拓?fù)鋵W(xué)Ⅰ

內(nèi)容概要

本書作者是拓?fù)鋵W(xué)領(lǐng)域最知名的專家之一，曾獲菲爾茲獎和沃爾夫數(shù)學(xué)獎。本書對整個拓?fù)鋵W(xué)領(lǐng)域(不包括一般拓?fù)鋵W(xué)(集論拓?fù)鋵W(xué)))作出最新綜述。依照諾維科夫自己的觀點(diǎn)，拓?fù)鋵W(xué)在19世紀(jì)末被稱為位置分析，隨后分為組合拓?fù)?、代?shù)拓?fù)?、微分拓?fù)?、同倫拓?fù)洹缀瓮負(fù)涞炔煌念I(lǐng)域。    本書從基本原理開始，隨之闡述當(dāng)前的研究前沿，概述這些領(lǐng)域；第二章介紹纖維空間；第三章論述CW-復(fù)形、同調(diào)和同倫理論、配邊理論、K-理論及亞當(dāng)斯-諾維科夫譜序列；第四章全面(而精要)地討論流形理論。本書附錄大致闡述了紐結(jié)和連接理論及低維拓?fù)渲械牧钊瞬毮康淖钚逻M(jìn)展。通過本書，讀者可以全面了解拓?fù)鋵W(xué)的概念。    本書具有指導(dǎo)意義，將促使不同的作者對這些拓?fù)鋵W(xué)領(lǐng)域給出更詳盡的綜述。

作者簡介

S.P.Novikov,Institute for Physical Science and Technology,University of Meryland,College Park,MD20742-2431,USA.e-mail:novikov@ipst.umd,edu.Landau Institute for Theoretical Physics of the Russian Academy of Sciences,Kosygin str2,117940Moscow,Russia.e-mail:novikov@landau.ac.ru.

書籍目錄

IntroductionIntroduction to the English TranslationChapter 1. The Simplest Topological PropertiesChapter 2. topological Spaces. Fibrations. Homotopies  1. Observations from general topology. Terminology  2. Homotopies. Homotopy type  3. Covering homotopies. Fibrations  4. Homotopy groups and fibrations. Exact sequences. ExamplesChapter 3. Simplicial Complexes and CW-complexes. Homology and Cohomology. Their Relation to Homotopy Theory. Obstructions  1. Simplicial complexes  2. The homology and cohomology groups.Poincare duality  3. Relative homology. The exact sequence of a pair. Axioms for homology theory. CW-complexes  4. Simplicial complexes and other homology theories. Singular homulogy. Coverings and sheaves. The exact sequence of sheaves and cohomology  5. Homology theory of non-simply-connected spaces. Complexes of modules. Reidemeister torsion. Simples homotopy type  6. Simplicial and cell bundles with a structure group. Obstructions. Universal lbjects: universal fiber bundles and the universal property of Eilenberg-MacLane complexes. Cohomology operations. The Steenrod algebra. The Adams spectral sequence  7. The classical apparatus of homotopy theory. The Laray spectral sequence. The homology theory of fiber bundles. The Cartan-Serre method. The Postnikov tower. The Adams spectral sequence  8. Definition and properties of K-theory. The Atiyah-Hirzebruch spectral sequence. Adams operations. Analogues of the Thom isomorphism and the Riemann-Roch theorem. Elliptic operators and K-theory. Transformation groups. Four-dimensional manifolds  9. Bordism and cobordism theory as generalized homology and cohomology. Cohomology operations in cobordism. The Adams-Novikov spectral sequence. Formal groups. Actions of cyclic groups and the circle on manifoldsChapter 4. Smooth Manifolds  1. Basic concepts. Smooth fiber bundles. Connexions. Characteristic classes  2. The homology theory of smooth manifolds. Complex manifolds. The classical global calculus of variations. H-spaces. Multi-valued functions and functionals  3. Smooth manifolds and homotopy theory. Framed manifolds. Bordisms. Thom spaces. The Hirzebruch formulae. Estimates of the orders of homotopy groups of spheres. Milnor's example. The integral properties of cobordisms  4. Classification problems in the theory of smooth manifolds. The theory of immersions. Manifolds with the homotopy type of a sphere. Relationships between smooth and PL-manifolds. Integral Pontryagin classes  5. The role of the fundamental group in topology. Manifolds of low dimension (n=2,3). Knots. The boundary of an open manifold. The classification invariance of the rational Pontryagin classes. The classification theory of non-simply-connected manifolds of dimension>5. Higher signatures. Hermitian K-theory. Geometric topology: the construction of non-smooth homeomorphisms. Milnor's example. The annulus conjecture. Topological and PL-structuresConcluding RemarksAppendix. Recent Developments in the Topology of 3-manifolds and Knots  1. Introduction: Recent developments in Topology  2. Knots: the classical and modern approaches to the Alexander polynomial. Jones-type polynomials  3. Vassiliev Invariants  4. New topological invariants for 3-manifolds. Topological Quantum Field TheoriesBibliographyIndex

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