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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, Pon- tryagin classes (Z coecients) and Chern-Weil classes (coecients in R) joining those of Stiefel Whitney (Z mod 2 coecients). Ren e 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. 1. Thom's isomorphism and his work on Steenrod's Problem Thom's (1951) thesis about spherical brations and cohomology (2) showed a new cohomology isomorphism between the (compact) base of a vector bundle and the one point compactication of its total space (now called the Thom space). Thom's isomorphism was eected algebraically by multiplying an arbitrary class in the total space (which deformation retracts to the base) with a certain cohomology class on the total space with compact support (now called the Thom class). Geometrically the isomorphism in terms of homology could be seen by intersecting a relative cycle in the total space (mod1) with the zero section to obtain a cycle in the base. The latter perspective shows that the Thom class of the oriented normal bun- dle of an oriented submanifold in an oriented manifold determines by pullback the cohomology class Poincar e dual to the homology class of the submanifold. Thom then showed how the Stiefel Whitney classes of the vector bundle could be obtained by applying Steenrod operations to the Thom class in the Thom space and then using the Thom isomorphism to get back to the base,

René Thom’s work on geometric homology and bordism