Locally semialgebraic spaces serve as an appropriate
framework for studying the topological properties of
varieties and semialgebraic sets over a real closed field.
This book contributes to the fundamental theory of
semialgebraic topology and falls into two main parts.
The first dealswith sheaves and their cohomology on spaces
which locally look like a constructible subset of a real
spectrum. Topics like families of support, homotopy, acyclic
sheaves, base-change theorems and cohomological dimension
are considered.
In the second part a homology theory for locally complete
locally semialgebraic spaces over a real closed field is
developed, the semialgebraic analogue of classical
Bore-Moore-homology. Topics include fundamental classes of
manifolds and varieties, Poincare duality, extensions of the
base field and a comparison with the classical theory.
Applying semialgebraic Borel-Moore-homology, a semialgebraic
("topological") approach to intersection theory on varieties
over an algebraically closed field of characteristic zero is
given. The book is addressed to researchers and advanced
students in real algebraic geometry and related areas.