The Topology of Real Algebraic Sets with Isolated Singularities

Abstract

In this paper we give a topological classification of real algebraic sets with isolated singularities, showing that they are exactly smooth closed manifolds with smooth subpolyhedra crushed to points. The question of which topological spaces are homeomorphic to real algebraic sets (solutions of polynomial equations in Euclidean space) has been long studied. In 1936 Seifert showed that any smooth compact stably parallelizable manifold is diffeomorphic to a component of an algebraic set [121 and in 1952 Nash extended this result to all smooth compact manifolds [11]. In 1973 Tognoli showed that any smooth compact manifold is diffeomorphic to a nonsingular algebraic set [13], so at least compact nonsingular algebraic sets are classified. Little has been done with singular algebraic sets however, since the transversality arguments used by Seifert, Nash and Tognoli no longer apply except in some special cases. One could use stability of singularities such as Kuiper [71 and Akbulut [1] used to show certain nonsmoothable PL manifolds are algebraic sets or one could use the projective version of Seifert-Nash-Tognoli as King did [6], but one could still not hope these techniques would allow even a characterization of isolated singularities. To get around this problem we take a cue from Hironaka's resolution of singularities [4]. The idea is to take a 'topological resolution' of a space if it exists. We can apply transversality techniques (SeifertNash-Tognoli) to the resolved space and then blow down algebraically and end up with the original space as an algebraic set. It seems likely that this technique allows one to classify all algebraic sets but in any case, we show that it classifies all algebraic sets with isolated singularities. In future papers we will use this technique to show, for instance, that all compact PL manifolds are homeomorphic to real algebraic sets [17] and that 2-dimensional real algebraic sets are topologically characterized as polyhedra satisfying Sullivan's even local Euler characteristic condition [16].