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When is a closed smooth connected submanifold M of R isotopic to an algebraic subset? This is an old misunderstood problem whose answer is often thought to be known. Evidently Seifert is the first one to make progress on this question. In his 1936 paper [S] he showed that if M has a trivial normal bundle, then it can be isotoped to a nonsingular component Z0 of an algebraic subset Z . Furthermore, he showed that one can take Z = ZQ if either n m l , o r n m = 2 and M is orientable. His method in fact gives Z to be a complete intersection in R" . This conclusion makes his result the best possible; because it turns out that there are homotopy theoretical obstructions to isotoping submanifolds with trivial normal bundle to complete intersections [AK6]. In 1952 in his celebrated paper Nash [N] generalized Seifert's result with a weaker conclusion. He proved that in general M can be isotoped to a nonsingular sheet of an algebraic subset of R" (the sheets might intersect each other). Then when n > 2m + 1, by a normalization process, he was able to separate the sheets in R" , thereby isotoping M to a nonsingular component of an algebraic subset of R" provided n > 2m + 1. He conjectured the same holds without the dimension restriction. Wallace's 1957 attempt [W] to prove this conjecture failed (see [K, page 823]). In 1973, Tognoli [Tl] improved Nash's result by getting rid of the extra unwanted components in the conclusion of the Nash's theorem, i.e., showing that M is in fact isotopic to a nonsingular algebraic subset of R" provided n > 2m + 1 (this result is sometimes incorrectly referred to as the solution of the Nash's conjecture). Later Tognoli [Tl] and Ivanov [I] improved this result to n > 3m/2. Most recently another attempt to prove

Some new results on the topology of nonsingular real algebraic sets