The manifold CP^2#-CP^2, the non-trivial S^2 bundle over S^2, and it is known to be diffeomorphic to the space that we now describe. Represent S^3 ⊂ C^2 as pairs of complex numbers (z1, z2) with |z1|^2 + |z2|^2 = 1. Let S^1 act on S^3 by (w,(z1, z2)) → (wz1, wz2), 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 = S^3 ×S^1 S^2 obtained by taking the quotient of S^3×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