A resolution theorem for homology cycles of real algebraic varieties

The theorem is the best possible in the sense that the components of nonsingular algebraic subsets in the conclusion cannot be replaced by nonsingular algebraic subsets (see Remark 14). It is well known that the topology of the singularities of 7/2-homology cycles is closely related to real algebraic structures on V [AK 2, AK3]. Therefore it is natural to study to what extent general homology cycles differ from the cycles induced by submanifolds. The main theorem of this paper implies that all the singularities of homology cycles can be smoothed by a blowing-up process. This result should be viewed as a homology version of the resolution theorem [H]. The reason this result is not an easy consequence of [H] is that not all homology cycles of V may be represented by algebraic subsets (there are real obstructions, e.g. [AK2]; even if they were, the map resolving them may introduce new bad homology cycles upstairs. The result turns out to be a useful device in topologically classifying real algebraic sets [AK6], it also gives interesting Corollaries in algebraic topology ([AK2]). Another result of this paper (Theorem 4) says that any smooth map f: W ~ V between nonsingular algebraic sets can be approximated by a rational map after changing W by a rational diffeomorphism, if the homotopy class of f contains a rational representative; this strengthens Proposition 2.3 of [AKs]. For given algebraic sets VcF , " and W e N m a function f: V--, W is called an entire rational function if f (x) = P(x)/Q (x) where P: ~," ~ IR" and Q: IR" ~ IR