Affiliation: College of Arts and Sciences, Department of Mathematics
The cohomology of the real Grassmann and flag manifolds is discussed at length, making use of Stiefel-Whitney classes. It is shown that the tangent bundle of the Grassmannian splits into line bundles over the flag manifold. Additionally, it is found that the cohomology of the flag manifold is exactly the polynomial algebra ℤ(subscript 2)[𝑒(subscript 1),...,𝑒(subscript 𝑛)], for one-dimensional classes 𝑒(subscript 𝑖), modulo the relation ∏(subscript 𝑖)(1 + 𝑒(subscript 𝑖))=1. Using facts about this ring to compute the dimension of the Stiefel-Whitney class of the normal bundle to the Grassmannian, we find a lower bound for immersions of certain real Grassmannians. In particular, G(subscript 𝑛)(ℝ(superscript 𝑛+𝑘)) with 𝑛≤2(superscript 𝑠)≤𝑘 and 𝑛+𝑘≤2(superscript 1+𝑠) cannot be immersed in dimension less than 𝑛(2(superscript 𝑠+1)-1).