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The conjecture that every $S^2 \subseteq \mathbb{C}P^2$ is standard if it is homologous to flat is implied by the smooth Poincaré conjecture in 4 dimensions. It also implies a special of smooth Poincaré that is accepted as an open problem, the case of Gluck surgery in $S^4$. I can't prove or disprove the question of course, but since the question is sandwiched between two open problems, I can prove "prove" that it is an open problem.

It is easier to consider $\mathbb{C}P^2$ minus a tubular neighborhood of the $S^2$, rather than to "blow it down". The condition on the homology class is equivalent to the condition that the boundary of this tube is $S^3$; the projection to the core is a Hopf fibration. The blowdown consists of attaching a 4-ball to this 3-sphere; let's skip this step. As Joel had in mind, the complement of the $S^2$ is simply connected. In fact, it is a homotopy 4-ball with boundary $S^3$. Thus, Freedman's theorem implies that it is homeomorphic to a 4-ball and smooth Poincaré would imply that it is diffeomorphic to a 4-ball. When it is, this 4-ball is still standard with its Hopf-fibered boundary (the Hopf fibration is unique up to orientation), so the $S^2$ is unknotted.

In the other direction, the $S^2$ could be the direct sum of a standard complex line in $\mathbb{C}P^2$ with a 2-knot $K$ in $S^4$. I argue that in this case, the blowdown is equivalent to the Gluck surgery along $K$. What is a Gluck surgery? It looks like Dehn surgery in 3 dimensions, except with peculiar behavior. The official definition is that you remove a neighborhood of $K$ (which here is $D^2 \times S^2$, not the twisted bundle in Joel's construction), then glue it back after applying the non-trivial diffeomorphism of $S^1 \times S^2$. That diffeomorphism comes from the non-trivial element in $\pi_1(\text{SO}(3)) = \mathbb{Z}/2$. One thing that is peculiar is that the Gluck surgery does not change the homotopy type of its 4-manifold, which is why it produces many candidate counterexamples to smooth Poincaré.

Again, it is easier to think about the closed complement to Joel's $S^2$ than the blowdown. The corresponding version of Gluck surgery is to remove all of $D^2 \times K$, but only glue back a thickened $D^2$ (a 2-handle) along an attaching circle, and not glue back in the remaining 4-ball along the rest of $K$. What is peculiar here is that the attaching circle does not change; it is still a vertical circle in $S^1 \times S^2$. What changes instead is that the framing of the attachment is twisted by 1. (Or it can be twisted by some other odd number, since $\pi_1(\text{SO}(3)) = \mathbb{Z}/2$ and not $\mathbb{Z}$. More prosaically, the "belt trick" lets you change the twisting by an even number.) Anyway, if Joel's sphere is $K$ connect summed with a complex line $L$, then you can represent this crucial 2-handle with another complex line $J$ in $\mathbb{C}P^2$. The question is whether the framing of its attachment to $L$ is odd or even. The fact that a perturbation $J'$ of $J$ intersects $J$ once tells me that the framing is odd. So the result is Gluck surgery.

The old version of this answer was less developed (and at first I made the $\pi_1$ mistake that is corrected in the comments and the edit to the question). But it is still worth noting that there are many open special cases of smooth Poincaré that consist of just one homotopy 4-sphere. Some topologists interpret this as strong evidence that smooth Poincaré is false. Others suppose that we just might not be very good at finding diffeomorphisms with $S^4$. A few examples, including some Gluck surgeries, were shown to be standard only after many years, for instance in this paper by Selman Akbulut.

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There are lots of knotted 2-spheres in

The conjecture that every $\mathbb{R}^4$ which are distinguished S^2 \subseteq \mathbb{C}P^2$is standard if it is homologous to flat is implied by their fundamental groups, among other informationthe smooth Poincaré conjecture in 4 dimensions. This It also implies a special of smooth Poincaré that is discussed in Rolfsen's bookaccepted as an open problem, for example. You could take the connected sum case of any Gluck surgery in$S^4$. I can't prove or disprove the question of these with course, but since the question is sandwiched between two open problems, I can prove that it is an open problem. It is easier to consider$\mathbb{C}P^2$minus a standard tubular neighborhood of the$S^2$with S^2$, rather than to "blow it down". The condition on the homology class is equivalent to the condition that the boundary of this tube is $\mathbb{C}P^2$. From Smooth 2-knots S^3$; the projection to the core is a Hopf fibration. The blowdown consists of attaching a 4-ball to this 3-sphere; let's skip this step. As Joel had in mind, the complement of the$S^2 \times S^2$..S^2$ is simply connected. by SatoIn fact, It it is a consequence of homotopy 4-ball with boundary $S^3$. Thus, Freedman's theorem implies that $(\mathbb{C}P^2,S)$, where it is homeomorphic to a 4-ball and smooth Poincaré would imply that it is diffeomorphic to a 4-ball. When it is, this 4-ball is still standard with its Hopf-fibered boundary (the Hopf fibration is unique up to orientation), so the $S$ S^2$is unknotted. In the other direction, the$S^2$could be the direct sum of a smoothly embedded 2-sphere standard complex line in$\mathbb{C}P^2$representing with a generator of 2-knot$H_2(\mathbb{C}P^2;\mathbb{Z})$, is pairwise homeomorphic to K$ in $(\mathbb{C}P^2, \mathbb{C}P^1)$". I'm not speaking with as much authority as I thought S^4$. I couldargue that in this case, but it indeed sounds like your problem the blowdown is open. My first thought was equivalent to take the Gluck surgery along$K$. What is a connected sum with something interesting Gluck surgery? It looks like Dehn surgery in a 4-ball and use van Kampen's theorem3 dimensions, but the van Kampen calculation actually says except with peculiar behavior. The official definition is that$\pi_1$you remove a neighborhood of the result$K$(which here is trivial. Since you have in mind$D^2 \times S^2$, not the smooth Poincare conjecture twisted bundle in 4 dimensionsJoel's construction), there are actually many examples then glue it back after applying the non-trivial diffeomorphism of smooth 4-manifolds$S^1 \times S^2$. That diffeomorphism comes from the non-trivial element in$\pi_1(\text{SO}(3)) = \mathbb{Z}/2$. One thing that are is peculiar is that the Gluck surgery does not change the homotopy type of its 4-manifold, which is why it produces many candidate counterexamples to smooth Poincaré. Again, it is easier to think about the closed complement to Joel's$S^4$'s, and thus homeomorphic S^2$ than the blowdown. The corresponding version of Gluck surgery is to remove all of $S^4$ by Freedman's theoremD^2 \times K$, but not known to be diffeomorphic to only glue back a thickened$S^4$. A traditional contruction D^2$ (but a 2-handle) along an attaching circle, and not glue back in the only construction by any means) remaining 4-ball along the rest of $K$. What is peculiar here is that looks something like your idea the attaching circle does not change; it is Gluck surgerystill a vertical circle in $S^1 \times S^2$. In factWhat changes instead is that the framing of the attachment is twisted by 1. (Or it can be twisted by some other odd number, since $\pi_1(\text{SO}(3)) = \mathbb{Z}/2$ and not $\mathbb{Z}$. More prosaically, the "belt trick" lets you change the twisting by an even number.) Anyway, if your 2-sphere Joel's sphere is only (putatively) knotted inside of $K$ connect summed with a 4-ball, complex line $L$, then your construction possibly just you can represent this crucial 2-handle with another complex line $J$ in $\mathbb{C}P^2$. The question is Gluck surgery on whether the framing of its attachment to $S^4$.L$is odd or even. The 4D smooth Poincare conjecture fact that a perturbation$J'$of$J$intersects$J$once tells me that the framing is in an ironic stateodd. Because So the result is Gluck surgery. 2 Major correction There are lots of knotted 2-spheres in$\mathbb{R}^4$which are distinguished by their fundamental groups, among other information. This is discussed in Rolfsen's book, for example. You could take the connected sum of any of these with a standard$S^2$with$\mathbb{C}P^2$. From Smooth 2-knots in$S^2 \times S^2$... by Sato, It is a consequence of Freedman's theorem that$(\mathbb{C}P^2,S)$, where$S$is a smoothly embedded 2-sphere in$\mathbb{C}P^2$representing a generator of$H_2(\mathbb{C}P^2;\mathbb{Z})$, is pairwise homeomorphic to$(\mathbb{C}P^2, \mathbb{C}P^1)$". I'm not speaking with as much authority as I thought I could, but it indeed sounds like your problem is open. My first thought was to take a connected sum with something interesting in a 4-ball and use van Kampen's theorem, but the van Kampen calculation actually says that$\pi_1$of the result is trivial. Since you have in mind the smooth Poincare conjecture in 4 dimensions, there are actually many examples of smooth 4-manifolds that are homotopy$S^4$'s (S^4$'s, and thus homeomorphic to $S^4$ by Freedman's theorem)theorem, but not known to be diffeomorphic to $S^4$. A traditional contruction (but not the only construction by any means) that looks something like your idea is Gluck surgery. In fact, if your 2-sphere is only (putatively) knotted inside of a 4-ball, then your construction possibly just is Gluck surgery on $S^4$.

The 4D smooth Poincare conjecture is in an ironic state. Because of these examples, many people think that it must be false, while other people still think that it has a good chance of being true. Every once in a blue moon, some particular favorite putative counterexamples are shown to be diffeomorphic to $S^4$ after all, for instance in this paper by Selman Akbulut.

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