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.
General
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