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The Whitney trick is an important step in Smale's proof of the Poincaré conjecture for smooth manifolds of dimension $n\geqslant 5$. It turns out however that such a trick does not work in dimension 4. However, as shown by Freedman (using previous work by Casson), it is possible (in a non-trivial way) to make this trick work for topological 4-manifolds. This partially explains the striking difference between topological and smooth manifolds in dimension 4.

As an example of this striking and exceptional difference between these two categories, we know that in every dimension $n\neq 4$ a topological closed manifold may admit only finitely many smooth structures. In dimension $n=4$ however there are 4-manifolds like the $K3$ surface or (very recently) $S^2\times S^2$ that admit infinitely many distinct smooth structures. As far as we know, it might as well be that any closed smoothable 4-manifold has infinitely many distinct structures!

The question is open for instance for $S^4$ itself, which might have any number of distinct differentiable structures ranging from 1 to $\infty$ (extremes included). That's why we say that the Poincaré Conjecture is true for topological 4-manifolds but is still (very) open for smooth 4-manifolds.

The Whitney trick is an important step in Smale's proof of the Poincaré conjecture for smooth manifolds of dimension $n\geqslant 5$. It turns out however that such a trick does not work in dimension 4. However, as shown by Freedman (using previous work by Casson), it is possible (in a non-trivial way) to make this trick work for topological 4-manifolds with "good" fundamental group. This partially explains the striking difference between topological and smooth manifolds in dimension 4.

As an example of this striking and exceptional difference between these two categories, we know that in every dimension $n\neq 4$ a topological closed manifold may admit only finitely many smooth structures. In dimension $n=4$ however there are 4-manifolds like the $K3$ surface or (very recently) $S^2\times S^2$ that admit infinitely many distinct smooth structures. As far as we know, it might as well be that any closed smoothable 4-manifold has infinitely many distinct structures!

The question is open for instance for $S^4$ itself, which might have any number of distinct differentiable structures ranging from 1 to $\infty$ (extremes included). That's why we say that the Poincaré Conjecture is true for topological 4-manifolds but is still (very) open for smooth 4-manifolds.

The Whitney trick is an important step in Smale's proof of the Poincaré conjecture for smooth manifolds of dimension $n\geqslant 5$. It turns out however that such a trick does not work in dimension 4. However, as shown by Freedman (using previous work by Casson), it is possible (in a non-trivial way) to make this trick work for topological 4-manifolds. This partially explains the striking difference between topological and smooth manifolds in dimension 4.

As an example of this striking and excepional difference between these two categories, we know that in every dimension $n\neq 4$ a topological closed manifold may admit only finitely many smooth structures. In dimension $n=4$ however there are 4-manifolds like the $K3$ surface or (very recently) $S^2\times S^2$ that admit infinitely many distinct smooth structures. As far as we know, it might as well be that any closed smoothable 4-manifold has infinitely many distinct structures!

The question is open for instance for $S^4$ itself, which might have any number of distinct differentiable structures ranging from 1 to $\infty$ (extremes included). That's why we say that the Poincaré Conjecture is true for topological 4-manifolds but is still (very) open for smooth 4-manifolds.

The Whitney trick is an important step in Smale's proof of the Poincaré conjecture for smooth manifolds of dimension $n\geqslant 5$. It turns out however that such a trick does not work in dimension 4. However, as shown by Freedman (using previous work by Casson), it is possible (in a non-trivial way) to make this trick work for topological 4-manifolds. This partially explains the striking difference between topological and smooth manifolds in dimension 4.

As an example of this striking and exceptional difference between these two categories, we know that in every dimension $n\neq 4$ a topological closed manifold may admit only finitely many smooth structures. In dimension $n=4$ however there are 4-manifolds like the $K3$ surface or (very recently) $S^2\times S^2$ that admit infinitely many distinct smooth structures. As far as we know, it might as well be that any closed smoothable 4-manifold has infinitely many distinct structures!

The question is open for instance for $S^4$ itself, which might have any number of distinct differentiable structures ranging from 1 to $\infty$ (extremes included). That's why we say that the Poincaré Conjecture is true for topological 4-manifolds but is still (very) open for smooth 4-manifolds.

The Whitney trick is an important step in Smale's proof of the Poincaré conjecture for smooth manifolds of dimension $n\geqslant 5$. It turns out however that such a trick does not work in dimension 4. However, as shown by Freeman (using previous work by Casson), it is possible (in a non-trivial way) to make this trick work for topological 4-manifolds. This partially explains the striking difference between topological and smooth manifolds in dimension 4.

As an example of this striking and excepional difference between these two categories, we know that in every dimension $n\neq 4$ a topological closed manifold may admit only finitely many smooth structures. In dimension $n=4$ however there are 4-manifolds like the $K3$ surface or (very recently) $S^2\times S^2$ that admit infinitely many distinct smooth structures. As far as we know, it might as well be that any closed smoothable 4-manifold has infinitely many distinct structures!

The question is open for instance for $S^4$ itself, which might have any number of distinct differentiable structures ranging from 1 to $\infty$ (extremes included). That's why we say that the Poincaré Conjecture is true for topological 4-manifolds but is still (very) open for smooth 4-manifolds.

The Whitney trick is an important step in Smale's proof of the Poincaré conjecture for smooth manifolds of dimension $n\geqslant 5$. It turns out however that such a trick does not work in dimension 4. However, as shown by Freedman (using previous work by Casson), it is possible (in a non-trivial way) to make this trick work for topological 4-manifolds. This partially explains the striking difference between topological and smooth manifolds in dimension 4.

As an example of this striking and excepional difference between these two categories, we know that in every dimension $n\neq 4$ a topological closed manifold may admit only finitely many smooth structures. In dimension $n=4$ however there are 4-manifolds like the $K3$ surface or (very recently) $S^2\times S^2$ that admit infinitely many distinct smooth structures. As far as we know, it might as well be that any closed smoothable 4-manifold has infinitely many distinct structures!

The question is open for instance for $S^4$ itself, which might have any number of distinct differentiable structures ranging from 1 to $\infty$ (extremes included). That's why we say that the Poincaré Conjecture is true for topological 4-manifolds but is still (very) open for smooth 4-manifolds.