I would like to add another geometric description of $\mathbb C\mathbb P^n \#\overline{\mathbb C\mathbb P^n}$ as sphere bundle (same works for $\mathbb C\mathbb P^n \# \mathbb C\mathbb P^n$):

Let $S^1 \to S^{2n+1} \to \mathbb C\mathbb P^n$ be the Hopf fibration. Then $S^1$ acts on $\mathbb R^2$ by rotations and the associated vector bundle $E$ is the normal bundle of $\mathbb C\mathbb P^{n-1}$ in $\mathbb C\mathbb P^n$. Thus the total space of $E$ is diffeomorphic to $\mathbb C\mathbb P^n$ with a disc removed. Hence if we glue the disc bundles of $E$ with $\overline E$ (which denotes here the the bundle with the reversed orientation) along their common boundary one obtains $\mathbb C\mathbb P^n\# \overline{\mathbb C\mathbb P^n}$.

Moreover one deduces from this description that the connected sum is given as the quotient $(S^{2n-1}\times S^2)/S^1$, where $S^1$ acts on $S^{2n-1}$ such that it induces the Hopf fibrations and on $S^2$ by rotations. This quotient has a projection map to $S^{2n-2}$ with fibre $S^2$. (This is just the associated $S^2$-bundle to the Hopf fibration)

I would like to add another geometric description of $\mathbb C\mathbb P^n \#\overline{\mathbb C\mathbb P^n}$ as sphere bundle (same works for $\mathbb C\mathbb P^n \# \mathbb C\mathbb P^n$):

Let $S^1 \to S^{2n+1} \to \mathbb C\mathbb P^n$ be the Hopf fibration. Then $S^1$ acts on $\mathbb R^2$ by rotations and the associated vector bundle $E$ is the normal bundle of $\mathbb C\mathbb P^{n-1}$ in $\mathbb C\mathbb P^n$. Thus the total space of $E$ is diffeomorphic to $\mathbb C\mathbb P^n$ with a disc removed. Hence if we glue the disc bundles of $E$ with $\overline E$ (which denotes here the the bundle with the reversed orientation) along their common boundary one obtains $\mathbb C\mathbb P^n\# \overline{\mathbb C\mathbb P^n}$.

Moreover one deduces from this description that the connected sum is given as the quotient $(S^{2n-1}\times S^2)/S^1$, where $S^1$ acts on $S^{2n-1}$ such that it induces the Hopf fibrations and on $S^2$ by rotations. This quotient has a projection map to $\mathbb C\mathbb P^{n-1}$ with fibre $S^2$. (This is just the associated $S^2$-bundle to the Hopf fibration)

I would like to add another geometric description of $\mathbb C\mathbb P^n \#\overline{\mathbb C\mathbb P^n}$ as sphere bundle (same works for $\mathbb C\mathbb P^n \# \mathbb C\mathbb P^n$):

Let $S^1 \to S^{2n+1} \to \mathbb C\mathbb P^n$ be the Hopf fibration. Then $S^1$ acts on $\mathbb R^2$ by rotations and the associated vector bundle $E$ is the normal bundle of $\mathbb C\mathbb P^{n-1}$ in $\mathbb C\mathbb P^n$. Thus the total space of $E$ is diffeomorphic to $\mathbb C\mathbb P^n$ with a disc removed. Hence if we glue the disc bundles of $E$ with $\overline E$ (which denotes here the the bundle with the reversed orientation) along their common boundary one obtains $\mathbb C\mathbb P^n\# \overline{\mathbb C\mathbb P^n}$.

Moreover one deduces from this descriprtion that the connected sum is given as the quotient $(S^{2n-1}\times S^2)/S^1$, where $S^1$ acts on $S^{2n-1}$ such that it induces the Hopf fibrations and on $S^2$ by rotations. This quotient has a projection map to $S^{2n-2}$ with fibre $S^2$. (This is just the associated $S^2$-bundle to the Hopf fibration)

I would like to add another geometric description of $\mathbb C\mathbb P^n \#\overline{\mathbb C\mathbb P^n}$ as sphere bundle (same works for $\mathbb C\mathbb P^n \# \mathbb C\mathbb P^n$):

Let $S^1 \to S^{2n+1} \to \mathbb C\mathbb P^n$ be the Hopf fibration. Then $S^1$ acts on $\mathbb R^2$ by rotations and the associated vector bundle $E$ is the normal bundle of $\mathbb C\mathbb P^{n-1}$ in $\mathbb C\mathbb P^n$. Thus the total space of $E$ is diffeomorphic to $\mathbb C\mathbb P^n$ with a disc removed. Hence if we glue the disc bundles of $E$ with $\overline E$ (which denotes here the the bundle with the reversed orientation) along their common boundary one obtains $\mathbb C\mathbb P^n\# \overline{\mathbb C\mathbb P^n}$.

Moreover one deduces from this description that the connected sum is given as the quotient $(S^{2n-1}\times S^2)/S^1$, where $S^1$ acts on $S^{2n-1}$ such that it induces the Hopf fibrations and on $S^2$ by rotations. This quotient has a projection map to $S^{2n-2}$ with fibre $S^2$. (This is just the associated $S^2$-bundle to the Hopf fibration)