Homology of locally semialgebraic spaces
This book is a contribution to the fundamentals of semi-algebraic topology over arbitrary real closed fields. One basic idea in the development of modern real algebraic geometry is that main algebraic and geometric results are valid not only for the reals but for more general real closed fields (including non archimedean). Naturally some of the definitions and proofs have to be modified. For example connected components of semi-algebraic sets do not exist for general real closed fields but semi-algebraically connected components do (and in the case of the reals they coincide with ordinary connected components) and enjoy a lot of nice properties. --- The situation for topology is still more complicated. The semialgebraic opens do not form a topology in the ordinary sense, but only a ``generalized topological space''. A topological space, the real spectrum, can be associated to these semi- algebraic spaces. This topological spaces are no more Hausdorff, but still enjoy a lot of nice properties. In chapter I a short introduction to the subject of abstract locally semialgebraic spaces is given. Chapter II is devoted to the study of sheaves on abstract locally semialgebraic spaces. One of the main results is the following: The cohomology groups of these sheaves do not change when the base field is extended. Chapter III deals with semi-algebraic Borel-Moore homology. Using combinatorial methods (with open simplices rather than closed ones) one can prove that every algebraic variety possesses a fundamental class. The use of sheaves allows to obtain results on Poincaré duality for arbitrary families of supports. Chapter IV is devoted to some aspects of intersection theory for a general algebraically closed field of characteristic zero and some generalizations to this case of classical results are proved. The book is very clear and well written. Many examples enlighten the results.
semi-algebraic topology over arbitrary real closed fields
scientific article; zbMATH DE number 51539