Related constructions and facts

The direct product of two vector bundles \({(E,M,\mathbb{K}^{m})}\) and \({(E^{\prime},M^{\prime},\mathbb{K}^{n})}\) is another vector bundle

\(\displaystyle (E\times E^{\prime},M\times M^{\prime},\mathbb{K}^{m+n}). \)

If we form the direct product of two vector bundles with the same base space, we can then restrict the base space to the diagonal via the pullback by \({f\colon M\times M\rightarrow M}\) defined by \({(x,x)\mapsto x}\). The resulting vector bundle is called the Whitney sum (AKA direct sum bundle), and is denoted

\(\displaystyle (E\oplus E^{\prime},M,\mathbb{K}^{m+n}). \)

The total Whitney class of a real vector bundle \({(E,M,\mathbb{R}^{n})}\) is defined as

\(\displaystyle w(E)\equiv1+w_{1}(E)+w_{2}(E)+\cdots+w_{n}(E). \)

The series is finite since \({w_{i}(E)}\) vanishes for \({i>n}\), and is thus an element of \({H^{*}(M,\mathbb{Z}_{2})}\). The total Whitney class is multiplicative over the Whitney sum, i.e.

\(\displaystyle w(E\oplus E^{\prime})=w(E)w(E^{\prime}). \)

The total Chern class is defined similarly, and has the same multiplicative property.

The flag manifold \({F_{n}(\mathbb{K}^{\infty})}\) is a limit of the finite-dimensional flag manifold \({F_{n}(\mathbb{K}^{k})}\), which is all ordered \({n}\)-tuples of orthogonal lines in \({\mathbb{K}^{k}}\) through the origin. The name is due to the fact that an ordered \({n}\)-tuple of orthogonal lines in \({\mathbb{K}^{k}}\) is equivalent to an n-flag, a sequence of subspaces \({V_{1}\subset\cdots\subset V_{n}}\) in \({\mathbb{K}^{k}}\) where each \({V_{i}}\) has dimension \({i}\).

An Illustrated Handbook