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The book is the second part of an intended three-volume treatise on semialgebraic topology over an arbitrary real closed field R. In the first volume (LNM 1173) the category LSA® or regular paracompact locally semialgebraic spaces over R was studied. The category WSA® of weakly semialgebraic spaces over R - the focus of this new volume - contains LSA® as a full subcategory. The book provides ample evidence that WSA® is the right cadre to understand homotopy and homology of semialgebraic sets, while LSA® seems to be more natural and beautiful from a geometric angle. The semialgebraic sets appear in LSA® and WSA® as the full subcategory SA® of affine semialgebraic spaces. The theory is new although it borrows from algebraic topology. A highlight is the proof that every generalized topological (co)homology theory has a counterpart in WSA® with in some sense the same , or even better, properties as the topological theory. Thus we may speak of ordinary (=singular) homology groups, orthogonal, unitary or symplectic K-groups, and various sorts of cobordism groups of a semialgebraic set over R. If R is not archimedean then it seems difficult to develop a satisfactory theory of these groups within the category of semialgebraic sets over R: with weakly semialgebraic spaces this becomes easy. It remains for us to interpret the elements of these groups in geometric terms: this is done here for ordinary (co)homology.
