Relations Between Cohomology and Vector Bundles

Abstract

In the present chapter we shall establish some relations between vector bundles over a space and the cohomology of the space. These relations are determined by the characteristic classes, which are called the Stiefel-Whitney classes in the case of real vector bundles and are called Chern classes in the complex case. To be more precise, we shall first rely on the fact that ℝℙ^∞ and ℂℙ^∞ are simultaneously Eilenberg-Mac Lane spaces (of type K(ℤ/2,1) and K(ℤ,2), respectively) and Grassmann manifolds (namely, G₁(ℝ^∞) and G₁(ℂ^∞), respectively). Here G₁(ℝ^∞) = G<Stack><Subscript>ℝ</Subscript><Superscript>1</Superscript></Stack> (ℝ^∞) denotes the Grassmann manifold of real one-dimensional subspaces of R^∞, while G₁(ℂ^∞) = G<Stack><Subscript>ℂ</Subscript><Superscript>1</Superscript></Stack> (C^∞) denotes the Grassmann manifold of complex one-dimensional subspaces of ℂ^∞. This means that on the one hand these two spaces determine the cohomology functors H ¹(-;ℤ/2) and H ²(-;ℤ), while on the other hand they classify real and complex line bundles, denoted functorially by Vect<Stack><Subscript>ℂ</Subscript><Superscript>1</Superscript></Stack> and Vect<Stack><Subscript> ℝ</Subscript><Superscript>1</Superscript></Stack>. In this way we shall define the first Stiefel-Whitney class and the first Chern class.