Algebra/Topology seminar

Abstract: Orientable manifolds have even Euler characteristic unless the dimension is a multiple of 4. I give a generalisation of this theorem: k-orientable manifolds have even Euler characteristic (and in fact vanishing top Wu class), unless their dimension is 2^{k+1}m for some m > 0. Here we call a manifold k-orientable if the i^{th} Stiefel-Whitney class vanishes for all i< 2^k. This theorem is strict for k=0,1,2,3, but whether there exist 4-orientable manifolds with an odd Euler characteristic is a new open question. An argument similar to Adams' work on the Hopf invariant one theorem yields that furthermore from k=4 on, m>1. This means that the lowest dimension in which we might hope to find a 4-orientable odd Euler characteristic manifold is 64. I present the results of calculations on the cohomology of the second Rosenfeld plane, a special 64-dimensional manifold with odd Euler characteristic.