By Manfred Knebusch
The e-book 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 normal paracompact in the neighborhood semialgebraic areas over R was once studied. the class WSA(R) of weakly semialgebraic areas over R - the point of interest of this new quantity - includes LSA(R) as a whole subcategory. The ebook offers considerable facts that WSA(R) is "the" correct cadre to appreciate homotopy and homology of semialgebraic units, whereas LSA(R) appears extra traditional and gorgeous from a geometrical attitude. The semialgebraic units seem 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 evidence that each generalized topological (co)homology conception has a counterpart in WSA(R) with in a few feel "the same", or perhaps higher, homes because the topological thought. therefore we could communicate of standard (=singular) homology teams, orthogonal, unitary or symplectic K-groups, and diverse different types of cobordism teams of a semialgebraic set over R. If R isn't archimedean then it kind of feels tough to advance a passable thought of those teams in the classification 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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