REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: In his famous lectures on characteristic classes John Milnor makes use of a cohomology class in H*(M x M, Δ(M)) of a closed manifold M, where Δ : M --> M x M is the diagonal embedding, to describe Poincar'e duality and define Stiefel--Whitney classes. In this talk I will make use of one of Milnor's lemmas to give a purely algebraic definition of an element u \in \A \tensor A , where A is a Poincar'e duality algebra, that plays the same role. I will use this Milnor diagonal element to describe a Macaulay dual for the kernel of the multiplication map A \tensor A --> A, relate u to the dimension of A as a vector space, and define Stiefel -- Whitney classes if A supports an unstable Steenrod algebra action. Using this definition of Stiefel--Whitney classes I will show that Wu's formula for the action of the Steenrod algebraon them holds, recovering a new proof of an old result of J.F. Adams.