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By Manfred Knebusch

The ebook is the second one a part of an meant three-volume treatise on semialgebraic topology over an arbitrary genuine closed box R. within the first quantity (LNM 1173) the class LSA(R) or ordinary paracompact in the community semialgebraic areas over R used to be 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 ebook offers considerable proof that WSA(R) is "the" correct cadre to appreciate homotopy and homology of semialgebraic units, whereas LSA(R) looks extra usual and lovely 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 concept has a counterpart in WSA(R) with in a few feel "the same", or perhaps greater, homes because the topological concept. therefore we might converse of normal (=singular) homology teams, orthogonal, unitary or symplectic K-groups, and numerous forms of cobordism teams of a semialgebraic set over R. If R isn't really archimedean then it sort of feels tricky to increase a passable conception of those teams in the class of semialgebraic units over R: with weakly semialgebraic areas this turns into effortless. It is still 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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