The manifold $\mathbb{CP}^2\#-\mathbb{CP}^2,$ the non-trivial $\mathbb{S}^2$ bundle over $\mathbb{S}^2$, and it is known to be diffeomorphic to the space that we now describe. Represent $\mathbb{S}^3$ ⊂ C^2 as pairs of complex numbers $(z_1, z_2)$ with $|z_1|^2 + |z_2|^2 = 1.$ Let $\mathbb{S}^1$ act on $\mathbb{S}^3$ by $(w,(z_1, z_2)) \mapsto (wz_1, wz_2),$ where w ∈ S^1 is a complex number with modulus one. Let S^1 also act on S^2 by rotations. Consider the space $M = \mathbb{S}^2 \times_{\mathbb{S}^1}\mathbb{S}^2$ obtained by taking the quotient of $\mathbb{S}^3 \times \mathbb{S}^2$ by the diagonal action of S^1.Then The manifold M is diffeomorphic to CP^2#CP^2.

here I cannot find the homeomorphism between M and CP^2 # -CP^2. Please give me some idea about this homeomorphism.

Thanks