TITLE: Decribe the $S^2$ fibration over $S^2$ that gives $\mathbf{CP}^2\#\overline{\mathbf{CP}}^2$
QUESTION [6 upvotes]: According to this MO post, there is two possible $S^2$ fibration over $S^2$. One is obviously $S^2\times S^2$, another one is the connected sum of two copies of $\mathbf {CP}^2$ with different orientations. Can someone explicitly describe the $S^2$ fibration over $S^2$ that gives $\mathbf{CP}^2\#\overline{\mathbf{CP}}^2$?

REPLY [5 votes]: Cheeger described in "Some examples of manifolds with nonnegative curvature" the connected sum $\mathbb C\mathbb P^n\#\overline{\mathbb C\mathbb P^n}$ as a biquotient: Take $(S^{2n-1}\times S^2)/S^1$, where $S^1$ is acting freely on $S^{2n-1}$ and by rotations on $S^2$. This is the associated bundle to the principal fibration $S^1\to S^{2n-1}\to\mathbb C\mathbb P^n$ with fiber $S^2$. See Example 3 in Cheeger's article, why the above quotient is $\mathbb C\mathbb P^n\#\overline{\mathbb C\mathbb P^n}$.