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(R) or regular paracompact
locally semialgebraic spaces over R was studied. The category WSA(R) of
weakly semialgebraic spaces over R - the focus of this new volume -
contains LSA(R) as a full subcategory. The book provides ample evidence
that WSA(R) is "the" right cadre to understand homotopy and homology of
semialgebraic sets, while LSA(R) seems to be more natural and beautiful
from a geometric angle. The semialgebraic sets appear in LSA(R) and
WSA(R) as the full subcategory SA(R) 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(R) 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.
