# $\mathbb{C}P^n$-bundle over $BU(n)$

The inclusion $$U(1)\times U(n-1)\to U(n)$$ induces a map $$BU(1)\times BU(n-1)\to BU(n)$$. This source claims that the fiber of this map is $$\mathbb{C}P^n$$. Could someone please explain why this is true?

There is a typo --- it should be $$\mathbb{C}P^{n-1}$$ (as you can also see from the cohomology ring in the next sentence).
$$BU(1)\times BU(n-1)\to BU(n)$$ is the classifying map for taking orthogonal direct sum of a rank-$$1$$ bundle and a rank-$$(n-1)$$ bundle. So the fibers are ways to split an $$n$$-dimensional vector space into the sum of $$1$$- and $$(n-1)$$-dimensional subspaces, but that is just choosing $$1$$-dimensional subspaces from a fixed $$n$$-dimensionsional one (the $$n-1$$ must then be its orthogonal complement), which is $$\mathbb{C}P^{n-1}$$.
• A follow up question: how do we know Leray-Hirsch applies here? i.e. Why is the generator of the cohomology of $\mathbb{C}P^{n−1}$ the pullback of some element of the cohomology of $BU(1)×BU(n−1)$? – Judson Kuhrman Jul 26 at 5:24