Locally semialgebraic spaces serve as an appropriateframework for studying the topological properties ofvarieties and semialgebraic sets over a real closed field.This book contributes to the fundamental theory ofsemialgebraic topology and falls into two main parts.The first dealswith sheaves and their cohomology on spaceswhich locally look like a constructible subset of a realspectrum. Topics like families of support, homotopy, acyclicsheaves, base-change theorems and cohomological dimensionare considered.In the second part a homology theory for locally completelocally semialgebraic spaces over a real closed field isdeveloped, the semialgebraic analogue of classicalBore-Moore-homology. Topics include fundamental classes ofmanifolds and varieties, Poincare duality, extensions of thebase field and a comparison with the classical theory.Applying semialgebraic Borel-Moore-homology, a semialgebraic("topological") approach to intersection theory on varietiesover an algebraically closed field of characteristic zero isgiven. The book is addressed to researchers and advancedstudents in real algebraic geometry and related areas.

Locally semialgebraic spaces serve as an appropriateframework for studying the topological properties ofvarieties and semialgebraic sets over a real closed field.This book contributes to the fundamental theory ofsemialgebraic topology and falls into two main parts.The first dealswith sheaves and their cohomology on spaceswhich locally look like a constructible subset of a realspectrum. Topics like families of support, homotopy, acyclicsheaves, base-change theorems and cohomological dimensionare considered.In the second part a homology theory for locally completelocally semialgebraic spaces over a real closed field isdeveloped, the semialgebraic analogue of classicalBore-Moore-homology. Topics include fundamental classes ofmanifolds and varieties, Poincare duality, extensions of thebase field and a comparison with the classical theory.Applying semialgebraic Borel-Moore-homology, a semialgebraic("topological") approach to intersection theory on varietiesover an algebraically closed field of characteristic zero isgiven. The book is addressed to researchers and advancedstudents in real algebraic geometry and related areas.