低 价：￥560.70

定 价：￥623.00

作 者：Mikhail G. Katz　著 著

出 版 社：

出版时间：2007-4-1

I S B N：9780821841778

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    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.

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    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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