Online Talk

For n-dimensional orientable (resp. Spin) manifolds M, we wish to find the largest k such that there exists M with kth dual Stiefel-Whitney class nonzero, and to find an explicit such M. Orientable real Bott manifolds work if n is nonzero mod 4. By computing the ko-homology of K(Z/2,2), we prove there is an n-dimensional Spin manifold with (n-2)nd dual Stiefel-Whitney class nonzero if and only if n is a 2-power greater than 4