By Manfred Knebusch
The booklet is the second one a part of an meant three-volume treatise on semialgebraic topology over an arbitrary actual closed box R. within the first quantity (LNM 1173) the class LSA(R) or normal paracompact in the neighborhood semialgebraic areas over R was once studied. the class WSA(R) of weakly semialgebraic areas over R - the focal point of this new quantity - comprises LSA(R) as an entire subcategory. The booklet presents abundant proof that WSA(R) is "the" correct cadre to appreciate homotopy and homology of semialgebraic units, whereas LSA(R) seems extra ordinary and gorgeous from a geometrical attitude. The semialgebraic units look in LSA(R) and WSA(R) because the complete subcategory SA(R) of affine semialgebraic areas. the speculation is new even though it borrows from algebraic topology. A spotlight is the facts that each generalized topological (co)homology idea has a counterpart in WSA(R) with in a few experience "the same", or maybe greater, homes because the topological idea. therefore we may well communicate of standard (=singular) homology teams, orthogonal, unitary or symplectic K-groups, and numerous types of cobordism teams of a semialgebraic set over R. If R isn't archimedean then it kind of feels tough to enhance a passable conception of those teams in the class of semialgebraic units over R: with weakly semialgebraic areas this turns into effortless. It continues to be for us to interpret the weather of those teams in geometric phrases: this can be performed the following for traditional (co)homology.
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