Ebook: Topology of Real Algebraic Sets

In the Fall of 1975 we started a joint project with the ultimate goal of topo­ logically classifying real algebraic sets. This has been a long happy collaboration (c.f., [K2)). In 1985 while visiting M.S.R.1. we organized and presented our classification results up to that point in the M.S.R.1. preprint series [AK14] -[AK17]. Since these results are interdependent and require some prerequisites as well as familiarity with real algebraic geometry, we decided to make them self contained by presenting them as a part of a book in real algebraic geometry. Even though we have not arrived to our final goal yet we feel that it is time to introduce them in a self contained coherent version and demonstrate their use by giving some applications. Chapter I gives the overview of the classification program. Chapter II has all the necessary background for the rest of the book, which therefore can be used as a course in real algebraic geometry. It starts with the elementary properties of real algebraic sets and ends with the recent solution of the Nash Conjecture. Chapter III and Chapter IV develop the theory of resolution towers. Resolution towers are basic topologically defined objects generalizing the notion of manifold.

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This book is intended to cover real algebraic varieties emphasizing the author's program to classify them topologically. The first chapter gives an overview of the classification program. The second chapter provides background material for the rest of the book. It covers subjects starting with the elementary properties of real algebraic sets and ending with the recent solution of the nash conjecture. Chapters three and four develop the theory of resolution towers, which are basic topologically defined objects generalizing the notion of manifold and enable us to study singular spaces in an organized way. Chapter five shows how to obtain algebraic sets from resolution towers. Chapter six explains how to put resolution tower structures on real or complex algebraic sets. Chapter seven applies this theory to real algebraic sets of dimensions less than four by giving their complete topological characterization.

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This book is intended to cover real algebraic varieties emphasizing the author's program to classify them topologically. The first chapter gives an overview of the classification program. The second chapter provides background material for the rest of the book. It covers subjects starting with the elementary properties of real algebraic sets and ending with the recent solution of the nash conjecture. Chapters three and four develop the theory of resolution towers, which are basic topologically defined objects generalizing the notion of manifold and enable us to study singular spaces in an organized way. Chapter five shows how to obtain algebraic sets from resolution towers. Chapter six explains how to put resolution tower structures on real or complex algebraic sets. Chapter seven applies this theory to real algebraic sets of dimensions less than four by giving their complete topological characterization.
Content:
Front Matter....Pages i-x
Introduction....Pages 1-16
Algebraic Sets....Pages 17-92
Ticos....Pages 93-135
Resolution Towers....Pages 136-157
Algebraic Structures on Resolution Towers....Pages 158-172
Resolution Tower Structures on Algebraic Sets....Pages 173-190
The Characterization of Three Dimensional Algebraic Sets....Pages 191-243
Back Matter....Pages 244-249

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This book is intended to cover real algebraic varieties emphasizing the author's program to classify them topologically. The first chapter gives an overview of the classification program. The second chapter provides background material for the rest of the book. It covers subjects starting with the elementary properties of real algebraic sets and ending with the recent solution of the nash conjecture. Chapters three and four develop the theory of resolution towers, which are basic topologically defined objects generalizing the notion of manifold and enable us to study singular spaces in an organized way. Chapter five shows how to obtain algebraic sets from resolution towers. Chapter six explains how to put resolution tower structures on real or complex algebraic sets. Chapter seven applies this theory to real algebraic sets of dimensions less than four by giving their complete topological characterization.
Content:
Front Matter....Pages i-x
Introduction....Pages 1-16
Algebraic Sets....Pages 17-92
Ticos....Pages 93-135
Resolution Towers....Pages 136-157
Algebraic Structures on Resolution Towers....Pages 158-172
Resolution Tower Structures on Algebraic Sets....Pages 173-190
The Characterization of Three Dimensional Algebraic Sets....Pages 191-243
Back Matter....Pages 244-249
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