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Espace quotient. --- Groupes finis. --- Géométrie algébrique.

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Histoire des mathematiques --- Histoire des mathematiques --- Histoire des mathematiques --- Geometrie algebrique --- 20e siecle --- 19e siecle --- Document --- Histoire des mathematiques --- Histoire des mathematiques --- Histoire des mathematiques --- Geometrie algebrique --- 20e siecle --- 19e siecle --- Document

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From the reviews: "This is a very interesting book containing material for a comprehensive study of the cyclid homological theory of algebras, cyclic sets and S1-spaces. Lie algebras and algebraic K-theory and an introduction to Connes'work and recent results on the Novikov conjecture. The book requires a knowledge of homological algebra and Lie algebra theory as well as basic technics coming from algebraic topology. The bibliographic comments at the end of each chapter offer good suggestions for further reading and research. The book can be strongly recommended to anybody interested in noncommutative geometry, contemporary algebraic topology and related topics." European Mathematical Society Newsletter In this second edition the authors have added a chapter 13 on MacLane (co)homology.

Homologie --- Homology theory --- Category theory. Homological algebra --- K-théorie. --- Homologie. --- K-theory --- Algebraic topology. --- K-theory. --- Mathematics. --- Algebraic Topology. --- K-Theory. --- Mathematics, general. --- Math --- Science --- Algebraic topology --- Topology --- Géométrie algébrique

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The present volume is a translation, revision and updating of our book (pub­ lished in French) with the title "Geometrie Algebrique Reelle". Since its pub­ lication in 1987 the theory has made advances in several directions. There have also been new insights into material already in the French edition. Many of these advances and insights have been incorporated in this English version of the book, so that it may be viewed as being substantially different from the original. We wish to thank Michael Buchner for his careful reading of the text and for his linguistic corrections and stylistic improvements. The initial Jb. TEiX file was prepared by Thierry van Effelterre. The three authors participate in the European research network "Real Algebraic and Analytic Geometry". The first author was partially supported by NATO Collaborative Research Grant 960011. Jacek Bochnak April 1998 Michel Coste Marie-Pranroise Roy Table of Contents Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Ordered Fields, Real Closed Fields . . . . . . . . . . . . . . . . . . . . . . . 7 1. 1 Ordered Fields, Real Fields . . . . . " . . . . . . . . . . . . . . . . . . . . . . . 7 1. 2 Real Closed Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1. 3 Real Closure of an Ordered Field. . . . . . . . . . . . . . . . . . . . . . . . . 14 1. 4 The Tarski-Seidenberg Principle. . . . . . . . . . . . . . . . . . . . . . . . . . 17 2. Semi-algebraic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2. 1 Algebraic and Semi-algebraic Sets. . . . . . . . . . . . . . . . . . . . . . . . 23 2. 2 Projection of Semi-algebraic Sets. Semi-algebraic Mappings. . 26 2. 3 Decomposition of Semi-algebraic Sets. . . . . . . . . . . . . . . . . . . . . 30 2. 4 Connectedness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2. 5 Closed and Bounded Semi-algebraic Sets. Curve-selection Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2. 6 Continuous Semi-algebraic Functions. Lojasiewicz's Inequality 42 2. 7 Separation of Closed Semi-algebraic Sets. . . . . . . . . . . . . . . . . .

Algebraic geometry --- Algebraïsche meetkunde --- Geometry [Algebraic ] --- Géométrie algébrique --- Meetkunde [Algebraïsche ] --- Geometry, Algebraic. --- Functions of real variables. --- 512.7 --- Real variables --- Functions of complex variables --- Geometry --- Algebraic geometry. Commutative rings and algebras --- 512.7 Algebraic geometry. Commutative rings and algebras --- Functions of real variables --- Geometry, Algebraic --- Algebraic geometry. --- Algebraic Geometry.

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Algebra --- Symmetric functions. --- Algèbre. --- Fonctions symétriques. --- Algebraic topology --- Topologie algébrique --- Transformation groups --- Transformations, Groupes de --- Group theory --- Groupes, Théorie des --- Symmetry groups --- Groupes de symétrie --- Algebra. --- Algèbre --- Algebraic topology. --- Topologie algébrique --- Transformation groups. --- Groupes, Théorie des --- Symmetry groups. --- Groupes de symétrie --- Analyse combinatoire --- Geometrie algebrique --- Varietes algebriques