Topology and Real Algebraic Geometry

The problem of solving real polynomial equations long fascinated people. Starting with a finite number of points on the real line one can get many complicated spaces as zero sets of polynomials. For example every closed P.L. manifold occurs as an algebraic set [3]. The main goal of understanding the topology of algebraic sets is to topologically characterize the image of the forgetful functor {Algebraic sets}-» {Topological spaces}. Rather than discussing the history and development of this problem we simply refer the reader to the surveys ([5], [7]) and instead describe the program which I developed jointly with H. King towards solving this problem. Lets take an algebraic set Xin R 3 given by (2x 2 H-2y 2 — \){{x — l) 4 — (x-l) 2 + y 2) 2 + 2z 2 = 0 X looks like this. By resolving singularities of X or just by inspecting, we see that X is obtained by a disjoint union of smooth manifolds by doing some identifications. More specifically, X = (Jf =0 F f-/~ where K 0 , V l9 V 2 are a point, a circle, and a 2-sphere respectively; ~ indicates that the equator of V 2 is folded onto V l9 and the two points of Vi are folded onto V 0 (i.e., identified).