By Manfred Knebusch

The ebook 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 average 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 a whole subcategory. The ebook presents abundant proof that WSA(R) is "the" correct cadre to appreciate homotopy and homology of semialgebraic units, whereas LSA(R) appears to be like extra usual and lovely from a geometrical perspective. The semialgebraic units look in LSA(R) and WSA(R) because the complete subcategory SA(R) of affine semialgebraic areas. the idea is new even though it borrows from algebraic topology. A spotlight is the evidence that each generalized topological (co)homology idea has a counterpart in WSA(R) with in a few experience "the same", or maybe greater, houses because the topological idea. therefore we may possibly converse of normal (=singular) homology teams, orthogonal, unitary or symplectic K-groups, and numerous kinds of cobordism teams of a semialgebraic set over R. If R isn't archimedean then it kind of feels tricky to boost a passable thought of those teams in the type 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 is often performed the following for usual (co)homology.

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