Topology of Real Algebraic Sets

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.