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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.
List of contents
I. Introduction.- 1. Overview.- 2. Stratified Sets.- 3. Ticos.- 4. Resolution Towers.- II. Algebraic Sets.- 1. Basic Properties of Algebraic Sets.- 2. Singularities of Real and Complex Algebraic Sets.- 3. Projective Algebraic Sets.- 4. Grassmannians.- 5. Blowing Up.- 6. Blowing Down.- 7. Algebraic Homology.- 8. Making Smooth Objects Algebraic.- 9. Homology of Blowups.- 10. Isotoping Submanifolds to Algebraic Subsets.- III. Ticos.- 1. Some Results about Smooth Functions.- 2. Ticos.- 3. Tico Blowups.- 4. Full Ticos.- 5. Type N Tico Maps.- 6. Submersive Tico Maps.- 7. Micos.- IV. Resolution Towers.- 1. Definiton of Resolution Towers.- 2. Blowing up Resolution Towers.- 3. Realizations of Resolution Towers.- V. Algebraic Structures on Resolution Towers.- 1. Making Tico Maps Algebraic.- 2. Nice Charts on Resolution Towers.- 3. Quasialgebraic Towers are Algebraic.- 4. RF Towers are Quasialgebraic.- VI. Resolution Tower Structures on Algebraic Sets.- 1. Uzunblowups and Fullness.- 2. Complex Ticos and Complexifications.- 3. Extending Algebraic Resolution Towers.- 4. Resolution Towers for Algebraic Sets.- VII. The Characterization of Three Dimensional Algebraic Sets.- 1. Obstructions.- 2. The Cobordism Groups.- 3. Characterization in Dimension 3.- 4. Algebraic Resolution of Real Algebraic Sets in Dimension Three.- 5. Bounding Resolution Towers.
