Abstract of talk for the UM/OSU Seminar
One important part of Voevodsky's work on the (generalized) Milnor Conjecture is the observation that minimal versal splitting varieties of symbols in Galois cohomology satisfy some non-triviality condition on their characteristic numbers. This plays an important role in Voevodsky's proof of the Milnor conjecture, as it allows to prove the vanishing of the Margolis homology of certain gadgets. It also plays an important role to get a hand on symmetric products of the splitting varieties. The aim is here to reduce problems for symbols of higher weight to symbols of weight 2. We call this the Chain Lemma for splitting fields of symbols.
In the talk we plan to discuss the Chain Lemma in some simple cases and draw its relation with characteristic numbers and Steenrod operations.
If time permits, we will also discuss the following "formula" for 1-dimensional real manifolds: