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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

The manifold $\mathbb{CP}^2 \# -\mathbb{CP}^2$, the non-trivial $\mathbb{S}^2$ bundle over $\mathbb{S}^2$, is known to be diffeomorphic to the space that we will now describe. Represent $\mathbb{S}^3 \subseteq \mathbb{C}^2$ 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 $w \in S^1$ is a complex number of modulus one. Let $S^1$ also act on $S^2$ by rotations around a fixed axis. 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 $\mathbb{CP}^2 \# -\mathbb{CP}^2$.

Here I cannot find the homeomorphism between $M$ and $\mathbb{CP}^2 \# -\mathbb{CP}^2$. Please give me some ideas about this homeomorphism.

Thanks.

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

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