Cohomology of Grassmannian of 2-planes in $\mathbb C^4$ The cohomology of the Grassmannian of 2-planes in $\mathbb C^4$ can be deduced from computations in section 4.D of Hatcher's book, using the Leray-Hirsch theorem for fiber bundles.
However, I was told this specific case is not that hard, because this Grassmannian can be given a simple CW structure, and then one can proceed directly (by computing cellular homology and then using universal coefficients, etc).
However, my geometric intuition is terrible, and I do not see it. How can one give $G(2,4)$ a nice CW structure?

This is a slight adaptation of the construction for real Grassmannians in chapter 6 of Milnor and Stasheff's Characteristic Classes. In particular, the cohomology is trivial to compute, since cells appear only in even dimensions.
