Introduction to the Papers of R. Thom and J. Mather

For any closed set E ⊂ R there exists a C∞ function f : R → R such that E = f−1(0). This includes the Cantor set, the Sierpiński sponge, the Snowflake, and other sets of fractional Hausdorff dimension. How does one prove that this sort of behavior cannot happen when f is an analytic function or an algebraic function? These questions were approached about 80 years ago when it was shown ([38, 15, 13, 16]) that algebraic sets could be triangulated. For many years the 1932 paper [13] was cited as the only known proof that complex algebraic sets were locally contractible. But these papers are difficult to follow, and for decades the triangulability of algebraic and analytic sets was treated with some suspicion. Later articles, such as [30, 3, 18] and especially [10, 11, 8], finally put these questions to rest. However, the interesting structure of the singularities of an algebraic set is not easily2 described using simplices, and people began to look for a more intelligent way to decompose an algebraic set into (fewer, larger) pieces, starting with the nonsingular part. Hassler Whitney struggled with these questions for decades. In 1946 he wrote Complexes of manifolds [40], in which he considered spaces that were glued together out of smooth manifolds much in the way that a cell complex is glued together from cells. In 1957 Whitney showed in [41] that it is possible, in any algebraic set, to choose an open dense nonsingular part such that its complement is an algebraic set of smaller dimension. Therefore this procedure can be repeated so as to give a finite filtration by closed subsets X0 ⊂ X1 ⊂ · · · ⊂ Xn such that Xr+1 is obtained from Xr by attaching a (possibly empty) smooth manifold Sr+1 := Xr+1 −Xr of dimension r + 1. One might hope that the resulting decomposition into piecesX = ∐ r Sr is locally trivial—that any two sufficiently nearby points x, y ∈ Sr should have neighborhoods that are isomorphic (in a sense to be made precise below) by an isomorphism (a homeomorphism, or perhaps, a diffeomorphism) that preserves the induced filtrations. In 1962 René Thom made a first attempt in [34] to make precise such a notion of a locally trivial stratification. He proposed that each stratum S should have a