Algebraic independence of topological Pontrjagin classes

Classical results of Sullivan and Kirby–Siebenmann may be used to see that the map from the space BO (classifying stable vector bundles) to the space BTop (classifying stable bundles of Euclidean spaces) is a rational homotopy equivalence. Therefore the familiar Pontrjagin classes of vector bundles must arise from more mysterious invariants of bundles of Euclidean spaces. For bundles of d-dimensional Euclidean spaces, one may ask whether the identities among Pontrjagin classes familiar from d-dimensional vector bundles continue to hold. To everyone’s great surprise, Weiss has shown that they need not. I will explain an elaboration of Weiss’ results, using completely unrelated methods: there are no relations at all among (Euler and) Pontrjagin classes for bundles of d-dimensional Euclidean spaces, whenever d ≥ 4. This is joint work with S. Galatius.