René Thom’s Work on Geometric Homology and Bordism

By the early 1950’s algebraic topology had reached great heights with Serre’s thesis and the calculations in the seminar of Henri Cartan of the cohomology of spaces with one nonzero homotopy group in terms of Steenrod operations. There was also the appearance of the new characteristic classes of vector bundles, Pontryagin classes (Z coefficients) and Chern-Weil classes (coefficients in R) joining those of Stiefel Whitney (Z mod 2 coefficients). René Thom absorbed all this structure, made vigorous use of it, and added a geometric perspective that combined to revolutionize topology, manifold theory, and algebraic geometry. For the unexpected and fertile results in bordism (closed manifolds mod boundaries of manifolds) of the 1954 paper [1], Thom received the Fields Medal at Edinburgh in 1958. Many more applications of Thom’s ideas came even later.