According to [this](https://mathoverflow.net/questions/53399/spaces-with-same-homotopy-and-homology-groups-that-are-not-homotopy-equivalent) 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 $\mathbf{CP}^2\#\overline{\mathbf{CP}}^2$?



Decribe the $S^2$ fibration over $S^2$ that gives $\mathbf{CP}^2\#\overline{\mathbf{CP}}^2$
$\mathbb{C}\mathbb{P}^{2} \# \bar{\mathbb{C}\mathbb{P}}^{2}$ is the blow-up of $\mathbb{C}\mathbb{P}^{2}$ at a point. Let $A$ be an affine chart of $ \mathbb{C}\mathbb{P}^{2}$ and $L\_{\infty} = \mathbb{C}\mathbb{P}^{2} \setminus A$ be the line at infinity.


Let $p$ be the origin in $A$. Consider the set of all (projective) lines through $p \in \mathbb{C}\mathbb{P}^{2}$, then they cover $\mathbb{C}\mathbb{P}^{2}$ and only intersect pair-wise at $p$. Blowing up $p$ "seperates" all of these lines and creates a $\mathbb{C}\mathbb{P}^{1}$-bundle structure; the fibres being strict transforms of lines through $p$. The base can also be identified with $\mathbb{C}\mathbb{P}^{1}$ since each of the lines containing $p$ intersects the line $L\_{\infty}$ at a unique point.


